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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Bounds for the regression parameters in dependently censored survival models</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2503.11210v2/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S1" title="In Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S1.SS1" title="In 1 Introduction ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Literature review</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S1.SS2" title="In 1 Introduction ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.2 </span>Paper outline</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2" title="In Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Model and methodology</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.SS1" title="In 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>A preliminary note on partial identification</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.SS2" title="In 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Methodology</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.SS3" title="In 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Unconditional moment restrictions</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.SS4" title="In 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.4 </span>Testing procedure</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.SS5" title="In 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.5 </span>Overview</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3" title="In Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Estimation procedure</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.SS1" title="In 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Modeling assumptions and theoretical results</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.SS2" title="In 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Instrumental functions</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.SS3" title="In 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>Time-independent effects of covariates</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.SS4" title="In 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.4 </span>Discussion</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S4" title="In Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Simulation study</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S4.SS1" title="In 4 Simulation study ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Summary of additional simulations</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S5" title="In Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Data applications</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S5.SS1" title="In 5 Data applications ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1 </span>Pancreas cancer data</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S5.SS2" title="In 5 Data applications ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2 </span>NLSY data</span></a></li> </ol> </li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <div class="ltx_para" id="p1"> <span class="ltx_ERROR undefined" id="p1.1">\externaldocument</span> <p class="ltx_p" id="p1.2">auxfile_suppArXiv </p> </div> <h1 class="ltx_title ltx_font_bold ltx_title_document">Bounds for the regression parameters in dependently censored survival models</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Ilias Willems <br class="ltx_break"/>ORSTAT, KU Leuven, Belgium <br class="ltx_break"/>and <br class="ltx_break"/>Jad Beyhum <br class="ltx_break"/>Department of Economics, KU Leuven, Belgium <br class="ltx_break"/>and <br class="ltx_break"/>Ingrid Van Keilegom <br class="ltx_break"/>ORSTAT, KU Leuven, Belgium </span><span class="ltx_author_notes">Corresponding author (ilias.willems@kuleuven.be).Jad Beyhum gratefully acknowledges financial support from the Research Fund KU Leuven through the grant STG/23/014.Ingrid Van Keilegom gratefully acknowledges funding from the FWO and F.R.S. - FNRS (Excellence of Science programme, project ASTeRISK, grant no. 40007517), and financial support from the FWO (senior research projects fundamental research, grant no. G047524N).</span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id1.id1">We propose a semiparametric model to study the effect of covariates on the distribution of a censored event time while making minimal assumptions about the censoring mechanism. The result is a partially identified model, in the sense that we obtain bounds on the covariate effects, which are allowed to be time-dependent. Moreover, these bounds can be interpreted as classical confidence intervals and are obtained by aggregating information in the conditional Peterson bounds over the entire covariate space. As a special case, our approach can be used to study the popular Cox proportional hazards model while leaving the censoring distribution as well as its dependence with the time of interest completely unspecified. A simulation study illustrates good finite sample performance of the method, and several data applications in both economics and medicine demonstrate its practicability on real data. All developed methodology is implemented in R and made available in the package <span class="ltx_text ltx_font_typewriter" id="id1.id1.1">depCensoring</span>.</p> </div> <div class="ltx_para" id="p2"> <br class="ltx_break"/> </div> <div class="ltx_para ltx_noindent" id="p3"> <p class="ltx_p" id="p3.1"><span class="ltx_text ltx_font_italic" id="p3.1.1">Keywords:</span> Cox proportional hazards model, set identification, duration analysis, moment inequality models, informative censoring.</p> </div> <div class="ltx_pagination ltx_role_newpage"></div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.2">In the context of event time modeling, the <cite class="ltx_cite ltx_citemacro_cite">Cox, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib6" title="">1972</a>)</cite> proportional hazards (PH) model has enjoyed great popularity among practitioners. However, it has the downside of having to impose that the event time <math alttext="T" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mi id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><ci id="S1.p1.1.m1.1.1.cmml" xref="S1.p1.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">italic_T</annotation></semantics></math> and censoring time <math alttext="C" class="ltx_Math" display="inline" id="S1.p1.2.m2.1"><semantics id="S1.p1.2.m2.1a"><mi id="S1.p1.2.m2.1.1" xref="S1.p1.2.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S1.p1.2.m2.1b"><ci id="S1.p1.2.m2.1.1.cmml" xref="S1.p1.2.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.2.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="S1.p1.2.m2.1d">italic_C</annotation></semantics></math> are independent, which is an assumption that deserves careful scrutiny <cite class="ltx_cite ltx_citemacro_citep">(Kaplan and Meier,, <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib21" title="">1958</a>)</cite>, as it may not hold in many applications. While several innovative works (<cite class="ltx_cite ltx_citemacro_cite">Huang and Zhang, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib19" title="">2008</a>)</cite>; <cite class="ltx_cite ltx_citemacro_cite">Deresa and Van Keilegom, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib13" title="">2024</a>)</cite>, …) have succeeded in relaxing this assumption since the seminal publication by <cite class="ltx_cite ltx_citemacro_cite">Cox, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib6" title="">1972</a>)</cite>, a truly censoring-agnostic method for doing inference on the Cox PH model has eluded the literature thus far. With this paper, we aim to fill that gap. More precisely, we propose a framework that, as a special case, is able to perform inference in the Cox PH model without making any assumptions on both the censoring distribution as well as its dependence with respect to the event of interest. The framework, however, reaches much further as it includes many more models (e.g. the proportional odds model), and can handle time-dependent effects of covariates.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">Under the generality of the proposed approach – specifically, by not imposing any assumptions on the censoring mechanism – the parameters in the Cox PH model are not identified <cite class="ltx_cite ltx_citemacro_citep">(Huang and Zhang,, <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib19" title="">2008</a>)</cite>. Therefore, we can only aim to <em class="ltx_emph ltx_font_italic" id="S1.p2.1.1">partially identify</em> them. In practice, this means that we will estimate bounds on the covariate effects instead of obtaining point estimates. These bounds can be interpreted as regular confidence intervals and can still be informative provided they are sufficiently narrow.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">From a technical point of view, the bounds are obtained starting from the conditional <cite class="ltx_cite ltx_citemacro_cite">Peterson, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib26" title="">1976</a>)</cite> bounds. These are generally known to be wide and uninformative, but by aggregating their information over the entire covariate space, we are able to convert them into meaningful bounds on the parameters. We achieve this by employing instrumental functions to cast the problem into one of testing moment restrictions and proving that the recent developments of <cite class="ltx_cite ltx_citemacro_cite">Bei, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib2" title="">2024</a>)</cite> are valid in this setting. The insights and structure used to construct the proofs are not specific to our setting and could be used by researchers who want to apply this test in a different setting. In particular, we replace a strong assumption made in <cite class="ltx_cite ltx_citemacro_cite">Bei, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib2" title="">2024</a>)</cite> by a more tractable one. This assumption from <cite class="ltx_cite ltx_citemacro_cite">Bei, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib2" title="">2024</a>)</cite> has already been the subject of detailed research <cite class="ltx_cite ltx_citemacro_citep">(Kaido et al.,, <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib20" title="">2022</a>)</cite>. As such, our theoretical results could be of independent interest.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">The finite sample performance of the methodology is assessed by means of a simulation study. To demonstrate the practicability on real data, we provide two data applications in economics and medicine, where the independent censoring assumption may be dubious in the presented cases. The software is made available on CRAN in the <span class="ltx_text ltx_font_typewriter" id="S1.p4.1.1">depCensoring</span> package in R, and details regarding the implementation can be found in Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Algorithm_and_implementation_details</span>.</p> </div> <section class="ltx_subsection" id="S1.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.1 </span>Literature review</h3> <div class="ltx_para" id="S1.SS1.p1"> <p class="ltx_p" id="S1.SS1.p1.3">Event time analysis under dependent censoring has received increasing amounts of attention over the last decade, and could be argued to have originated from the work by <cite class="ltx_cite ltx_citemacro_cite">Tsiatis, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib33" title="">1975</a>)</cite>, who shows that the distribution of <math alttext="T" class="ltx_Math" display="inline" id="S1.SS1.p1.1.m1.1"><semantics id="S1.SS1.p1.1.m1.1a"><mi id="S1.SS1.p1.1.m1.1.1" xref="S1.SS1.p1.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.1.m1.1b"><ci id="S1.SS1.p1.1.m1.1.1.cmml" xref="S1.SS1.p1.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.1.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.1.m1.1d">italic_T</annotation></semantics></math> is in general not identifiable in a fully nonparametric way. Therefore, additional assumption on the data generating process – and, particularly, on the dependence between <math alttext="T" class="ltx_Math" display="inline" id="S1.SS1.p1.2.m2.1"><semantics id="S1.SS1.p1.2.m2.1a"><mi id="S1.SS1.p1.2.m2.1.1" xref="S1.SS1.p1.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.2.m2.1b"><ci id="S1.SS1.p1.2.m2.1.1.cmml" xref="S1.SS1.p1.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.2.m2.1d">italic_T</annotation></semantics></math> and <math alttext="C" class="ltx_Math" display="inline" id="S1.SS1.p1.3.m3.1"><semantics id="S1.SS1.p1.3.m3.1a"><mi id="S1.SS1.p1.3.m3.1.1" xref="S1.SS1.p1.3.m3.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.3.m3.1b"><ci id="S1.SS1.p1.3.m3.1.1.cmml" xref="S1.SS1.p1.3.m3.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.3.m3.1c">C</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.3.m3.1d">italic_C</annotation></semantics></math> – often have to be made.</p> </div> <div class="ltx_para" id="S1.SS1.p2"> <p class="ltx_p" id="S1.SS1.p2.2">Many approaches are centered around copulas to model the dependence between <math alttext="T" class="ltx_Math" display="inline" id="S1.SS1.p2.1.m1.1"><semantics id="S1.SS1.p2.1.m1.1a"><mi id="S1.SS1.p2.1.m1.1.1" xref="S1.SS1.p2.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.1.m1.1b"><ci id="S1.SS1.p2.1.m1.1.1.cmml" xref="S1.SS1.p2.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.1.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.1.m1.1d">italic_T</annotation></semantics></math> and <math alttext="C" class="ltx_Math" display="inline" id="S1.SS1.p2.2.m2.1"><semantics id="S1.SS1.p2.2.m2.1a"><mi id="S1.SS1.p2.2.m2.1.1" xref="S1.SS1.p2.2.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.2.m2.1b"><ci id="S1.SS1.p2.2.m2.1.1.cmml" xref="S1.SS1.p2.2.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.2.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.2.m2.1d">italic_C</annotation></semantics></math>, either specifying a fully known copula (<cite class="ltx_cite ltx_citemacro_cite">Zheng and Klein, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib34" title="">1995</a>); Rivest and Wells, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib27" title="">2001</a>); Braekers and Veraverbeke, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib3" title="">2005</a>); Huang and Zhang, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib19" title="">2008</a>); Sujica and Van Keilegom, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib31" title="">2018</a>)</cite>, …), or allowing for a parametric copula class and estimating the dependence parameter (<cite class="ltx_cite ltx_citemacro_cite">Czado and Van Keilegom, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib9" title="">2023</a>); Lo and Wilke, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib24" title="">2023</a>); Deresa and Van Keilegom, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib13" title="">2024</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib14" title="">2025</a>); Ding and Van Keilegom, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib15" title="">2025</a>)</cite>, …). We refer to <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib8" title="">Crommen et al., 2024b </a></cite> for an in-depth review on copula-based methods for dependent censoring, as well as some other approaches, including transformation models (<cite class="ltx_cite ltx_citemacro_cite">Deresa and Van Keilegom, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib11" title="">2020</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib12" title="">2021</a>); <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib7" title="">Crommen et al., 2024a </a>; Rutten et al., (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib28" title="">2024</a>)</cite>, …). Notably, all works discussed in their review paper impose restrictions on the dependence between the censoring time and the event time in order to obtain point identification. In contrast, we are fully agnostic on this dependence, but only reach partial identification.</p> </div> <div class="ltx_para" id="S1.SS1.p3"> <p class="ltx_p" id="S1.SS1.p3.1">Partial identification is mostly studied in the field of econometrics. By allowing models to only be partially identified, one is often able to omit stringent modeling assumptions which would otherwise be needed to attain point identification. The body of literature on partial identification can partly be categorized into two branches. One branch focuses on the special case of compact, convex identified sets and makes use of support functions, which allows for many computational advantages. The other branch considers more general identified sets, but as a trade-off needs to use a <em class="ltx_emph ltx_font_italic" id="S1.SS1.p3.1.1">test inversion</em> method (cf. infra), which is computationally more burdensome. We refer to <cite class="ltx_cite ltx_citemacro_cite">Molinari, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib25" title="">2020</a>)</cite> for an extensive review on this literature.</p> </div> <div class="ltx_para" id="S1.SS1.p4"> <p class="ltx_p" id="S1.SS1.p4.1">Of most interest for this paper is the literature situated at the intersection of survival analysis and partial identification. In this line, several works exist, though in many, restrictive assumptions persist. <cite class="ltx_cite ltx_citemacro_cite">Horowitz and Manski, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib18" title="">1998</a>)</cite> use weighting or imputation to deal with the missing information that results from censoring, but this requires the specification of a weight function or imputation method, which in turn imposes assumptions on the censoring distribution. Other authors (<cite class="ltx_cite ltx_citemacro_cite">Szydlowski, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib32" title="">2019</a>)</cite>; <cite class="ltx_cite ltx_citemacro_cite">Kim, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib22" title="">2023</a>)</cite>, …) avoid this approach, but impose that covariates and/or outcomes are discrete, which does not allow practitioners to include typical continuous covariates such as age into their analysis. To our knowledge, the most general approach is provided by <cite class="ltx_cite ltx_citemacro_cite">Sakaguchi, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib29" title="">2024</a>)</cite>, who is able to partially identify several parameters of interest, including covariate effects, while leaving the censoring mechanism unspecified. Since his work is based on the assumption of time-independent effects and employs a different methodology as in this paper, our work is not comparable, but should rather be viewed as complementary. Finally, as an overarching remark, the works discussed in this paragraph have an econometric focus, whereas partially identified methods for statistics and biostatistics have received far less attention.</p> </div> </section> <section class="ltx_subsection" id="S1.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.2 </span>Paper outline</h3> <div class="ltx_para" id="S1.SS2.p1"> <p class="ltx_p" id="S1.SS2.p1.1">The remainder of this paper is structured as follows. In Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2" title="2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">2</span></a> we introduce the model and explain the methodology used to estimate it. The section is self-contained and should suffice for researchers who want to apply our model in a practical setting. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3" title="3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">3</span></a> contains the theoretical underpinnings, elaborates on several aspects of the method and provides a detailed discussion. Following that, we investigate finite sample performance in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S4" title="4 Simulation study ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">4</span></a> using simulation studies. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S5" title="5 Data applications ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">5</span></a> applies our methodology to two real data sets. Many additional analyses, remarks and discussions are deferred to the Supplementary material.</p> </div> </section> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Model and methodology</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.18">Let <math alttext="T" class="ltx_Math" display="inline" id="S2.p1.1.m1.1"><semantics id="S2.p1.1.m1.1a"><mi id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.1b"><ci id="S2.p1.1.m1.1.1.cmml" xref="S2.p1.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.1d">italic_T</annotation></semantics></math> and <math alttext="C" class="ltx_Math" display="inline" id="S2.p1.2.m2.1"><semantics id="S2.p1.2.m2.1a"><mi id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S2.p1.2.m2.1b"><ci id="S2.p1.2.m2.1.1.cmml" xref="S2.p1.2.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m2.1d">italic_C</annotation></semantics></math> be real-valued random variables. Suppose we have <math alttext="n" class="ltx_Math" display="inline" id="S2.p1.3.m3.1"><semantics id="S2.p1.3.m3.1a"><mi id="S2.p1.3.m3.1.1" xref="S2.p1.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.p1.3.m3.1b"><ci id="S2.p1.3.m3.1.1.cmml" xref="S2.p1.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.m3.1d">italic_n</annotation></semantics></math> observations <math alttext="\{W_{i}\}_{i=1}^{n}" class="ltx_Math" display="inline" id="S2.p1.4.m4.1"><semantics id="S2.p1.4.m4.1a"><msubsup id="S2.p1.4.m4.1.1" xref="S2.p1.4.m4.1.1.cmml"><mrow id="S2.p1.4.m4.1.1.1.1.1" xref="S2.p1.4.m4.1.1.1.1.2.cmml"><mo id="S2.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="S2.p1.4.m4.1.1.1.1.2.cmml">{</mo><msub id="S2.p1.4.m4.1.1.1.1.1.1" xref="S2.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S2.p1.4.m4.1.1.1.1.1.1.2" xref="S2.p1.4.m4.1.1.1.1.1.1.2.cmml">W</mi><mi id="S2.p1.4.m4.1.1.1.1.1.1.3" xref="S2.p1.4.m4.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="S2.p1.4.m4.1.1.1.1.2.cmml">}</mo></mrow><mrow id="S2.p1.4.m4.1.1.1.3" xref="S2.p1.4.m4.1.1.1.3.cmml"><mi id="S2.p1.4.m4.1.1.1.3.2" xref="S2.p1.4.m4.1.1.1.3.2.cmml">i</mi><mo id="S2.p1.4.m4.1.1.1.3.1" xref="S2.p1.4.m4.1.1.1.3.1.cmml">=</mo><mn id="S2.p1.4.m4.1.1.1.3.3" xref="S2.p1.4.m4.1.1.1.3.3.cmml">1</mn></mrow><mi id="S2.p1.4.m4.1.1.3" xref="S2.p1.4.m4.1.1.3.cmml">n</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.p1.4.m4.1b"><apply id="S2.p1.4.m4.1.1.cmml" xref="S2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.p1.4.m4.1.1.2.cmml" xref="S2.p1.4.m4.1.1">superscript</csymbol><apply id="S2.p1.4.m4.1.1.1.cmml" xref="S2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.p1.4.m4.1.1.1.2.cmml" xref="S2.p1.4.m4.1.1">subscript</csymbol><set id="S2.p1.4.m4.1.1.1.1.2.cmml" xref="S2.p1.4.m4.1.1.1.1.1"><apply id="S2.p1.4.m4.1.1.1.1.1.1.cmml" 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end_POSTSUPERSCRIPT</annotation></semantics></math> which are independent and identically distributed according to <math alttext="W=(Y,\Delta,\tilde{X})" class="ltx_Math" display="inline" id="S2.p1.5.m5.3"><semantics id="S2.p1.5.m5.3a"><mrow id="S2.p1.5.m5.3.4" xref="S2.p1.5.m5.3.4.cmml"><mi id="S2.p1.5.m5.3.4.2" xref="S2.p1.5.m5.3.4.2.cmml">W</mi><mo id="S2.p1.5.m5.3.4.1" xref="S2.p1.5.m5.3.4.1.cmml">=</mo><mrow id="S2.p1.5.m5.3.4.3.2" xref="S2.p1.5.m5.3.4.3.1.cmml"><mo id="S2.p1.5.m5.3.4.3.2.1" stretchy="false" xref="S2.p1.5.m5.3.4.3.1.cmml">(</mo><mi id="S2.p1.5.m5.1.1" xref="S2.p1.5.m5.1.1.cmml">Y</mi><mo id="S2.p1.5.m5.3.4.3.2.2" xref="S2.p1.5.m5.3.4.3.1.cmml">,</mo><mi id="S2.p1.5.m5.2.2" mathvariant="normal" xref="S2.p1.5.m5.2.2.cmml">Δ</mi><mo id="S2.p1.5.m5.3.4.3.2.3" xref="S2.p1.5.m5.3.4.3.1.cmml">,</mo><mover accent="true" id="S2.p1.5.m5.3.3" xref="S2.p1.5.m5.3.3.cmml"><mi id="S2.p1.5.m5.3.3.2" xref="S2.p1.5.m5.3.3.2.cmml">X</mi><mo id="S2.p1.5.m5.3.3.1" xref="S2.p1.5.m5.3.3.1.cmml">~</mo></mover><mo id="S2.p1.5.m5.3.4.3.2.4" stretchy="false" xref="S2.p1.5.m5.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.5.m5.3b"><apply id="S2.p1.5.m5.3.4.cmml" xref="S2.p1.5.m5.3.4"><eq id="S2.p1.5.m5.3.4.1.cmml" xref="S2.p1.5.m5.3.4.1"></eq><ci id="S2.p1.5.m5.3.4.2.cmml" xref="S2.p1.5.m5.3.4.2">𝑊</ci><vector id="S2.p1.5.m5.3.4.3.1.cmml" xref="S2.p1.5.m5.3.4.3.2"><ci id="S2.p1.5.m5.1.1.cmml" xref="S2.p1.5.m5.1.1">𝑌</ci><ci id="S2.p1.5.m5.2.2.cmml" xref="S2.p1.5.m5.2.2">Δ</ci><apply id="S2.p1.5.m5.3.3.cmml" xref="S2.p1.5.m5.3.3"><ci id="S2.p1.5.m5.3.3.1.cmml" xref="S2.p1.5.m5.3.3.1">~</ci><ci id="S2.p1.5.m5.3.3.2.cmml" xref="S2.p1.5.m5.3.3.2">𝑋</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.5.m5.3c">W=(Y,\Delta,\tilde{X})</annotation><annotation encoding="application/x-llamapun" id="S2.p1.5.m5.3d">italic_W = ( italic_Y , roman_Δ , over~ start_ARG italic_X end_ARG )</annotation></semantics></math>, where <math alttext="Y=\min(T,C)" class="ltx_Math" display="inline" id="S2.p1.6.m6.3"><semantics id="S2.p1.6.m6.3a"><mrow id="S2.p1.6.m6.3.4" xref="S2.p1.6.m6.3.4.cmml"><mi id="S2.p1.6.m6.3.4.2" xref="S2.p1.6.m6.3.4.2.cmml">Y</mi><mo id="S2.p1.6.m6.3.4.1" xref="S2.p1.6.m6.3.4.1.cmml">=</mo><mrow id="S2.p1.6.m6.3.4.3.2" xref="S2.p1.6.m6.3.4.3.1.cmml"><mi id="S2.p1.6.m6.1.1" xref="S2.p1.6.m6.1.1.cmml">min</mi><mo id="S2.p1.6.m6.3.4.3.2a" xref="S2.p1.6.m6.3.4.3.1.cmml">⁡</mo><mrow id="S2.p1.6.m6.3.4.3.2.1" xref="S2.p1.6.m6.3.4.3.1.cmml"><mo id="S2.p1.6.m6.3.4.3.2.1.1" stretchy="false" xref="S2.p1.6.m6.3.4.3.1.cmml">(</mo><mi id="S2.p1.6.m6.2.2" xref="S2.p1.6.m6.2.2.cmml">T</mi><mo id="S2.p1.6.m6.3.4.3.2.1.2" xref="S2.p1.6.m6.3.4.3.1.cmml">,</mo><mi id="S2.p1.6.m6.3.3" xref="S2.p1.6.m6.3.3.cmml">C</mi><mo id="S2.p1.6.m6.3.4.3.2.1.3" stretchy="false" xref="S2.p1.6.m6.3.4.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.6.m6.3b"><apply id="S2.p1.6.m6.3.4.cmml" xref="S2.p1.6.m6.3.4"><eq id="S2.p1.6.m6.3.4.1.cmml" xref="S2.p1.6.m6.3.4.1"></eq><ci id="S2.p1.6.m6.3.4.2.cmml" xref="S2.p1.6.m6.3.4.2">𝑌</ci><apply id="S2.p1.6.m6.3.4.3.1.cmml" xref="S2.p1.6.m6.3.4.3.2"><min id="S2.p1.6.m6.1.1.cmml" xref="S2.p1.6.m6.1.1"></min><ci id="S2.p1.6.m6.2.2.cmml" xref="S2.p1.6.m6.2.2">𝑇</ci><ci id="S2.p1.6.m6.3.3.cmml" xref="S2.p1.6.m6.3.3">𝐶</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.6.m6.3c">Y=\min(T,C)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.6.m6.3d">italic_Y = roman_min ( italic_T , italic_C )</annotation></semantics></math>, <math alttext="\Delta=\mathbbm{1}(Y=T)" class="ltx_Math" display="inline" id="S2.p1.7.m7.1"><semantics id="S2.p1.7.m7.1a"><mrow id="S2.p1.7.m7.1.1" xref="S2.p1.7.m7.1.1.cmml"><mi id="S2.p1.7.m7.1.1.3" mathvariant="normal" xref="S2.p1.7.m7.1.1.3.cmml">Δ</mi><mo id="S2.p1.7.m7.1.1.2" xref="S2.p1.7.m7.1.1.2.cmml">=</mo><mrow id="S2.p1.7.m7.1.1.1" xref="S2.p1.7.m7.1.1.1.cmml"><mn id="S2.p1.7.m7.1.1.1.3" xref="S2.p1.7.m7.1.1.1.3.cmml">𝟙</mn><mo id="S2.p1.7.m7.1.1.1.2" xref="S2.p1.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S2.p1.7.m7.1.1.1.1.1" xref="S2.p1.7.m7.1.1.1.1.1.1.cmml"><mo id="S2.p1.7.m7.1.1.1.1.1.2" stretchy="false" xref="S2.p1.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.p1.7.m7.1.1.1.1.1.1" xref="S2.p1.7.m7.1.1.1.1.1.1.cmml"><mi id="S2.p1.7.m7.1.1.1.1.1.1.2" xref="S2.p1.7.m7.1.1.1.1.1.1.2.cmml">Y</mi><mo id="S2.p1.7.m7.1.1.1.1.1.1.1" xref="S2.p1.7.m7.1.1.1.1.1.1.1.cmml">=</mo><mi id="S2.p1.7.m7.1.1.1.1.1.1.3" xref="S2.p1.7.m7.1.1.1.1.1.1.3.cmml">T</mi></mrow><mo id="S2.p1.7.m7.1.1.1.1.1.3" stretchy="false" xref="S2.p1.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.7.m7.1b"><apply id="S2.p1.7.m7.1.1.cmml" xref="S2.p1.7.m7.1.1"><eq id="S2.p1.7.m7.1.1.2.cmml" xref="S2.p1.7.m7.1.1.2"></eq><ci id="S2.p1.7.m7.1.1.3.cmml" xref="S2.p1.7.m7.1.1.3">Δ</ci><apply id="S2.p1.7.m7.1.1.1.cmml" xref="S2.p1.7.m7.1.1.1"><times id="S2.p1.7.m7.1.1.1.2.cmml" xref="S2.p1.7.m7.1.1.1.2"></times><cn id="S2.p1.7.m7.1.1.1.3.cmml" type="integer" xref="S2.p1.7.m7.1.1.1.3">1</cn><apply id="S2.p1.7.m7.1.1.1.1.1.1.cmml" xref="S2.p1.7.m7.1.1.1.1.1"><eq id="S2.p1.7.m7.1.1.1.1.1.1.1.cmml" xref="S2.p1.7.m7.1.1.1.1.1.1.1"></eq><ci id="S2.p1.7.m7.1.1.1.1.1.1.2.cmml" xref="S2.p1.7.m7.1.1.1.1.1.1.2">𝑌</ci><ci id="S2.p1.7.m7.1.1.1.1.1.1.3.cmml" xref="S2.p1.7.m7.1.1.1.1.1.1.3">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.7.m7.1c">\Delta=\mathbbm{1}(Y=T)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.7.m7.1d">roman_Δ = blackboard_1 ( italic_Y = italic_T )</annotation></semantics></math> and <math alttext="\tilde{X}" class="ltx_Math" display="inline" id="S2.p1.8.m8.1"><semantics id="S2.p1.8.m8.1a"><mover accent="true" id="S2.p1.8.m8.1.1" xref="S2.p1.8.m8.1.1.cmml"><mi id="S2.p1.8.m8.1.1.2" xref="S2.p1.8.m8.1.1.2.cmml">X</mi><mo id="S2.p1.8.m8.1.1.1" xref="S2.p1.8.m8.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S2.p1.8.m8.1b"><apply id="S2.p1.8.m8.1.1.cmml" xref="S2.p1.8.m8.1.1"><ci id="S2.p1.8.m8.1.1.1.cmml" xref="S2.p1.8.m8.1.1.1">~</ci><ci id="S2.p1.8.m8.1.1.2.cmml" xref="S2.p1.8.m8.1.1.2">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.8.m8.1c">\tilde{X}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.8.m8.1d">over~ start_ARG italic_X end_ARG</annotation></semantics></math> a <math alttext="d" class="ltx_Math" display="inline" id="S2.p1.9.m9.1"><semantics id="S2.p1.9.m9.1a"><mi id="S2.p1.9.m9.1.1" xref="S2.p1.9.m9.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S2.p1.9.m9.1b"><ci id="S2.p1.9.m9.1.1.cmml" xref="S2.p1.9.m9.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.9.m9.1c">d</annotation><annotation encoding="application/x-llamapun" id="S2.p1.9.m9.1d">italic_d</annotation></semantics></math>-dimensional vector of covariates, each element of which can either be discrete or continuous. Let <math alttext="X=(1,\tilde{X}^{\top})^{\top}" class="ltx_Math" display="inline" id="S2.p1.10.m10.2"><semantics id="S2.p1.10.m10.2a"><mrow id="S2.p1.10.m10.2.2" xref="S2.p1.10.m10.2.2.cmml"><mi id="S2.p1.10.m10.2.2.3" xref="S2.p1.10.m10.2.2.3.cmml">X</mi><mo id="S2.p1.10.m10.2.2.2" xref="S2.p1.10.m10.2.2.2.cmml">=</mo><msup id="S2.p1.10.m10.2.2.1" xref="S2.p1.10.m10.2.2.1.cmml"><mrow id="S2.p1.10.m10.2.2.1.1.1" xref="S2.p1.10.m10.2.2.1.1.2.cmml"><mo id="S2.p1.10.m10.2.2.1.1.1.2" stretchy="false" xref="S2.p1.10.m10.2.2.1.1.2.cmml">(</mo><mn id="S2.p1.10.m10.1.1" xref="S2.p1.10.m10.1.1.cmml">1</mn><mo id="S2.p1.10.m10.2.2.1.1.1.3" xref="S2.p1.10.m10.2.2.1.1.2.cmml">,</mo><msup id="S2.p1.10.m10.2.2.1.1.1.1" xref="S2.p1.10.m10.2.2.1.1.1.1.cmml"><mover accent="true" id="S2.p1.10.m10.2.2.1.1.1.1.2" xref="S2.p1.10.m10.2.2.1.1.1.1.2.cmml"><mi id="S2.p1.10.m10.2.2.1.1.1.1.2.2" xref="S2.p1.10.m10.2.2.1.1.1.1.2.2.cmml">X</mi><mo id="S2.p1.10.m10.2.2.1.1.1.1.2.1" xref="S2.p1.10.m10.2.2.1.1.1.1.2.1.cmml">~</mo></mover><mo id="S2.p1.10.m10.2.2.1.1.1.1.3" xref="S2.p1.10.m10.2.2.1.1.1.1.3.cmml">⊤</mo></msup><mo id="S2.p1.10.m10.2.2.1.1.1.4" stretchy="false" xref="S2.p1.10.m10.2.2.1.1.2.cmml">)</mo></mrow><mo id="S2.p1.10.m10.2.2.1.3" xref="S2.p1.10.m10.2.2.1.3.cmml">⊤</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.10.m10.2b"><apply id="S2.p1.10.m10.2.2.cmml" xref="S2.p1.10.m10.2.2"><eq id="S2.p1.10.m10.2.2.2.cmml" xref="S2.p1.10.m10.2.2.2"></eq><ci id="S2.p1.10.m10.2.2.3.cmml" xref="S2.p1.10.m10.2.2.3">𝑋</ci><apply id="S2.p1.10.m10.2.2.1.cmml" xref="S2.p1.10.m10.2.2.1"><csymbol cd="ambiguous" id="S2.p1.10.m10.2.2.1.2.cmml" xref="S2.p1.10.m10.2.2.1">superscript</csymbol><interval closure="open" id="S2.p1.10.m10.2.2.1.1.2.cmml" xref="S2.p1.10.m10.2.2.1.1.1"><cn id="S2.p1.10.m10.1.1.cmml" type="integer" xref="S2.p1.10.m10.1.1">1</cn><apply id="S2.p1.10.m10.2.2.1.1.1.1.cmml" xref="S2.p1.10.m10.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.p1.10.m10.2.2.1.1.1.1.1.cmml" xref="S2.p1.10.m10.2.2.1.1.1.1">superscript</csymbol><apply id="S2.p1.10.m10.2.2.1.1.1.1.2.cmml" xref="S2.p1.10.m10.2.2.1.1.1.1.2"><ci id="S2.p1.10.m10.2.2.1.1.1.1.2.1.cmml" xref="S2.p1.10.m10.2.2.1.1.1.1.2.1">~</ci><ci id="S2.p1.10.m10.2.2.1.1.1.1.2.2.cmml" xref="S2.p1.10.m10.2.2.1.1.1.1.2.2">𝑋</ci></apply><csymbol cd="latexml" id="S2.p1.10.m10.2.2.1.1.1.1.3.cmml" xref="S2.p1.10.m10.2.2.1.1.1.1.3">top</csymbol></apply></interval><csymbol cd="latexml" id="S2.p1.10.m10.2.2.1.3.cmml" xref="S2.p1.10.m10.2.2.1.3">top</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.10.m10.2c">X=(1,\tilde{X}^{\top})^{\top}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.10.m10.2d">italic_X = ( 1 , over~ start_ARG italic_X end_ARG start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>, which maps into the covariate space <math alttext="\mathcal{X}" class="ltx_Math" display="inline" id="S2.p1.11.m11.1"><semantics id="S2.p1.11.m11.1a"><mi class="ltx_font_mathcaligraphic" id="S2.p1.11.m11.1.1" xref="S2.p1.11.m11.1.1.cmml">𝒳</mi><annotation-xml encoding="MathML-Content" id="S2.p1.11.m11.1b"><ci id="S2.p1.11.m11.1.1.cmml" xref="S2.p1.11.m11.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.11.m11.1c">\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.11.m11.1d">caligraphic_X</annotation></semantics></math>. For convenience, we will index the elements of <math alttext="X" class="ltx_Math" display="inline" id="S2.p1.12.m12.1"><semantics id="S2.p1.12.m12.1a"><mi id="S2.p1.12.m12.1.1" xref="S2.p1.12.m12.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.p1.12.m12.1b"><ci id="S2.p1.12.m12.1.1.cmml" xref="S2.p1.12.m12.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.12.m12.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.p1.12.m12.1d">italic_X</annotation></semantics></math> starting from zero, so that <math alttext="X_{0}\equiv 1" class="ltx_Math" display="inline" id="S2.p1.13.m13.1"><semantics id="S2.p1.13.m13.1a"><mrow id="S2.p1.13.m13.1.1" xref="S2.p1.13.m13.1.1.cmml"><msub id="S2.p1.13.m13.1.1.2" xref="S2.p1.13.m13.1.1.2.cmml"><mi id="S2.p1.13.m13.1.1.2.2" xref="S2.p1.13.m13.1.1.2.2.cmml">X</mi><mn id="S2.p1.13.m13.1.1.2.3" xref="S2.p1.13.m13.1.1.2.3.cmml">0</mn></msub><mo id="S2.p1.13.m13.1.1.1" xref="S2.p1.13.m13.1.1.1.cmml">≡</mo><mn id="S2.p1.13.m13.1.1.3" xref="S2.p1.13.m13.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.13.m13.1b"><apply id="S2.p1.13.m13.1.1.cmml" xref="S2.p1.13.m13.1.1"><equivalent id="S2.p1.13.m13.1.1.1.cmml" xref="S2.p1.13.m13.1.1.1"></equivalent><apply id="S2.p1.13.m13.1.1.2.cmml" xref="S2.p1.13.m13.1.1.2"><csymbol cd="ambiguous" id="S2.p1.13.m13.1.1.2.1.cmml" xref="S2.p1.13.m13.1.1.2">subscript</csymbol><ci id="S2.p1.13.m13.1.1.2.2.cmml" xref="S2.p1.13.m13.1.1.2.2">𝑋</ci><cn id="S2.p1.13.m13.1.1.2.3.cmml" type="integer" xref="S2.p1.13.m13.1.1.2.3">0</cn></apply><cn id="S2.p1.13.m13.1.1.3.cmml" type="integer" xref="S2.p1.13.m13.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.13.m13.1c">X_{0}\equiv 1</annotation><annotation encoding="application/x-llamapun" id="S2.p1.13.m13.1d">italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≡ 1</annotation></semantics></math> and <math alttext="X_{k}=\tilde{X}_{k}" class="ltx_Math" display="inline" id="S2.p1.14.m14.1"><semantics id="S2.p1.14.m14.1a"><mrow id="S2.p1.14.m14.1.1" xref="S2.p1.14.m14.1.1.cmml"><msub id="S2.p1.14.m14.1.1.2" xref="S2.p1.14.m14.1.1.2.cmml"><mi id="S2.p1.14.m14.1.1.2.2" xref="S2.p1.14.m14.1.1.2.2.cmml">X</mi><mi id="S2.p1.14.m14.1.1.2.3" xref="S2.p1.14.m14.1.1.2.3.cmml">k</mi></msub><mo id="S2.p1.14.m14.1.1.1" xref="S2.p1.14.m14.1.1.1.cmml">=</mo><msub id="S2.p1.14.m14.1.1.3" xref="S2.p1.14.m14.1.1.3.cmml"><mover accent="true" id="S2.p1.14.m14.1.1.3.2" xref="S2.p1.14.m14.1.1.3.2.cmml"><mi id="S2.p1.14.m14.1.1.3.2.2" xref="S2.p1.14.m14.1.1.3.2.2.cmml">X</mi><mo id="S2.p1.14.m14.1.1.3.2.1" xref="S2.p1.14.m14.1.1.3.2.1.cmml">~</mo></mover><mi id="S2.p1.14.m14.1.1.3.3" xref="S2.p1.14.m14.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.14.m14.1b"><apply id="S2.p1.14.m14.1.1.cmml" xref="S2.p1.14.m14.1.1"><eq id="S2.p1.14.m14.1.1.1.cmml" xref="S2.p1.14.m14.1.1.1"></eq><apply id="S2.p1.14.m14.1.1.2.cmml" xref="S2.p1.14.m14.1.1.2"><csymbol cd="ambiguous" id="S2.p1.14.m14.1.1.2.1.cmml" xref="S2.p1.14.m14.1.1.2">subscript</csymbol><ci id="S2.p1.14.m14.1.1.2.2.cmml" xref="S2.p1.14.m14.1.1.2.2">𝑋</ci><ci id="S2.p1.14.m14.1.1.2.3.cmml" xref="S2.p1.14.m14.1.1.2.3">𝑘</ci></apply><apply id="S2.p1.14.m14.1.1.3.cmml" xref="S2.p1.14.m14.1.1.3"><csymbol cd="ambiguous" id="S2.p1.14.m14.1.1.3.1.cmml" xref="S2.p1.14.m14.1.1.3">subscript</csymbol><apply id="S2.p1.14.m14.1.1.3.2.cmml" xref="S2.p1.14.m14.1.1.3.2"><ci id="S2.p1.14.m14.1.1.3.2.1.cmml" xref="S2.p1.14.m14.1.1.3.2.1">~</ci><ci id="S2.p1.14.m14.1.1.3.2.2.cmml" xref="S2.p1.14.m14.1.1.3.2.2">𝑋</ci></apply><ci id="S2.p1.14.m14.1.1.3.3.cmml" xref="S2.p1.14.m14.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.14.m14.1c">X_{k}=\tilde{X}_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.14.m14.1d">italic_X start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, for all <math alttext="k\in\{1,\dots,d\}" class="ltx_Math" display="inline" id="S2.p1.15.m15.3"><semantics id="S2.p1.15.m15.3a"><mrow id="S2.p1.15.m15.3.4" xref="S2.p1.15.m15.3.4.cmml"><mi id="S2.p1.15.m15.3.4.2" xref="S2.p1.15.m15.3.4.2.cmml">k</mi><mo id="S2.p1.15.m15.3.4.1" xref="S2.p1.15.m15.3.4.1.cmml">∈</mo><mrow id="S2.p1.15.m15.3.4.3.2" xref="S2.p1.15.m15.3.4.3.1.cmml"><mo id="S2.p1.15.m15.3.4.3.2.1" stretchy="false" xref="S2.p1.15.m15.3.4.3.1.cmml">{</mo><mn id="S2.p1.15.m15.1.1" xref="S2.p1.15.m15.1.1.cmml">1</mn><mo id="S2.p1.15.m15.3.4.3.2.2" xref="S2.p1.15.m15.3.4.3.1.cmml">,</mo><mi id="S2.p1.15.m15.2.2" mathvariant="normal" xref="S2.p1.15.m15.2.2.cmml">…</mi><mo id="S2.p1.15.m15.3.4.3.2.3" xref="S2.p1.15.m15.3.4.3.1.cmml">,</mo><mi id="S2.p1.15.m15.3.3" xref="S2.p1.15.m15.3.3.cmml">d</mi><mo id="S2.p1.15.m15.3.4.3.2.4" stretchy="false" xref="S2.p1.15.m15.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.15.m15.3b"><apply id="S2.p1.15.m15.3.4.cmml" xref="S2.p1.15.m15.3.4"><in id="S2.p1.15.m15.3.4.1.cmml" xref="S2.p1.15.m15.3.4.1"></in><ci id="S2.p1.15.m15.3.4.2.cmml" xref="S2.p1.15.m15.3.4.2">𝑘</ci><set id="S2.p1.15.m15.3.4.3.1.cmml" xref="S2.p1.15.m15.3.4.3.2"><cn id="S2.p1.15.m15.1.1.cmml" type="integer" xref="S2.p1.15.m15.1.1">1</cn><ci id="S2.p1.15.m15.2.2.cmml" xref="S2.p1.15.m15.2.2">…</ci><ci id="S2.p1.15.m15.3.3.cmml" xref="S2.p1.15.m15.3.3">𝑑</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.15.m15.3c">k\in\{1,\dots,d\}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.15.m15.3d">italic_k ∈ { 1 , … , italic_d }</annotation></semantics></math>. In this way, we can use <math alttext="X_{k}" class="ltx_Math" display="inline" id="S2.p1.16.m16.1"><semantics id="S2.p1.16.m16.1a"><msub id="S2.p1.16.m16.1.1" xref="S2.p1.16.m16.1.1.cmml"><mi id="S2.p1.16.m16.1.1.2" xref="S2.p1.16.m16.1.1.2.cmml">X</mi><mi id="S2.p1.16.m16.1.1.3" xref="S2.p1.16.m16.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.16.m16.1b"><apply id="S2.p1.16.m16.1.1.cmml" xref="S2.p1.16.m16.1.1"><csymbol cd="ambiguous" id="S2.p1.16.m16.1.1.1.cmml" xref="S2.p1.16.m16.1.1">subscript</csymbol><ci id="S2.p1.16.m16.1.1.2.cmml" xref="S2.p1.16.m16.1.1.2">𝑋</ci><ci id="S2.p1.16.m16.1.1.3.cmml" xref="S2.p1.16.m16.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.16.m16.1c">X_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.16.m16.1d">italic_X start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\tilde{X}_{k}" class="ltx_Math" display="inline" id="S2.p1.17.m17.1"><semantics id="S2.p1.17.m17.1a"><msub id="S2.p1.17.m17.1.1" xref="S2.p1.17.m17.1.1.cmml"><mover accent="true" id="S2.p1.17.m17.1.1.2" xref="S2.p1.17.m17.1.1.2.cmml"><mi id="S2.p1.17.m17.1.1.2.2" xref="S2.p1.17.m17.1.1.2.2.cmml">X</mi><mo id="S2.p1.17.m17.1.1.2.1" xref="S2.p1.17.m17.1.1.2.1.cmml">~</mo></mover><mi id="S2.p1.17.m17.1.1.3" xref="S2.p1.17.m17.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.17.m17.1b"><apply id="S2.p1.17.m17.1.1.cmml" xref="S2.p1.17.m17.1.1"><csymbol cd="ambiguous" id="S2.p1.17.m17.1.1.1.cmml" xref="S2.p1.17.m17.1.1">subscript</csymbol><apply id="S2.p1.17.m17.1.1.2.cmml" xref="S2.p1.17.m17.1.1.2"><ci id="S2.p1.17.m17.1.1.2.1.cmml" xref="S2.p1.17.m17.1.1.2.1">~</ci><ci id="S2.p1.17.m17.1.1.2.2.cmml" xref="S2.p1.17.m17.1.1.2.2">𝑋</ci></apply><ci id="S2.p1.17.m17.1.1.3.cmml" xref="S2.p1.17.m17.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.17.m17.1c">\tilde{X}_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.17.m17.1d">over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> interchangeably. We will refer to random variables with uppercase letters and to realizations with lowercase letters. Fix a time point of interest <math alttext="t" class="ltx_Math" display="inline" id="S2.p1.18.m18.1"><semantics id="S2.p1.18.m18.1a"><mi id="S2.p1.18.m18.1.1" xref="S2.p1.18.m18.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.p1.18.m18.1b"><ci id="S2.p1.18.m18.1.1.cmml" xref="S2.p1.18.m18.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.18.m18.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.p1.18.m18.1d">italic_t</annotation></semantics></math>. We consider the model</p> <table class="ltx_equation ltx_eqn_table" id="S2.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathbb{P}(T>t|X=x)=1-\Lambda_{t}(x^{\top}\beta_{\text{true},t})," class="ltx_Math" display="block" id="S2.E1.m1.3"><semantics id="S2.E1.m1.3a"><mrow id="S2.E1.m1.3.3.1" xref="S2.E1.m1.3.3.1.1.cmml"><mrow id="S2.E1.m1.3.3.1.1" xref="S2.E1.m1.3.3.1.1.cmml"><mrow id="S2.E1.m1.3.3.1.1.1" xref="S2.E1.m1.3.3.1.1.1.cmml"><mi id="S2.E1.m1.3.3.1.1.1.3" xref="S2.E1.m1.3.3.1.1.1.3.cmml">ℙ</mi><mo id="S2.E1.m1.3.3.1.1.1.2" xref="S2.E1.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S2.E1.m1.3.3.1.1.1.1.1" xref="S2.E1.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S2.E1.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S2.E1.m1.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E1.m1.3.3.1.1.1.1.1.1" xref="S2.E1.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.2" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.cmml">T</mi><mo id="S2.E1.m1.3.3.1.1.1.1.1.1.3" xref="S2.E1.m1.3.3.1.1.1.1.1.1.3.cmml">></mo><mrow id="S2.E1.m1.3.3.1.1.1.1.1.1.4" xref="S2.E1.m1.3.3.1.1.1.1.1.1.4.cmml"><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.4.2" xref="S2.E1.m1.3.3.1.1.1.1.1.1.4.2.cmml">t</mi><mo fence="false" id="S2.E1.m1.3.3.1.1.1.1.1.1.4.1" xref="S2.E1.m1.3.3.1.1.1.1.1.1.4.1.cmml">|</mo><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.4.3" xref="S2.E1.m1.3.3.1.1.1.1.1.1.4.3.cmml">X</mi></mrow><mo id="S2.E1.m1.3.3.1.1.1.1.1.1.5" xref="S2.E1.m1.3.3.1.1.1.1.1.1.5.cmml">=</mo><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.6" xref="S2.E1.m1.3.3.1.1.1.1.1.1.6.cmml">x</mi></mrow><mo id="S2.E1.m1.3.3.1.1.1.1.1.3" stretchy="false" xref="S2.E1.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.3.3.1.1.3" xref="S2.E1.m1.3.3.1.1.3.cmml">=</mo><mrow id="S2.E1.m1.3.3.1.1.2" xref="S2.E1.m1.3.3.1.1.2.cmml"><mn id="S2.E1.m1.3.3.1.1.2.3" xref="S2.E1.m1.3.3.1.1.2.3.cmml">1</mn><mo id="S2.E1.m1.3.3.1.1.2.2" xref="S2.E1.m1.3.3.1.1.2.2.cmml">−</mo><mrow id="S2.E1.m1.3.3.1.1.2.1" xref="S2.E1.m1.3.3.1.1.2.1.cmml"><msub id="S2.E1.m1.3.3.1.1.2.1.3" xref="S2.E1.m1.3.3.1.1.2.1.3.cmml"><mi id="S2.E1.m1.3.3.1.1.2.1.3.2" mathvariant="normal" xref="S2.E1.m1.3.3.1.1.2.1.3.2.cmml">Λ</mi><mi id="S2.E1.m1.3.3.1.1.2.1.3.3" xref="S2.E1.m1.3.3.1.1.2.1.3.3.cmml">t</mi></msub><mo id="S2.E1.m1.3.3.1.1.2.1.2" xref="S2.E1.m1.3.3.1.1.2.1.2.cmml">⁢</mo><mrow id="S2.E1.m1.3.3.1.1.2.1.1.1" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.cmml"><mo id="S2.E1.m1.3.3.1.1.2.1.1.1.2" stretchy="false" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.cmml">(</mo><mrow id="S2.E1.m1.3.3.1.1.2.1.1.1.1" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.cmml"><msup id="S2.E1.m1.3.3.1.1.2.1.1.1.1.2" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.2.cmml"><mi id="S2.E1.m1.3.3.1.1.2.1.1.1.1.2.2" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.2.2.cmml">x</mi><mo id="S2.E1.m1.3.3.1.1.2.1.1.1.1.2.3" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.2.3.cmml">⊤</mo></msup><mo id="S2.E1.m1.3.3.1.1.2.1.1.1.1.1" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.1.cmml">⁢</mo><msub id="S2.E1.m1.3.3.1.1.2.1.1.1.1.3" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.3.cmml"><mi id="S2.E1.m1.3.3.1.1.2.1.1.1.1.3.2" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.3.2.cmml">β</mi><mrow id="S2.E1.m1.2.2.2.4" xref="S2.E1.m1.2.2.2.3.cmml"><mtext id="S2.E1.m1.1.1.1.1" xref="S2.E1.m1.1.1.1.1a.cmml">true</mtext><mo id="S2.E1.m1.2.2.2.4.1" xref="S2.E1.m1.2.2.2.3.cmml">,</mo><mi id="S2.E1.m1.2.2.2.2" xref="S2.E1.m1.2.2.2.2.cmml">t</mi></mrow></msub></mrow><mo id="S2.E1.m1.3.3.1.1.2.1.1.1.3" stretchy="false" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.E1.m1.3.3.1.2" xref="S2.E1.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E1.m1.3b"><apply id="S2.E1.m1.3.3.1.1.cmml" xref="S2.E1.m1.3.3.1"><eq id="S2.E1.m1.3.3.1.1.3.cmml" xref="S2.E1.m1.3.3.1.1.3"></eq><apply id="S2.E1.m1.3.3.1.1.1.cmml" xref="S2.E1.m1.3.3.1.1.1"><times id="S2.E1.m1.3.3.1.1.1.2.cmml" xref="S2.E1.m1.3.3.1.1.1.2"></times><ci id="S2.E1.m1.3.3.1.1.1.3.cmml" xref="S2.E1.m1.3.3.1.1.1.3">ℙ</ci><apply id="S2.E1.m1.3.3.1.1.1.1.1.1.cmml" xref="S2.E1.m1.3.3.1.1.1.1.1"><and id="S2.E1.m1.3.3.1.1.1.1.1.1a.cmml" xref="S2.E1.m1.3.3.1.1.1.1.1"></and><apply id="S2.E1.m1.3.3.1.1.1.1.1.1b.cmml" xref="S2.E1.m1.3.3.1.1.1.1.1"><gt id="S2.E1.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S2.E1.m1.3.3.1.1.1.1.1.1.3"></gt><ci id="S2.E1.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2">𝑇</ci><apply id="S2.E1.m1.3.3.1.1.1.1.1.1.4.cmml" xref="S2.E1.m1.3.3.1.1.1.1.1.1.4"><csymbol cd="latexml" id="S2.E1.m1.3.3.1.1.1.1.1.1.4.1.cmml" xref="S2.E1.m1.3.3.1.1.1.1.1.1.4.1">conditional</csymbol><ci id="S2.E1.m1.3.3.1.1.1.1.1.1.4.2.cmml" xref="S2.E1.m1.3.3.1.1.1.1.1.1.4.2">𝑡</ci><ci id="S2.E1.m1.3.3.1.1.1.1.1.1.4.3.cmml" xref="S2.E1.m1.3.3.1.1.1.1.1.1.4.3">𝑋</ci></apply></apply><apply id="S2.E1.m1.3.3.1.1.1.1.1.1c.cmml" xref="S2.E1.m1.3.3.1.1.1.1.1"><eq id="S2.E1.m1.3.3.1.1.1.1.1.1.5.cmml" 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xref="S2.E1.m1.3.3.1.1.2.1.3.3">𝑡</ci></apply><apply id="S2.E1.m1.3.3.1.1.2.1.1.1.1.cmml" xref="S2.E1.m1.3.3.1.1.2.1.1.1"><times id="S2.E1.m1.3.3.1.1.2.1.1.1.1.1.cmml" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.1"></times><apply id="S2.E1.m1.3.3.1.1.2.1.1.1.1.2.cmml" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E1.m1.3.3.1.1.2.1.1.1.1.2.1.cmml" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.2">superscript</csymbol><ci id="S2.E1.m1.3.3.1.1.2.1.1.1.1.2.2.cmml" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.2.2">𝑥</ci><csymbol cd="latexml" id="S2.E1.m1.3.3.1.1.2.1.1.1.1.2.3.cmml" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.2.3">top</csymbol></apply><apply id="S2.E1.m1.3.3.1.1.2.1.1.1.1.3.cmml" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.E1.m1.3.3.1.1.2.1.1.1.1.3.1.cmml" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.3">subscript</csymbol><ci id="S2.E1.m1.3.3.1.1.2.1.1.1.1.3.2.cmml" xref="S2.E1.m1.3.3.1.1.2.1.1.1.1.3.2">𝛽</ci><list id="S2.E1.m1.2.2.2.3.cmml" xref="S2.E1.m1.2.2.2.4"><ci id="S2.E1.m1.1.1.1.1a.cmml" xref="S2.E1.m1.1.1.1.1"><mtext id="S2.E1.m1.1.1.1.1.cmml" mathsize="70%" xref="S2.E1.m1.1.1.1.1">true</mtext></ci><ci id="S2.E1.m1.2.2.2.2.cmml" xref="S2.E1.m1.2.2.2.2">𝑡</ci></list></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.3c">\mathbb{P}(T>t|X=x)=1-\Lambda_{t}(x^{\top}\beta_{\text{true},t}),</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.3d">blackboard_P ( italic_T > italic_t | italic_X = italic_x ) = 1 - roman_Λ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_x start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT italic_β start_POSTSUBSCRIPT true , italic_t end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p1.26">where <math alttext="\Lambda_{t}:\mathbb{R}\to(0,1)" class="ltx_Math" display="inline" id="S2.p1.19.m1.2"><semantics id="S2.p1.19.m1.2a"><mrow id="S2.p1.19.m1.2.3" xref="S2.p1.19.m1.2.3.cmml"><msub id="S2.p1.19.m1.2.3.2" xref="S2.p1.19.m1.2.3.2.cmml"><mi id="S2.p1.19.m1.2.3.2.2" mathvariant="normal" xref="S2.p1.19.m1.2.3.2.2.cmml">Λ</mi><mi id="S2.p1.19.m1.2.3.2.3" xref="S2.p1.19.m1.2.3.2.3.cmml">t</mi></msub><mo id="S2.p1.19.m1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S2.p1.19.m1.2.3.1.cmml">:</mo><mrow id="S2.p1.19.m1.2.3.3" xref="S2.p1.19.m1.2.3.3.cmml"><mi id="S2.p1.19.m1.2.3.3.2" xref="S2.p1.19.m1.2.3.3.2.cmml">ℝ</mi><mo id="S2.p1.19.m1.2.3.3.1" stretchy="false" xref="S2.p1.19.m1.2.3.3.1.cmml">→</mo><mrow id="S2.p1.19.m1.2.3.3.3.2" xref="S2.p1.19.m1.2.3.3.3.1.cmml"><mo id="S2.p1.19.m1.2.3.3.3.2.1" stretchy="false" xref="S2.p1.19.m1.2.3.3.3.1.cmml">(</mo><mn id="S2.p1.19.m1.1.1" xref="S2.p1.19.m1.1.1.cmml">0</mn><mo id="S2.p1.19.m1.2.3.3.3.2.2" 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xref="S2.p1.20.m2.11.11"></share><apply id="S2.p1.20.m2.11.11.7.cmml" xref="S2.p1.20.m2.11.11.7"><csymbol cd="ambiguous" id="S2.p1.20.m2.11.11.7.1.cmml" xref="S2.p1.20.m2.11.11.7">superscript</csymbol><ci id="S2.p1.20.m2.11.11.7.2.cmml" xref="S2.p1.20.m2.11.11.7.2">ℝ</ci><apply id="S2.p1.20.m2.11.11.7.3.cmml" xref="S2.p1.20.m2.11.11.7.3"><plus id="S2.p1.20.m2.11.11.7.3.1.cmml" xref="S2.p1.20.m2.11.11.7.3.1"></plus><ci id="S2.p1.20.m2.11.11.7.3.2.cmml" xref="S2.p1.20.m2.11.11.7.3.2">𝑑</ci><cn id="S2.p1.20.m2.11.11.7.3.3.cmml" type="integer" xref="S2.p1.20.m2.11.11.7.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.20.m2.11c">\beta_{\text{true},t}=(\beta_{\text{true},t,0},\dots,\beta_{\text{true},t,d})% \in\mathbb{R}^{d+1}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.20.m2.11d">italic_β start_POSTSUBSCRIPT true , italic_t end_POSTSUBSCRIPT = ( italic_β start_POSTSUBSCRIPT true , italic_t , 0 end_POSTSUBSCRIPT , … , italic_β start_POSTSUBSCRIPT true , italic_t , italic_d end_POSTSUBSCRIPT ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_d + 1 end_POSTSUPERSCRIPT</annotation></semantics></math> is the true parameter vector. Under certain choices for <math alttext="\Lambda_{t}" class="ltx_Math" display="inline" id="S2.p1.21.m3.1"><semantics id="S2.p1.21.m3.1a"><msub id="S2.p1.21.m3.1.1" xref="S2.p1.21.m3.1.1.cmml"><mi id="S2.p1.21.m3.1.1.2" mathvariant="normal" xref="S2.p1.21.m3.1.1.2.cmml">Λ</mi><mi id="S2.p1.21.m3.1.1.3" xref="S2.p1.21.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.21.m3.1b"><apply id="S2.p1.21.m3.1.1.cmml" xref="S2.p1.21.m3.1.1"><csymbol cd="ambiguous" id="S2.p1.21.m3.1.1.1.cmml" xref="S2.p1.21.m3.1.1">subscript</csymbol><ci id="S2.p1.21.m3.1.1.2.cmml" xref="S2.p1.21.m3.1.1.2">Λ</ci><ci id="S2.p1.21.m3.1.1.3.cmml" xref="S2.p1.21.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.21.m3.1c">\Lambda_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.21.m3.1d">roman_Λ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, this formulation includes some well-known models, such as the semiparametric Cox PH model (using <math alttext="\Lambda_{t}(\cdot)=1-\exp(-\exp(\cdot)" class="ltx_math_unparsed" display="inline" id="S2.p1.22.m4.3"><semantics id="S2.p1.22.m4.3a"><mrow id="S2.p1.22.m4.3b"><msub id="S2.p1.22.m4.3.4"><mi id="S2.p1.22.m4.3.4.2" mathvariant="normal">Λ</mi><mi id="S2.p1.22.m4.3.4.3">t</mi></msub><mrow id="S2.p1.22.m4.3.5"><mo id="S2.p1.22.m4.3.5.1" stretchy="false">(</mo><mo id="S2.p1.22.m4.1.1" lspace="0em" rspace="0em">⋅</mo><mo id="S2.p1.22.m4.3.5.2" stretchy="false">)</mo></mrow><mo id="S2.p1.22.m4.3.6">=</mo><mn id="S2.p1.22.m4.3.7">1</mn><mo id="S2.p1.22.m4.3.8">−</mo><mi id="S2.p1.22.m4.3.9">exp</mi><mrow id="S2.p1.22.m4.3.10"><mo id="S2.p1.22.m4.3.10.1" stretchy="false">(</mo><mo id="S2.p1.22.m4.3.10.2" lspace="0em">−</mo><mi id="S2.p1.22.m4.2.2">exp</mi><mrow id="S2.p1.22.m4.3.10.3"><mo id="S2.p1.22.m4.3.10.3.1" stretchy="false">(</mo><mo id="S2.p1.22.m4.3.3" lspace="0em" rspace="0em">⋅</mo><mo id="S2.p1.22.m4.3.10.3.2" stretchy="false">)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex" id="S2.p1.22.m4.3c">\Lambda_{t}(\cdot)=1-\exp(-\exp(\cdot)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.22.m4.3d">roman_Λ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( ⋅ ) = 1 - roman_exp ( - roman_exp ( ⋅ )</annotation></semantics></math>) or the proportional odds model (cf. Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Link_function</span>), and has been studied before, for example by <cite class="ltx_cite ltx_citemacro_cite">Delgado et al., (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib10" title="">2022</a>)</cite>. However, in contrast to these models, we stress again that we <em class="ltx_emph ltx_font_italic" id="S2.p1.26.1">only</em> impose our model at a <em class="ltx_emph ltx_font_italic" id="S2.p1.26.2">fixed</em> time point <math alttext="t" class="ltx_Math" display="inline" id="S2.p1.23.m5.1"><semantics id="S2.p1.23.m5.1a"><mi id="S2.p1.23.m5.1.1" xref="S2.p1.23.m5.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.p1.23.m5.1b"><ci id="S2.p1.23.m5.1.1.cmml" xref="S2.p1.23.m5.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.23.m5.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.p1.23.m5.1d">italic_t</annotation></semantics></math>. As a consequence, when choosing <math alttext="\Lambda_{t}" class="ltx_Math" display="inline" id="S2.p1.24.m6.1"><semantics id="S2.p1.24.m6.1a"><msub id="S2.p1.24.m6.1.1" xref="S2.p1.24.m6.1.1.cmml"><mi id="S2.p1.24.m6.1.1.2" mathvariant="normal" xref="S2.p1.24.m6.1.1.2.cmml">Λ</mi><mi id="S2.p1.24.m6.1.1.3" xref="S2.p1.24.m6.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.24.m6.1b"><apply id="S2.p1.24.m6.1.1.cmml" xref="S2.p1.24.m6.1.1"><csymbol cd="ambiguous" id="S2.p1.24.m6.1.1.1.cmml" xref="S2.p1.24.m6.1.1">subscript</csymbol><ci id="S2.p1.24.m6.1.1.2.cmml" xref="S2.p1.24.m6.1.1.2">Λ</ci><ci id="S2.p1.24.m6.1.1.3.cmml" xref="S2.p1.24.m6.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.24.m6.1c">\Lambda_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.24.m6.1d">roman_Λ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> in order to specify model (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E1" title="In 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">1</span></a>) in the form of a Cox PH or proportional odds model, we emphasize that the proportional hazards or odds assumption are not actually imposed. Indeed, the model is only imposed at a single time point <math alttext="t" class="ltx_Math" display="inline" id="S2.p1.25.m7.1"><semantics id="S2.p1.25.m7.1a"><mi id="S2.p1.25.m7.1.1" xref="S2.p1.25.m7.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.p1.25.m7.1b"><ci id="S2.p1.25.m7.1.1.cmml" xref="S2.p1.25.m7.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.25.m7.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.p1.25.m7.1d">italic_t</annotation></semantics></math>, and hence the covariate effects are allowed to vary over time (more generally, the model can even be completely different at other time points). For practical settings where these proportionality assumptions would be valid, we extend the approach in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.SS3" title="3.3 Time-independent effects of covariates ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">3.3</span></a> so that it can include the knowledge of covariate effects that are constant over time. For simplicity, we will suppress the subscript <math alttext="t" class="ltx_Math" display="inline" id="S2.p1.26.m8.1"><semantics id="S2.p1.26.m8.1a"><mi id="S2.p1.26.m8.1.1" xref="S2.p1.26.m8.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.p1.26.m8.1b"><ci id="S2.p1.26.m8.1.1.cmml" xref="S2.p1.26.m8.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.26.m8.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.p1.26.m8.1d">italic_t</annotation></semantics></math> from the notation throughout the rest of the paper.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.1">Specifying a choice for <math alttext="\Lambda" class="ltx_Math" display="inline" id="S2.p2.1.m1.1"><semantics id="S2.p2.1.m1.1a"><mi id="S2.p2.1.m1.1.1" mathvariant="normal" xref="S2.p2.1.m1.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.1b"><ci id="S2.p2.1.m1.1.1.cmml" xref="S2.p2.1.m1.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.1d">roman_Λ</annotation></semantics></math> is not innocuous and should be motivated based on the practical setting. However, the mathematical structure provided by this minimal modeling assumption enables us to improve upon the generally uniformative <cite class="ltx_cite ltx_citemacro_cite">Peterson, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib26" title="">1976</a>)</cite> bounds, and provides the model with interpretable regression coefficients. Moreover, we obtain a specification test in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.SS4" title="3.4 Discussion ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">3.4</span></a>.</p> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.6">Naturally, the choice of <math alttext="\Lambda" class="ltx_Math" display="inline" id="S2.p3.1.m1.1"><semantics id="S2.p3.1.m1.1a"><mi id="S2.p3.1.m1.1.1" mathvariant="normal" xref="S2.p3.1.m1.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S2.p3.1.m1.1b"><ci id="S2.p3.1.m1.1.1.cmml" xref="S2.p3.1.m1.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.1.m1.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S2.p3.1.m1.1d">roman_Λ</annotation></semantics></math> determines the way in which the coefficients should be interpreted. For <math alttext="\Lambda" class="ltx_Math" display="inline" id="S2.p3.2.m2.1"><semantics id="S2.p3.2.m2.1a"><mi id="S2.p3.2.m2.1.1" mathvariant="normal" xref="S2.p3.2.m2.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S2.p3.2.m2.1b"><ci id="S2.p3.2.m2.1.1.cmml" xref="S2.p3.2.m2.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.2.m2.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S2.p3.2.m2.1d">roman_Λ</annotation></semantics></math> related to the Cox PH model as explained above, we have <math alttext="\beta_{\text{true}}=(\log(H_{0}(t)),\beta_{\text{true},1},\dots,\beta_{\text{% true},d})" class="ltx_Math" display="inline" id="S2.p3.3.m3.10"><semantics id="S2.p3.3.m3.10a"><mrow id="S2.p3.3.m3.10.10" xref="S2.p3.3.m3.10.10.cmml"><msub id="S2.p3.3.m3.10.10.5" xref="S2.p3.3.m3.10.10.5.cmml"><mi id="S2.p3.3.m3.10.10.5.2" xref="S2.p3.3.m3.10.10.5.2.cmml">β</mi><mtext id="S2.p3.3.m3.10.10.5.3" xref="S2.p3.3.m3.10.10.5.3a.cmml">true</mtext></msub><mo id="S2.p3.3.m3.10.10.4" xref="S2.p3.3.m3.10.10.4.cmml">=</mo><mrow id="S2.p3.3.m3.10.10.3.3" xref="S2.p3.3.m3.10.10.3.4.cmml"><mo id="S2.p3.3.m3.10.10.3.3.4" stretchy="false" xref="S2.p3.3.m3.10.10.3.4.cmml">(</mo><mrow id="S2.p3.3.m3.8.8.1.1.1.1" xref="S2.p3.3.m3.8.8.1.1.1.2.cmml"><mi id="S2.p3.3.m3.6.6" xref="S2.p3.3.m3.6.6.cmml">log</mi><mo id="S2.p3.3.m3.8.8.1.1.1.1a" xref="S2.p3.3.m3.8.8.1.1.1.2.cmml">⁡</mo><mrow id="S2.p3.3.m3.8.8.1.1.1.1.1" xref="S2.p3.3.m3.8.8.1.1.1.2.cmml"><mo id="S2.p3.3.m3.8.8.1.1.1.1.1.2" stretchy="false" xref="S2.p3.3.m3.8.8.1.1.1.2.cmml">(</mo><mrow id="S2.p3.3.m3.8.8.1.1.1.1.1.1" xref="S2.p3.3.m3.8.8.1.1.1.1.1.1.cmml"><msub id="S2.p3.3.m3.8.8.1.1.1.1.1.1.2" xref="S2.p3.3.m3.8.8.1.1.1.1.1.1.2.cmml"><mi id="S2.p3.3.m3.8.8.1.1.1.1.1.1.2.2" xref="S2.p3.3.m3.8.8.1.1.1.1.1.1.2.2.cmml">H</mi><mn id="S2.p3.3.m3.8.8.1.1.1.1.1.1.2.3" xref="S2.p3.3.m3.8.8.1.1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S2.p3.3.m3.8.8.1.1.1.1.1.1.1" xref="S2.p3.3.m3.8.8.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.p3.3.m3.8.8.1.1.1.1.1.1.3.2" xref="S2.p3.3.m3.8.8.1.1.1.1.1.1.cmml"><mo id="S2.p3.3.m3.8.8.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S2.p3.3.m3.8.8.1.1.1.1.1.1.cmml">(</mo><mi id="S2.p3.3.m3.5.5" xref="S2.p3.3.m3.5.5.cmml">t</mi><mo id="S2.p3.3.m3.8.8.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S2.p3.3.m3.8.8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p3.3.m3.8.8.1.1.1.1.1.3" stretchy="false" xref="S2.p3.3.m3.8.8.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.p3.3.m3.10.10.3.3.5" xref="S2.p3.3.m3.10.10.3.4.cmml">,</mo><msub id="S2.p3.3.m3.9.9.2.2.2" xref="S2.p3.3.m3.9.9.2.2.2.cmml"><mi id="S2.p3.3.m3.9.9.2.2.2.2" xref="S2.p3.3.m3.9.9.2.2.2.2.cmml">β</mi><mrow id="S2.p3.3.m3.2.2.2.4" xref="S2.p3.3.m3.2.2.2.3.cmml"><mtext id="S2.p3.3.m3.1.1.1.1" xref="S2.p3.3.m3.1.1.1.1a.cmml">true</mtext><mo id="S2.p3.3.m3.2.2.2.4.1" xref="S2.p3.3.m3.2.2.2.3.cmml">,</mo><mn id="S2.p3.3.m3.2.2.2.2" xref="S2.p3.3.m3.2.2.2.2.cmml">1</mn></mrow></msub><mo id="S2.p3.3.m3.10.10.3.3.6" xref="S2.p3.3.m3.10.10.3.4.cmml">,</mo><mi id="S2.p3.3.m3.7.7" mathvariant="normal" xref="S2.p3.3.m3.7.7.cmml">…</mi><mo id="S2.p3.3.m3.10.10.3.3.7" xref="S2.p3.3.m3.10.10.3.4.cmml">,</mo><msub id="S2.p3.3.m3.10.10.3.3.3" xref="S2.p3.3.m3.10.10.3.3.3.cmml"><mi id="S2.p3.3.m3.10.10.3.3.3.2" xref="S2.p3.3.m3.10.10.3.3.3.2.cmml">β</mi><mrow id="S2.p3.3.m3.4.4.2.4" xref="S2.p3.3.m3.4.4.2.3.cmml"><mtext id="S2.p3.3.m3.3.3.1.1" xref="S2.p3.3.m3.3.3.1.1a.cmml">true</mtext><mo id="S2.p3.3.m3.4.4.2.4.1" xref="S2.p3.3.m3.4.4.2.3.cmml">,</mo><mi id="S2.p3.3.m3.4.4.2.2" xref="S2.p3.3.m3.4.4.2.2.cmml">d</mi></mrow></msub><mo id="S2.p3.3.m3.10.10.3.3.8" stretchy="false" xref="S2.p3.3.m3.10.10.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.3.m3.10b"><apply id="S2.p3.3.m3.10.10.cmml" xref="S2.p3.3.m3.10.10"><eq id="S2.p3.3.m3.10.10.4.cmml" xref="S2.p3.3.m3.10.10.4"></eq><apply id="S2.p3.3.m3.10.10.5.cmml" xref="S2.p3.3.m3.10.10.5"><csymbol cd="ambiguous" id="S2.p3.3.m3.10.10.5.1.cmml" xref="S2.p3.3.m3.10.10.5">subscript</csymbol><ci id="S2.p3.3.m3.10.10.5.2.cmml" xref="S2.p3.3.m3.10.10.5.2">𝛽</ci><ci id="S2.p3.3.m3.10.10.5.3a.cmml" xref="S2.p3.3.m3.10.10.5.3"><mtext id="S2.p3.3.m3.10.10.5.3.cmml" mathsize="70%" xref="S2.p3.3.m3.10.10.5.3">true</mtext></ci></apply><vector id="S2.p3.3.m3.10.10.3.4.cmml" xref="S2.p3.3.m3.10.10.3.3"><apply id="S2.p3.3.m3.8.8.1.1.1.2.cmml" xref="S2.p3.3.m3.8.8.1.1.1.1"><log id="S2.p3.3.m3.6.6.cmml" xref="S2.p3.3.m3.6.6"></log><apply id="S2.p3.3.m3.8.8.1.1.1.1.1.1.cmml" xref="S2.p3.3.m3.8.8.1.1.1.1.1.1"><times id="S2.p3.3.m3.8.8.1.1.1.1.1.1.1.cmml" xref="S2.p3.3.m3.8.8.1.1.1.1.1.1.1"></times><apply id="S2.p3.3.m3.8.8.1.1.1.1.1.1.2.cmml" xref="S2.p3.3.m3.8.8.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.p3.3.m3.8.8.1.1.1.1.1.1.2.1.cmml" xref="S2.p3.3.m3.8.8.1.1.1.1.1.1.2">subscript</csymbol><ci id="S2.p3.3.m3.8.8.1.1.1.1.1.1.2.2.cmml" xref="S2.p3.3.m3.8.8.1.1.1.1.1.1.2.2">𝐻</ci><cn id="S2.p3.3.m3.8.8.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S2.p3.3.m3.8.8.1.1.1.1.1.1.2.3">0</cn></apply><ci id="S2.p3.3.m3.5.5.cmml" xref="S2.p3.3.m3.5.5">𝑡</ci></apply></apply><apply id="S2.p3.3.m3.9.9.2.2.2.cmml" xref="S2.p3.3.m3.9.9.2.2.2"><csymbol cd="ambiguous" id="S2.p3.3.m3.9.9.2.2.2.1.cmml" xref="S2.p3.3.m3.9.9.2.2.2">subscript</csymbol><ci id="S2.p3.3.m3.9.9.2.2.2.2.cmml" xref="S2.p3.3.m3.9.9.2.2.2.2">𝛽</ci><list id="S2.p3.3.m3.2.2.2.3.cmml" xref="S2.p3.3.m3.2.2.2.4"><ci id="S2.p3.3.m3.1.1.1.1a.cmml" xref="S2.p3.3.m3.1.1.1.1"><mtext id="S2.p3.3.m3.1.1.1.1.cmml" mathsize="70%" xref="S2.p3.3.m3.1.1.1.1">true</mtext></ci><cn id="S2.p3.3.m3.2.2.2.2.cmml" type="integer" xref="S2.p3.3.m3.2.2.2.2">1</cn></list></apply><ci id="S2.p3.3.m3.7.7.cmml" xref="S2.p3.3.m3.7.7">…</ci><apply id="S2.p3.3.m3.10.10.3.3.3.cmml" xref="S2.p3.3.m3.10.10.3.3.3"><csymbol cd="ambiguous" id="S2.p3.3.m3.10.10.3.3.3.1.cmml" xref="S2.p3.3.m3.10.10.3.3.3">subscript</csymbol><ci id="S2.p3.3.m3.10.10.3.3.3.2.cmml" xref="S2.p3.3.m3.10.10.3.3.3.2">𝛽</ci><list id="S2.p3.3.m3.4.4.2.3.cmml" xref="S2.p3.3.m3.4.4.2.4"><ci id="S2.p3.3.m3.3.3.1.1a.cmml" xref="S2.p3.3.m3.3.3.1.1"><mtext id="S2.p3.3.m3.3.3.1.1.cmml" mathsize="70%" xref="S2.p3.3.m3.3.3.1.1">true</mtext></ci><ci id="S2.p3.3.m3.4.4.2.2.cmml" xref="S2.p3.3.m3.4.4.2.2">𝑑</ci></list></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.3.m3.10c">\beta_{\text{true}}=(\log(H_{0}(t)),\beta_{\text{true},1},\dots,\beta_{\text{% true},d})</annotation><annotation encoding="application/x-llamapun" id="S2.p3.3.m3.10d">italic_β start_POSTSUBSCRIPT true end_POSTSUBSCRIPT = ( roman_log ( italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) ) , italic_β start_POSTSUBSCRIPT true , 1 end_POSTSUBSCRIPT , … , italic_β start_POSTSUBSCRIPT true , italic_d end_POSTSUBSCRIPT )</annotation></semantics></math>, where <math alttext="H_{0}(t)=\int_{-\infty}^{t}h_{0}(s)ds" class="ltx_Math" display="inline" id="S2.p3.4.m4.2"><semantics id="S2.p3.4.m4.2a"><mrow id="S2.p3.4.m4.2.3" xref="S2.p3.4.m4.2.3.cmml"><mrow id="S2.p3.4.m4.2.3.2" xref="S2.p3.4.m4.2.3.2.cmml"><msub id="S2.p3.4.m4.2.3.2.2" xref="S2.p3.4.m4.2.3.2.2.cmml"><mi id="S2.p3.4.m4.2.3.2.2.2" xref="S2.p3.4.m4.2.3.2.2.2.cmml">H</mi><mn id="S2.p3.4.m4.2.3.2.2.3" xref="S2.p3.4.m4.2.3.2.2.3.cmml">0</mn></msub><mo id="S2.p3.4.m4.2.3.2.1" xref="S2.p3.4.m4.2.3.2.1.cmml">⁢</mo><mrow id="S2.p3.4.m4.2.3.2.3.2" xref="S2.p3.4.m4.2.3.2.cmml"><mo id="S2.p3.4.m4.2.3.2.3.2.1" stretchy="false" xref="S2.p3.4.m4.2.3.2.cmml">(</mo><mi id="S2.p3.4.m4.1.1" xref="S2.p3.4.m4.1.1.cmml">t</mi><mo id="S2.p3.4.m4.2.3.2.3.2.2" stretchy="false" xref="S2.p3.4.m4.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.p3.4.m4.2.3.1" rspace="0.111em" xref="S2.p3.4.m4.2.3.1.cmml">=</mo><mrow id="S2.p3.4.m4.2.3.3" xref="S2.p3.4.m4.2.3.3.cmml"><msubsup id="S2.p3.4.m4.2.3.3.1" xref="S2.p3.4.m4.2.3.3.1.cmml"><mo id="S2.p3.4.m4.2.3.3.1.2.2" xref="S2.p3.4.m4.2.3.3.1.2.2.cmml">∫</mo><mrow id="S2.p3.4.m4.2.3.3.1.2.3" xref="S2.p3.4.m4.2.3.3.1.2.3.cmml"><mo id="S2.p3.4.m4.2.3.3.1.2.3a" xref="S2.p3.4.m4.2.3.3.1.2.3.cmml">−</mo><mi id="S2.p3.4.m4.2.3.3.1.2.3.2" mathvariant="normal" xref="S2.p3.4.m4.2.3.3.1.2.3.2.cmml">∞</mi></mrow><mi id="S2.p3.4.m4.2.3.3.1.3" xref="S2.p3.4.m4.2.3.3.1.3.cmml">t</mi></msubsup><mrow id="S2.p3.4.m4.2.3.3.2" xref="S2.p3.4.m4.2.3.3.2.cmml"><msub id="S2.p3.4.m4.2.3.3.2.2" xref="S2.p3.4.m4.2.3.3.2.2.cmml"><mi id="S2.p3.4.m4.2.3.3.2.2.2" xref="S2.p3.4.m4.2.3.3.2.2.2.cmml">h</mi><mn id="S2.p3.4.m4.2.3.3.2.2.3" xref="S2.p3.4.m4.2.3.3.2.2.3.cmml">0</mn></msub><mo id="S2.p3.4.m4.2.3.3.2.1" xref="S2.p3.4.m4.2.3.3.2.1.cmml">⁢</mo><mrow id="S2.p3.4.m4.2.3.3.2.3.2" xref="S2.p3.4.m4.2.3.3.2.cmml"><mo id="S2.p3.4.m4.2.3.3.2.3.2.1" stretchy="false" xref="S2.p3.4.m4.2.3.3.2.cmml">(</mo><mi id="S2.p3.4.m4.2.2" xref="S2.p3.4.m4.2.2.cmml">s</mi><mo id="S2.p3.4.m4.2.3.3.2.3.2.2" stretchy="false" xref="S2.p3.4.m4.2.3.3.2.cmml">)</mo></mrow><mo id="S2.p3.4.m4.2.3.3.2.1a" lspace="0em" xref="S2.p3.4.m4.2.3.3.2.1.cmml">⁢</mo><mrow id="S2.p3.4.m4.2.3.3.2.4" xref="S2.p3.4.m4.2.3.3.2.4.cmml"><mo id="S2.p3.4.m4.2.3.3.2.4.1" rspace="0em" xref="S2.p3.4.m4.2.3.3.2.4.1.cmml">𝑑</mo><mi id="S2.p3.4.m4.2.3.3.2.4.2" xref="S2.p3.4.m4.2.3.3.2.4.2.cmml">s</mi></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.4.m4.2b"><apply id="S2.p3.4.m4.2.3.cmml" xref="S2.p3.4.m4.2.3"><eq id="S2.p3.4.m4.2.3.1.cmml" xref="S2.p3.4.m4.2.3.1"></eq><apply id="S2.p3.4.m4.2.3.2.cmml" xref="S2.p3.4.m4.2.3.2"><times id="S2.p3.4.m4.2.3.2.1.cmml" xref="S2.p3.4.m4.2.3.2.1"></times><apply id="S2.p3.4.m4.2.3.2.2.cmml" xref="S2.p3.4.m4.2.3.2.2"><csymbol cd="ambiguous" id="S2.p3.4.m4.2.3.2.2.1.cmml" xref="S2.p3.4.m4.2.3.2.2">subscript</csymbol><ci id="S2.p3.4.m4.2.3.2.2.2.cmml" xref="S2.p3.4.m4.2.3.2.2.2">𝐻</ci><cn id="S2.p3.4.m4.2.3.2.2.3.cmml" type="integer" xref="S2.p3.4.m4.2.3.2.2.3">0</cn></apply><ci id="S2.p3.4.m4.1.1.cmml" xref="S2.p3.4.m4.1.1">𝑡</ci></apply><apply id="S2.p3.4.m4.2.3.3.cmml" xref="S2.p3.4.m4.2.3.3"><apply id="S2.p3.4.m4.2.3.3.1.cmml" xref="S2.p3.4.m4.2.3.3.1"><csymbol cd="ambiguous" id="S2.p3.4.m4.2.3.3.1.1.cmml" xref="S2.p3.4.m4.2.3.3.1">superscript</csymbol><apply id="S2.p3.4.m4.2.3.3.1.2.cmml" xref="S2.p3.4.m4.2.3.3.1"><csymbol cd="ambiguous" id="S2.p3.4.m4.2.3.3.1.2.1.cmml" xref="S2.p3.4.m4.2.3.3.1">subscript</csymbol><int id="S2.p3.4.m4.2.3.3.1.2.2.cmml" xref="S2.p3.4.m4.2.3.3.1.2.2"></int><apply id="S2.p3.4.m4.2.3.3.1.2.3.cmml" xref="S2.p3.4.m4.2.3.3.1.2.3"><minus id="S2.p3.4.m4.2.3.3.1.2.3.1.cmml" xref="S2.p3.4.m4.2.3.3.1.2.3"></minus><infinity id="S2.p3.4.m4.2.3.3.1.2.3.2.cmml" xref="S2.p3.4.m4.2.3.3.1.2.3.2"></infinity></apply></apply><ci id="S2.p3.4.m4.2.3.3.1.3.cmml" xref="S2.p3.4.m4.2.3.3.1.3">𝑡</ci></apply><apply id="S2.p3.4.m4.2.3.3.2.cmml" xref="S2.p3.4.m4.2.3.3.2"><times id="S2.p3.4.m4.2.3.3.2.1.cmml" xref="S2.p3.4.m4.2.3.3.2.1"></times><apply id="S2.p3.4.m4.2.3.3.2.2.cmml" xref="S2.p3.4.m4.2.3.3.2.2"><csymbol cd="ambiguous" id="S2.p3.4.m4.2.3.3.2.2.1.cmml" xref="S2.p3.4.m4.2.3.3.2.2">subscript</csymbol><ci id="S2.p3.4.m4.2.3.3.2.2.2.cmml" xref="S2.p3.4.m4.2.3.3.2.2.2">ℎ</ci><cn id="S2.p3.4.m4.2.3.3.2.2.3.cmml" type="integer" xref="S2.p3.4.m4.2.3.3.2.2.3">0</cn></apply><ci id="S2.p3.4.m4.2.2.cmml" xref="S2.p3.4.m4.2.2">𝑠</ci><apply id="S2.p3.4.m4.2.3.3.2.4.cmml" xref="S2.p3.4.m4.2.3.3.2.4"><csymbol cd="latexml" id="S2.p3.4.m4.2.3.3.2.4.1.cmml" xref="S2.p3.4.m4.2.3.3.2.4.1">differential-d</csymbol><ci id="S2.p3.4.m4.2.3.3.2.4.2.cmml" xref="S2.p3.4.m4.2.3.3.2.4.2">𝑠</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.4.m4.2c">H_{0}(t)=\int_{-\infty}^{t}h_{0}(s)ds</annotation><annotation encoding="application/x-llamapun" id="S2.p3.4.m4.2d">italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) = ∫ start_POSTSUBSCRIPT - ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_s ) italic_d italic_s</annotation></semantics></math> and the true baseline hazard <math alttext="h_{0}(s)" class="ltx_Math" display="inline" id="S2.p3.5.m5.1"><semantics id="S2.p3.5.m5.1a"><mrow id="S2.p3.5.m5.1.2" xref="S2.p3.5.m5.1.2.cmml"><msub id="S2.p3.5.m5.1.2.2" xref="S2.p3.5.m5.1.2.2.cmml"><mi id="S2.p3.5.m5.1.2.2.2" xref="S2.p3.5.m5.1.2.2.2.cmml">h</mi><mn id="S2.p3.5.m5.1.2.2.3" xref="S2.p3.5.m5.1.2.2.3.cmml">0</mn></msub><mo id="S2.p3.5.m5.1.2.1" xref="S2.p3.5.m5.1.2.1.cmml">⁢</mo><mrow id="S2.p3.5.m5.1.2.3.2" xref="S2.p3.5.m5.1.2.cmml"><mo id="S2.p3.5.m5.1.2.3.2.1" stretchy="false" xref="S2.p3.5.m5.1.2.cmml">(</mo><mi id="S2.p3.5.m5.1.1" xref="S2.p3.5.m5.1.1.cmml">s</mi><mo id="S2.p3.5.m5.1.2.3.2.2" stretchy="false" xref="S2.p3.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.5.m5.1b"><apply id="S2.p3.5.m5.1.2.cmml" xref="S2.p3.5.m5.1.2"><times id="S2.p3.5.m5.1.2.1.cmml" xref="S2.p3.5.m5.1.2.1"></times><apply id="S2.p3.5.m5.1.2.2.cmml" xref="S2.p3.5.m5.1.2.2"><csymbol cd="ambiguous" id="S2.p3.5.m5.1.2.2.1.cmml" xref="S2.p3.5.m5.1.2.2">subscript</csymbol><ci id="S2.p3.5.m5.1.2.2.2.cmml" xref="S2.p3.5.m5.1.2.2.2">ℎ</ci><cn id="S2.p3.5.m5.1.2.2.3.cmml" type="integer" xref="S2.p3.5.m5.1.2.2.3">0</cn></apply><ci id="S2.p3.5.m5.1.1.cmml" xref="S2.p3.5.m5.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.5.m5.1c">h_{0}(s)</annotation><annotation encoding="application/x-llamapun" id="S2.p3.5.m5.1d">italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_s )</annotation></semantics></math> is allowed to vary in <math alttext="s" class="ltx_Math" display="inline" id="S2.p3.6.m6.1"><semantics id="S2.p3.6.m6.1a"><mi id="S2.p3.6.m6.1.1" xref="S2.p3.6.m6.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S2.p3.6.m6.1b"><ci id="S2.p3.6.m6.1.1.cmml" xref="S2.p3.6.m6.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.6.m6.1c">s</annotation><annotation encoding="application/x-llamapun" id="S2.p3.6.m6.1d">italic_s</annotation></semantics></math> nonparametrically. This indeed results in a model that includes the Cox PH model:</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="S(t|x)=\exp(-\exp(\log(H_{0}(t))+\tilde{x}^{\top}\tilde{\beta}_{\text{true}}))% =\exp(-H_{0}(t)\exp(\tilde{x}^{\top}\tilde{\beta}_{\text{true}}))," class="ltx_Math" display="block" id="S2.Ex1.m1.8"><semantics id="S2.Ex1.m1.8a"><mrow id="S2.Ex1.m1.8.8.1" xref="S2.Ex1.m1.8.8.1.1.cmml"><mrow id="S2.Ex1.m1.8.8.1.1" xref="S2.Ex1.m1.8.8.1.1.cmml"><mrow id="S2.Ex1.m1.8.8.1.1.1" xref="S2.Ex1.m1.8.8.1.1.1.cmml"><mi id="S2.Ex1.m1.8.8.1.1.1.3" xref="S2.Ex1.m1.8.8.1.1.1.3.cmml">S</mi><mo id="S2.Ex1.m1.8.8.1.1.1.2" xref="S2.Ex1.m1.8.8.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex1.m1.8.8.1.1.1.1.1" xref="S2.Ex1.m1.8.8.1.1.1.1.1.1.cmml"><mo id="S2.Ex1.m1.8.8.1.1.1.1.1.2" stretchy="false" xref="S2.Ex1.m1.8.8.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex1.m1.8.8.1.1.1.1.1.1" xref="S2.Ex1.m1.8.8.1.1.1.1.1.1.cmml"><mi id="S2.Ex1.m1.8.8.1.1.1.1.1.1.2" xref="S2.Ex1.m1.8.8.1.1.1.1.1.1.2.cmml">t</mi><mo fence="false" id="S2.Ex1.m1.8.8.1.1.1.1.1.1.1" xref="S2.Ex1.m1.8.8.1.1.1.1.1.1.1.cmml">|</mo><mi id="S2.Ex1.m1.8.8.1.1.1.1.1.1.3" xref="S2.Ex1.m1.8.8.1.1.1.1.1.1.3.cmml">x</mi></mrow><mo id="S2.Ex1.m1.8.8.1.1.1.1.1.3" stretchy="false" xref="S2.Ex1.m1.8.8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex1.m1.8.8.1.1.5" xref="S2.Ex1.m1.8.8.1.1.5.cmml">=</mo><mrow id="S2.Ex1.m1.8.8.1.1.2.1" xref="S2.Ex1.m1.8.8.1.1.2.2.cmml"><mi id="S2.Ex1.m1.4.4" xref="S2.Ex1.m1.4.4.cmml">exp</mi><mo id="S2.Ex1.m1.8.8.1.1.2.1a" xref="S2.Ex1.m1.8.8.1.1.2.2.cmml">⁡</mo><mrow id="S2.Ex1.m1.8.8.1.1.2.1.1" xref="S2.Ex1.m1.8.8.1.1.2.2.cmml"><mo id="S2.Ex1.m1.8.8.1.1.2.1.1.2" stretchy="false" xref="S2.Ex1.m1.8.8.1.1.2.2.cmml">(</mo><mrow id="S2.Ex1.m1.8.8.1.1.2.1.1.1" 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id="S2.Ex1.m1.8.8.1.1.3.1.1.1.1.1.1.1.1.3.3a.cmml" xref="S2.Ex1.m1.8.8.1.1.3.1.1.1.1.1.1.1.1.3.3"><mtext id="S2.Ex1.m1.8.8.1.1.3.1.1.1.1.1.1.1.1.3.3.cmml" mathsize="70%" xref="S2.Ex1.m1.8.8.1.1.3.1.1.1.1.1.1.1.1.3.3">true</mtext></ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m1.8c">S(t|x)=\exp(-\exp(\log(H_{0}(t))+\tilde{x}^{\top}\tilde{\beta}_{\text{true}}))% =\exp(-H_{0}(t)\exp(\tilde{x}^{\top}\tilde{\beta}_{\text{true}})),</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m1.8d">italic_S ( italic_t | italic_x ) = roman_exp ( - roman_exp ( roman_log ( italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) ) + over~ start_ARG italic_x end_ARG start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT over~ start_ARG italic_β end_ARG start_POSTSUBSCRIPT true end_POSTSUBSCRIPT ) ) = roman_exp ( - italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) roman_exp ( over~ start_ARG italic_x end_ARG start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT over~ start_ARG italic_β end_ARG start_POSTSUBSCRIPT true end_POSTSUBSCRIPT ) ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p3.8">where, analogously to the definition of <math alttext="X=(1,\tilde{X})" class="ltx_Math" display="inline" id="S2.p3.7.m1.2"><semantics id="S2.p3.7.m1.2a"><mrow id="S2.p3.7.m1.2.3" xref="S2.p3.7.m1.2.3.cmml"><mi id="S2.p3.7.m1.2.3.2" xref="S2.p3.7.m1.2.3.2.cmml">X</mi><mo id="S2.p3.7.m1.2.3.1" xref="S2.p3.7.m1.2.3.1.cmml">=</mo><mrow id="S2.p3.7.m1.2.3.3.2" xref="S2.p3.7.m1.2.3.3.1.cmml"><mo id="S2.p3.7.m1.2.3.3.2.1" stretchy="false" xref="S2.p3.7.m1.2.3.3.1.cmml">(</mo><mn id="S2.p3.7.m1.1.1" xref="S2.p3.7.m1.1.1.cmml">1</mn><mo id="S2.p3.7.m1.2.3.3.2.2" xref="S2.p3.7.m1.2.3.3.1.cmml">,</mo><mover accent="true" id="S2.p3.7.m1.2.2" xref="S2.p3.7.m1.2.2.cmml"><mi id="S2.p3.7.m1.2.2.2" xref="S2.p3.7.m1.2.2.2.cmml">X</mi><mo id="S2.p3.7.m1.2.2.1" xref="S2.p3.7.m1.2.2.1.cmml">~</mo></mover><mo id="S2.p3.7.m1.2.3.3.2.3" stretchy="false" xref="S2.p3.7.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.7.m1.2b"><apply id="S2.p3.7.m1.2.3.cmml" xref="S2.p3.7.m1.2.3"><eq id="S2.p3.7.m1.2.3.1.cmml" xref="S2.p3.7.m1.2.3.1"></eq><ci id="S2.p3.7.m1.2.3.2.cmml" xref="S2.p3.7.m1.2.3.2">𝑋</ci><interval closure="open" id="S2.p3.7.m1.2.3.3.1.cmml" xref="S2.p3.7.m1.2.3.3.2"><cn id="S2.p3.7.m1.1.1.cmml" type="integer" xref="S2.p3.7.m1.1.1">1</cn><apply id="S2.p3.7.m1.2.2.cmml" xref="S2.p3.7.m1.2.2"><ci id="S2.p3.7.m1.2.2.1.cmml" xref="S2.p3.7.m1.2.2.1">~</ci><ci id="S2.p3.7.m1.2.2.2.cmml" xref="S2.p3.7.m1.2.2.2">𝑋</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.7.m1.2c">X=(1,\tilde{X})</annotation><annotation encoding="application/x-llamapun" id="S2.p3.7.m1.2d">italic_X = ( 1 , over~ start_ARG italic_X end_ARG )</annotation></semantics></math>, we define <math alttext="\tilde{\beta}_{\text{true}}=(\beta_{\text{true},1},\dots,\beta_{\text{true},d})" class="ltx_Math" display="inline" id="S2.p3.8.m2.7"><semantics id="S2.p3.8.m2.7a"><mrow id="S2.p3.8.m2.7.7" xref="S2.p3.8.m2.7.7.cmml"><msub id="S2.p3.8.m2.7.7.4" xref="S2.p3.8.m2.7.7.4.cmml"><mover accent="true" id="S2.p3.8.m2.7.7.4.2" xref="S2.p3.8.m2.7.7.4.2.cmml"><mi id="S2.p3.8.m2.7.7.4.2.2" xref="S2.p3.8.m2.7.7.4.2.2.cmml">β</mi><mo id="S2.p3.8.m2.7.7.4.2.1" xref="S2.p3.8.m2.7.7.4.2.1.cmml">~</mo></mover><mtext id="S2.p3.8.m2.7.7.4.3" xref="S2.p3.8.m2.7.7.4.3a.cmml">true</mtext></msub><mo id="S2.p3.8.m2.7.7.3" xref="S2.p3.8.m2.7.7.3.cmml">=</mo><mrow id="S2.p3.8.m2.7.7.2.2" xref="S2.p3.8.m2.7.7.2.3.cmml"><mo id="S2.p3.8.m2.7.7.2.2.3" stretchy="false" xref="S2.p3.8.m2.7.7.2.3.cmml">(</mo><msub id="S2.p3.8.m2.6.6.1.1.1" xref="S2.p3.8.m2.6.6.1.1.1.cmml"><mi id="S2.p3.8.m2.6.6.1.1.1.2" xref="S2.p3.8.m2.6.6.1.1.1.2.cmml">β</mi><mrow id="S2.p3.8.m2.2.2.2.4" xref="S2.p3.8.m2.2.2.2.3.cmml"><mtext id="S2.p3.8.m2.1.1.1.1" xref="S2.p3.8.m2.1.1.1.1a.cmml">true</mtext><mo id="S2.p3.8.m2.2.2.2.4.1" xref="S2.p3.8.m2.2.2.2.3.cmml">,</mo><mn id="S2.p3.8.m2.2.2.2.2" xref="S2.p3.8.m2.2.2.2.2.cmml">1</mn></mrow></msub><mo id="S2.p3.8.m2.7.7.2.2.4" xref="S2.p3.8.m2.7.7.2.3.cmml">,</mo><mi id="S2.p3.8.m2.5.5" mathvariant="normal" xref="S2.p3.8.m2.5.5.cmml">…</mi><mo id="S2.p3.8.m2.7.7.2.2.5" xref="S2.p3.8.m2.7.7.2.3.cmml">,</mo><msub id="S2.p3.8.m2.7.7.2.2.2" xref="S2.p3.8.m2.7.7.2.2.2.cmml"><mi id="S2.p3.8.m2.7.7.2.2.2.2" xref="S2.p3.8.m2.7.7.2.2.2.2.cmml">β</mi><mrow id="S2.p3.8.m2.4.4.2.4" xref="S2.p3.8.m2.4.4.2.3.cmml"><mtext id="S2.p3.8.m2.3.3.1.1" xref="S2.p3.8.m2.3.3.1.1a.cmml">true</mtext><mo id="S2.p3.8.m2.4.4.2.4.1" xref="S2.p3.8.m2.4.4.2.3.cmml">,</mo><mi id="S2.p3.8.m2.4.4.2.2" xref="S2.p3.8.m2.4.4.2.2.cmml">d</mi></mrow></msub><mo id="S2.p3.8.m2.7.7.2.2.6" stretchy="false" xref="S2.p3.8.m2.7.7.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.8.m2.7b"><apply id="S2.p3.8.m2.7.7.cmml" xref="S2.p3.8.m2.7.7"><eq id="S2.p3.8.m2.7.7.3.cmml" xref="S2.p3.8.m2.7.7.3"></eq><apply id="S2.p3.8.m2.7.7.4.cmml" xref="S2.p3.8.m2.7.7.4"><csymbol cd="ambiguous" id="S2.p3.8.m2.7.7.4.1.cmml" xref="S2.p3.8.m2.7.7.4">subscript</csymbol><apply id="S2.p3.8.m2.7.7.4.2.cmml" xref="S2.p3.8.m2.7.7.4.2"><ci id="S2.p3.8.m2.7.7.4.2.1.cmml" xref="S2.p3.8.m2.7.7.4.2.1">~</ci><ci id="S2.p3.8.m2.7.7.4.2.2.cmml" xref="S2.p3.8.m2.7.7.4.2.2">𝛽</ci></apply><ci id="S2.p3.8.m2.7.7.4.3a.cmml" xref="S2.p3.8.m2.7.7.4.3"><mtext id="S2.p3.8.m2.7.7.4.3.cmml" mathsize="70%" xref="S2.p3.8.m2.7.7.4.3">true</mtext></ci></apply><vector id="S2.p3.8.m2.7.7.2.3.cmml" xref="S2.p3.8.m2.7.7.2.2"><apply id="S2.p3.8.m2.6.6.1.1.1.cmml" xref="S2.p3.8.m2.6.6.1.1.1"><csymbol cd="ambiguous" id="S2.p3.8.m2.6.6.1.1.1.1.cmml" xref="S2.p3.8.m2.6.6.1.1.1">subscript</csymbol><ci id="S2.p3.8.m2.6.6.1.1.1.2.cmml" xref="S2.p3.8.m2.6.6.1.1.1.2">𝛽</ci><list id="S2.p3.8.m2.2.2.2.3.cmml" xref="S2.p3.8.m2.2.2.2.4"><ci id="S2.p3.8.m2.1.1.1.1a.cmml" xref="S2.p3.8.m2.1.1.1.1"><mtext id="S2.p3.8.m2.1.1.1.1.cmml" mathsize="70%" xref="S2.p3.8.m2.1.1.1.1">true</mtext></ci><cn id="S2.p3.8.m2.2.2.2.2.cmml" type="integer" xref="S2.p3.8.m2.2.2.2.2">1</cn></list></apply><ci id="S2.p3.8.m2.5.5.cmml" xref="S2.p3.8.m2.5.5">…</ci><apply id="S2.p3.8.m2.7.7.2.2.2.cmml" xref="S2.p3.8.m2.7.7.2.2.2"><csymbol cd="ambiguous" id="S2.p3.8.m2.7.7.2.2.2.1.cmml" xref="S2.p3.8.m2.7.7.2.2.2">subscript</csymbol><ci id="S2.p3.8.m2.7.7.2.2.2.2.cmml" xref="S2.p3.8.m2.7.7.2.2.2.2">𝛽</ci><list id="S2.p3.8.m2.4.4.2.3.cmml" xref="S2.p3.8.m2.4.4.2.4"><ci id="S2.p3.8.m2.3.3.1.1a.cmml" xref="S2.p3.8.m2.3.3.1.1"><mtext id="S2.p3.8.m2.3.3.1.1.cmml" mathsize="70%" xref="S2.p3.8.m2.3.3.1.1">true</mtext></ci><ci id="S2.p3.8.m2.4.4.2.2.cmml" xref="S2.p3.8.m2.4.4.2.2">𝑑</ci></list></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.8.m2.7c">\tilde{\beta}_{\text{true}}=(\beta_{\text{true},1},\dots,\beta_{\text{true},d})</annotation><annotation encoding="application/x-llamapun" id="S2.p3.8.m2.7d">over~ start_ARG italic_β end_ARG start_POSTSUBSCRIPT true end_POSTSUBSCRIPT = ( italic_β start_POSTSUBSCRIPT true , 1 end_POSTSUBSCRIPT , … , italic_β start_POSTSUBSCRIPT true , italic_d end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.1">To achieve point identification of <math alttext="\beta_{\text{true}}" class="ltx_Math" display="inline" id="S2.p4.1.m1.1"><semantics id="S2.p4.1.m1.1a"><msub id="S2.p4.1.m1.1.1" xref="S2.p4.1.m1.1.1.cmml"><mi id="S2.p4.1.m1.1.1.2" xref="S2.p4.1.m1.1.1.2.cmml">β</mi><mtext id="S2.p4.1.m1.1.1.3" xref="S2.p4.1.m1.1.1.3a.cmml">true</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.p4.1.m1.1b"><apply id="S2.p4.1.m1.1.1.cmml" xref="S2.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p4.1.m1.1.1.1.cmml" xref="S2.p4.1.m1.1.1">subscript</csymbol><ci id="S2.p4.1.m1.1.1.2.cmml" xref="S2.p4.1.m1.1.1.2">𝛽</ci><ci id="S2.p4.1.m1.1.1.3a.cmml" xref="S2.p4.1.m1.1.1.3"><mtext id="S2.p4.1.m1.1.1.3.cmml" mathsize="70%" xref="S2.p4.1.m1.1.1.3">true</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.1.m1.1c">\beta_{\text{true}}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.1.m1.1d">italic_β start_POSTSUBSCRIPT true end_POSTSUBSCRIPT</annotation></semantics></math>, additional assumptions on the data generating process will have to be made. For example, <cite class="ltx_cite ltx_citemacro_cite">Delgado et al., (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib10" title="">2022</a>)</cite> impose the independence assumption. As already mentioned in the introduction, such identifying assumptions are stringent. We will therefore avoid them and allow model (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E1" title="In 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">1</span></a>) to be only partially identified instead of aiming for point identification.</p> </div> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1 </span>A preliminary note on partial identification</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.7">We start by introducing some concepts regarding model identification. For comprehensive overview, we refer to <cite class="ltx_cite ltx_citemacro_cite">Lewbel, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib23" title="">2019</a>)</cite>. We will say that a conditional distribution <math alttext="F_{T|X}" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.1"><semantics id="S2.SS1.p1.1.m1.1a"><msub id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml"><mi id="S2.SS1.p1.1.m1.1.1.2" xref="S2.SS1.p1.1.m1.1.1.2.cmml">F</mi><mrow id="S2.SS1.p1.1.m1.1.1.3" xref="S2.SS1.p1.1.m1.1.1.3.cmml"><mi id="S2.SS1.p1.1.m1.1.1.3.2" xref="S2.SS1.p1.1.m1.1.1.3.2.cmml">T</mi><mo fence="false" id="S2.SS1.p1.1.m1.1.1.3.1" xref="S2.SS1.p1.1.m1.1.1.3.1.cmml">|</mo><mi id="S2.SS1.p1.1.m1.1.1.3.3" xref="S2.SS1.p1.1.m1.1.1.3.3.cmml">X</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.1b"><apply id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.1.m1.1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1">subscript</csymbol><ci id="S2.SS1.p1.1.m1.1.1.2.cmml" xref="S2.SS1.p1.1.m1.1.1.2">𝐹</ci><apply id="S2.SS1.p1.1.m1.1.1.3.cmml" xref="S2.SS1.p1.1.m1.1.1.3"><csymbol cd="latexml" id="S2.SS1.p1.1.m1.1.1.3.1.cmml" xref="S2.SS1.p1.1.m1.1.1.3.1">conditional</csymbol><ci id="S2.SS1.p1.1.m1.1.1.3.2.cmml" xref="S2.SS1.p1.1.m1.1.1.3.2">𝑇</ci><ci id="S2.SS1.p1.1.m1.1.1.3.3.cmml" xref="S2.SS1.p1.1.m1.1.1.3.3">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.1c">F_{T|X}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.1d">italic_F start_POSTSUBSCRIPT italic_T | italic_X end_POSTSUBSCRIPT</annotation></semantics></math> implied by model (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E1" title="In 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">1</span></a>) is <em class="ltx_emph ltx_font_italic" id="S2.SS1.p1.7.1">consistent</em> with the observed data distribution <math alttext="F_{Y,\Delta,X}" class="ltx_Math" display="inline" id="S2.SS1.p1.2.m2.3"><semantics 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end_POSTSUBSCRIPT</annotation></semantics></math> and copula <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m4.1"><semantics id="S2.SS1.p1.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.4.m4.1.1" xref="S2.SS1.p1.4.m4.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m4.1b"><ci id="S2.SS1.p1.4.m4.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m4.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m4.1d">caligraphic_C</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">(Sklar,, <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib30" title="">1959</a>)</cite> such that <math alttext="(\mathcal{C}(F_{T|X},F_{C|X}),F_{X})" class="ltx_Math" display="inline" id="S2.SS1.p1.5.m5.2"><semantics id="S2.SS1.p1.5.m5.2a"><mrow id="S2.SS1.p1.5.m5.2.2.2" xref="S2.SS1.p1.5.m5.2.2.3.cmml"><mo 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start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT )</annotation></semantics></math> together imply the observed distribution of <math alttext="(Y,\Delta,X)" class="ltx_Math" display="inline" id="S2.SS1.p1.6.m6.3"><semantics id="S2.SS1.p1.6.m6.3a"><mrow id="S2.SS1.p1.6.m6.3.4.2" xref="S2.SS1.p1.6.m6.3.4.1.cmml"><mo id="S2.SS1.p1.6.m6.3.4.2.1" stretchy="false" xref="S2.SS1.p1.6.m6.3.4.1.cmml">(</mo><mi id="S2.SS1.p1.6.m6.1.1" xref="S2.SS1.p1.6.m6.1.1.cmml">Y</mi><mo id="S2.SS1.p1.6.m6.3.4.2.2" xref="S2.SS1.p1.6.m6.3.4.1.cmml">,</mo><mi id="S2.SS1.p1.6.m6.2.2" mathvariant="normal" xref="S2.SS1.p1.6.m6.2.2.cmml">Δ</mi><mo id="S2.SS1.p1.6.m6.3.4.2.3" xref="S2.SS1.p1.6.m6.3.4.1.cmml">,</mo><mi id="S2.SS1.p1.6.m6.3.3" xref="S2.SS1.p1.6.m6.3.3.cmml">X</mi><mo id="S2.SS1.p1.6.m6.3.4.2.4" stretchy="false" xref="S2.SS1.p1.6.m6.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.6.m6.3b"><vector id="S2.SS1.p1.6.m6.3.4.1.cmml" xref="S2.SS1.p1.6.m6.3.4.2"><ci 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The set of all distributions <math alttext="F_{T|X}" class="ltx_Math" display="inline" id="S2.SS1.p1.7.m7.1"><semantics id="S2.SS1.p1.7.m7.1a"><msub id="S2.SS1.p1.7.m7.1.1" xref="S2.SS1.p1.7.m7.1.1.cmml"><mi id="S2.SS1.p1.7.m7.1.1.2" xref="S2.SS1.p1.7.m7.1.1.2.cmml">F</mi><mrow id="S2.SS1.p1.7.m7.1.1.3" xref="S2.SS1.p1.7.m7.1.1.3.cmml"><mi id="S2.SS1.p1.7.m7.1.1.3.2" xref="S2.SS1.p1.7.m7.1.1.3.2.cmml">T</mi><mo fence="false" id="S2.SS1.p1.7.m7.1.1.3.1" xref="S2.SS1.p1.7.m7.1.1.3.1.cmml">|</mo><mi id="S2.SS1.p1.7.m7.1.1.3.3" xref="S2.SS1.p1.7.m7.1.1.3.3.cmml">X</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.7.m7.1b"><apply id="S2.SS1.p1.7.m7.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.7.m7.1.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1">subscript</csymbol><ci id="S2.SS1.p1.7.m7.1.1.2.cmml" xref="S2.SS1.p1.7.m7.1.1.2">𝐹</ci><apply id="S2.SS1.p1.7.m7.1.1.3.cmml" xref="S2.SS1.p1.7.m7.1.1.3"><csymbol cd="latexml" id="S2.SS1.p1.7.m7.1.1.3.1.cmml" xref="S2.SS1.p1.7.m7.1.1.3.1">conditional</csymbol><ci id="S2.SS1.p1.7.m7.1.1.3.2.cmml" xref="S2.SS1.p1.7.m7.1.1.3.2">𝑇</ci><ci id="S2.SS1.p1.7.m7.1.1.3.3.cmml" xref="S2.SS1.p1.7.m7.1.1.3.3">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.7.m7.1c">F_{T|X}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.7.m7.1d">italic_F start_POSTSUBSCRIPT italic_T | italic_X end_POSTSUBSCRIPT</annotation></semantics></math> that are consistent with the observed data is referred to as the <em class="ltx_emph ltx_font_italic" id="S2.SS1.p1.7.2">identified set of (conditional) distributions</em>. Each pair of elements in this set is called <em class="ltx_emph ltx_font_italic" id="S2.SS1.p1.7.3">observationally equivalent</em>, since one cannot further distinguish these elements in terms of validity solely based on the observed data.</p> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.3">By model (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E1" title="In 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">1</span></a>) we can express the identified set of conditional distributions in terms of the parameter vector. That is, the identified set of interest becomes the set of candidate parameter vectors <math alttext="\beta" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.1"><semantics id="S2.SS1.p2.1.m1.1a"><mi id="S2.SS1.p2.1.m1.1.1" xref="S2.SS1.p2.1.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.1.m1.1b"><ci id="S2.SS1.p2.1.m1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.1d">italic_β</annotation></semantics></math> for <math alttext="\beta_{\text{true}}" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.1"><semantics id="S2.SS1.p2.2.m2.1a"><msub id="S2.SS1.p2.2.m2.1.1" xref="S2.SS1.p2.2.m2.1.1.cmml"><mi id="S2.SS1.p2.2.m2.1.1.2" xref="S2.SS1.p2.2.m2.1.1.2.cmml">β</mi><mtext id="S2.SS1.p2.2.m2.1.1.3" xref="S2.SS1.p2.2.m2.1.1.3a.cmml">true</mtext></msub><annotation-xml encoding="MathML-Content" 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id="S2.SS2.p1.1.m1.1.1.3.1.cmml" xref="S2.SS2.p1.1.m1.1.1.3">superscript</csymbol><ci id="S2.SS2.p1.1.m1.1.1.3.2.cmml" xref="S2.SS2.p1.1.m1.1.1.3.2">ℝ</ci><apply id="S2.SS2.p1.1.m1.1.1.3.3.cmml" xref="S2.SS2.p1.1.m1.1.1.3.3"><plus id="S2.SS2.p1.1.m1.1.1.3.3.1.cmml" xref="S2.SS2.p1.1.m1.1.1.3.3.1"></plus><ci id="S2.SS2.p1.1.m1.1.1.3.3.2.cmml" xref="S2.SS2.p1.1.m1.1.1.3.3.2">𝑑</ci><cn id="S2.SS2.p1.1.m1.1.1.3.3.3.cmml" type="integer" xref="S2.SS2.p1.1.m1.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.1.m1.1c">\mathcal{B}\subset\mathbb{R}^{d+1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.1.m1.1d">caligraphic_B ⊂ blackboard_R start_POSTSUPERSCRIPT italic_d + 1 end_POSTSUPERSCRIPT</annotation></semantics></math> denote the parameter space. Since our model is not identified, <math alttext="\beta_{\text{true}}" class="ltx_Math" display="inline" id="S2.SS2.p1.2.m2.1"><semantics id="S2.SS2.p1.2.m2.1a"><msub id="S2.SS2.p1.2.m2.1.1" xref="S2.SS2.p1.2.m2.1.1.cmml"><mi id="S2.SS2.p1.2.m2.1.1.2" xref="S2.SS2.p1.2.m2.1.1.2.cmml">β</mi><mtext id="S2.SS2.p1.2.m2.1.1.3" xref="S2.SS2.p1.2.m2.1.1.3a.cmml">true</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.2.m2.1b"><apply id="S2.SS2.p1.2.m2.1.1.cmml" xref="S2.SS2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.2.m2.1.1.1.cmml" xref="S2.SS2.p1.2.m2.1.1">subscript</csymbol><ci id="S2.SS2.p1.2.m2.1.1.2.cmml" xref="S2.SS2.p1.2.m2.1.1.2">𝛽</ci><ci id="S2.SS2.p1.2.m2.1.1.3a.cmml" xref="S2.SS2.p1.2.m2.1.1.3"><mtext id="S2.SS2.p1.2.m2.1.1.3.cmml" mathsize="70%" xref="S2.SS2.p1.2.m2.1.1.3">true</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.2.m2.1c">\beta_{\text{true}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.2.m2.1d">italic_β start_POSTSUBSCRIPT true end_POSTSUBSCRIPT</annotation></semantics></math> might not be the only parameter for which the model is consistent with the observed data generating process. That is, a different parameter vector <math alttext="\beta^{*}" class="ltx_Math" display="inline" id="S2.SS2.p1.3.m3.1"><semantics id="S2.SS2.p1.3.m3.1a"><msup id="S2.SS2.p1.3.m3.1.1" xref="S2.SS2.p1.3.m3.1.1.cmml"><mi id="S2.SS2.p1.3.m3.1.1.2" xref="S2.SS2.p1.3.m3.1.1.2.cmml">β</mi><mo id="S2.SS2.p1.3.m3.1.1.3" xref="S2.SS2.p1.3.m3.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.3.m3.1b"><apply id="S2.SS2.p1.3.m3.1.1.cmml" xref="S2.SS2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.3.m3.1.1.1.cmml" xref="S2.SS2.p1.3.m3.1.1">superscript</csymbol><ci id="S2.SS2.p1.3.m3.1.1.2.cmml" xref="S2.SS2.p1.3.m3.1.1.2">𝛽</ci><times id="S2.SS2.p1.3.m3.1.1.3.cmml" xref="S2.SS2.p1.3.m3.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.3.m3.1c">\beta^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.3.m3.1d">italic_β start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> might correspond to a distribution of <math alttext="T|X" class="ltx_Math" display="inline" id="S2.SS2.p1.4.m4.1"><semantics id="S2.SS2.p1.4.m4.1a"><mrow id="S2.SS2.p1.4.m4.1.1" xref="S2.SS2.p1.4.m4.1.1.cmml"><mi id="S2.SS2.p1.4.m4.1.1.2" xref="S2.SS2.p1.4.m4.1.1.2.cmml">T</mi><mo fence="false" id="S2.SS2.p1.4.m4.1.1.1" xref="S2.SS2.p1.4.m4.1.1.1.cmml">|</mo><mi id="S2.SS2.p1.4.m4.1.1.3" xref="S2.SS2.p1.4.m4.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.4.m4.1b"><apply id="S2.SS2.p1.4.m4.1.1.cmml" xref="S2.SS2.p1.4.m4.1.1"><csymbol cd="latexml" id="S2.SS2.p1.4.m4.1.1.1.cmml" xref="S2.SS2.p1.4.m4.1.1.1">conditional</csymbol><ci id="S2.SS2.p1.4.m4.1.1.2.cmml" xref="S2.SS2.p1.4.m4.1.1.2">𝑇</ci><ci id="S2.SS2.p1.4.m4.1.1.3.cmml" xref="S2.SS2.p1.4.m4.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.4.m4.1c">T|X</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.4.m4.1d">italic_T | italic_X</annotation></semantics></math> which is also consistent with the observed data. Likewise, we call such values <math alttext="\beta^{*}" class="ltx_Math" display="inline" id="S2.SS2.p1.5.m5.1"><semantics id="S2.SS2.p1.5.m5.1a"><msup id="S2.SS2.p1.5.m5.1.1" xref="S2.SS2.p1.5.m5.1.1.cmml"><mi id="S2.SS2.p1.5.m5.1.1.2" xref="S2.SS2.p1.5.m5.1.1.2.cmml">β</mi><mo id="S2.SS2.p1.5.m5.1.1.3" xref="S2.SS2.p1.5.m5.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.5.m5.1b"><apply id="S2.SS2.p1.5.m5.1.1.cmml" xref="S2.SS2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.5.m5.1.1.1.cmml" xref="S2.SS2.p1.5.m5.1.1">superscript</csymbol><ci id="S2.SS2.p1.5.m5.1.1.2.cmml" xref="S2.SS2.p1.5.m5.1.1.2">𝛽</ci><times id="S2.SS2.p1.5.m5.1.1.3.cmml" xref="S2.SS2.p1.5.m5.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.5.m5.1c">\beta^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.5.m5.1d">italic_β start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> observationally equivalent to <math alttext="\beta_{\text{true}}" class="ltx_Math" display="inline" id="S2.SS2.p1.6.m6.1"><semantics id="S2.SS2.p1.6.m6.1a"><msub id="S2.SS2.p1.6.m6.1.1" xref="S2.SS2.p1.6.m6.1.1.cmml"><mi id="S2.SS2.p1.6.m6.1.1.2" xref="S2.SS2.p1.6.m6.1.1.2.cmml">β</mi><mtext id="S2.SS2.p1.6.m6.1.1.3" xref="S2.SS2.p1.6.m6.1.1.3a.cmml">true</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.6.m6.1b"><apply id="S2.SS2.p1.6.m6.1.1.cmml" xref="S2.SS2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.6.m6.1.1.1.cmml" xref="S2.SS2.p1.6.m6.1.1">subscript</csymbol><ci id="S2.SS2.p1.6.m6.1.1.2.cmml" xref="S2.SS2.p1.6.m6.1.1.2">𝛽</ci><ci id="S2.SS2.p1.6.m6.1.1.3a.cmml" xref="S2.SS2.p1.6.m6.1.1.3"><mtext id="S2.SS2.p1.6.m6.1.1.3.cmml" mathsize="70%" xref="S2.SS2.p1.6.m6.1.1.3">true</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.6.m6.1c">\beta_{\text{true}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.6.m6.1d">italic_β start_POSTSUBSCRIPT true end_POSTSUBSCRIPT</annotation></semantics></math>. Consequently, there is no way to distinguish <math alttext="\beta^{*}" class="ltx_Math" display="inline" id="S2.SS2.p1.7.m7.1"><semantics id="S2.SS2.p1.7.m7.1a"><msup id="S2.SS2.p1.7.m7.1.1" xref="S2.SS2.p1.7.m7.1.1.cmml"><mi id="S2.SS2.p1.7.m7.1.1.2" xref="S2.SS2.p1.7.m7.1.1.2.cmml">β</mi><mo id="S2.SS2.p1.7.m7.1.1.3" xref="S2.SS2.p1.7.m7.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.7.m7.1b"><apply id="S2.SS2.p1.7.m7.1.1.cmml" xref="S2.SS2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.7.m7.1.1.1.cmml" xref="S2.SS2.p1.7.m7.1.1">superscript</csymbol><ci id="S2.SS2.p1.7.m7.1.1.2.cmml" xref="S2.SS2.p1.7.m7.1.1.2">𝛽</ci><times id="S2.SS2.p1.7.m7.1.1.3.cmml" xref="S2.SS2.p1.7.m7.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.7.m7.1c">\beta^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.7.m7.1d">italic_β start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> from <math alttext="\beta_{\text{true}}" class="ltx_Math" display="inline" id="S2.SS2.p1.8.m8.1"><semantics id="S2.SS2.p1.8.m8.1a"><msub id="S2.SS2.p1.8.m8.1.1" xref="S2.SS2.p1.8.m8.1.1.cmml"><mi id="S2.SS2.p1.8.m8.1.1.2" xref="S2.SS2.p1.8.m8.1.1.2.cmml">β</mi><mtext id="S2.SS2.p1.8.m8.1.1.3" xref="S2.SS2.p1.8.m8.1.1.3a.cmml">true</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.8.m8.1b"><apply id="S2.SS2.p1.8.m8.1.1.cmml" xref="S2.SS2.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.8.m8.1.1.1.cmml" xref="S2.SS2.p1.8.m8.1.1">subscript</csymbol><ci id="S2.SS2.p1.8.m8.1.1.2.cmml" xref="S2.SS2.p1.8.m8.1.1.2">𝛽</ci><ci id="S2.SS2.p1.8.m8.1.1.3a.cmml" xref="S2.SS2.p1.8.m8.1.1.3"><mtext id="S2.SS2.p1.8.m8.1.1.3.cmml" mathsize="70%" xref="S2.SS2.p1.8.m8.1.1.3">true</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.8.m8.1c">\beta_{\text{true}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.8.m8.1d">italic_β start_POSTSUBSCRIPT true end_POSTSUBSCRIPT</annotation></semantics></math> in terms of validity and hence the best we can aim for in this context is to find the set <math alttext="\mathcal{B}_{I}^{*}" class="ltx_Math" display="inline" id="S2.SS2.p1.9.m9.1"><semantics id="S2.SS2.p1.9.m9.1a"><msubsup id="S2.SS2.p1.9.m9.1.1" xref="S2.SS2.p1.9.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p1.9.m9.1.1.2.2" xref="S2.SS2.p1.9.m9.1.1.2.2.cmml">ℬ</mi><mi id="S2.SS2.p1.9.m9.1.1.2.3" xref="S2.SS2.p1.9.m9.1.1.2.3.cmml">I</mi><mo id="S2.SS2.p1.9.m9.1.1.3" xref="S2.SS2.p1.9.m9.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.9.m9.1b"><apply id="S2.SS2.p1.9.m9.1.1.cmml" xref="S2.SS2.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.9.m9.1.1.1.cmml" xref="S2.SS2.p1.9.m9.1.1">superscript</csymbol><apply id="S2.SS2.p1.9.m9.1.1.2.cmml" xref="S2.SS2.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.9.m9.1.1.2.1.cmml" xref="S2.SS2.p1.9.m9.1.1">subscript</csymbol><ci id="S2.SS2.p1.9.m9.1.1.2.2.cmml" xref="S2.SS2.p1.9.m9.1.1.2.2">ℬ</ci><ci id="S2.SS2.p1.9.m9.1.1.2.3.cmml" xref="S2.SS2.p1.9.m9.1.1.2.3">𝐼</ci></apply><times id="S2.SS2.p1.9.m9.1.1.3.cmml" xref="S2.SS2.p1.9.m9.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.9.m9.1c">\mathcal{B}_{I}^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.9.m9.1d">caligraphic_B start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> of all parameter vectors <math alttext="\beta" class="ltx_Math" display="inline" id="S2.SS2.p1.10.m10.1"><semantics id="S2.SS2.p1.10.m10.1a"><mi id="S2.SS2.p1.10.m10.1.1" xref="S2.SS2.p1.10.m10.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.10.m10.1b"><ci id="S2.SS2.p1.10.m10.1.1.cmml" xref="S2.SS2.p1.10.m10.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.10.m10.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.10.m10.1d">italic_β</annotation></semantics></math> that are observationally equivalent to <math alttext="\beta_{\text{true}}" class="ltx_Math" display="inline" id="S2.SS2.p1.11.m11.1"><semantics id="S2.SS2.p1.11.m11.1a"><msub id="S2.SS2.p1.11.m11.1.1" xref="S2.SS2.p1.11.m11.1.1.cmml"><mi id="S2.SS2.p1.11.m11.1.1.2" xref="S2.SS2.p1.11.m11.1.1.2.cmml">β</mi><mtext id="S2.SS2.p1.11.m11.1.1.3" xref="S2.SS2.p1.11.m11.1.1.3a.cmml">true</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.11.m11.1b"><apply id="S2.SS2.p1.11.m11.1.1.cmml" xref="S2.SS2.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.11.m11.1.1.1.cmml" xref="S2.SS2.p1.11.m11.1.1">subscript</csymbol><ci id="S2.SS2.p1.11.m11.1.1.2.cmml" xref="S2.SS2.p1.11.m11.1.1.2">𝛽</ci><ci id="S2.SS2.p1.11.m11.1.1.3a.cmml" xref="S2.SS2.p1.11.m11.1.1.3"><mtext id="S2.SS2.p1.11.m11.1.1.3.cmml" mathsize="70%" xref="S2.SS2.p1.11.m11.1.1.3">true</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.11.m11.1c">\beta_{\text{true}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.11.m11.1d">italic_β start_POSTSUBSCRIPT true end_POSTSUBSCRIPT</annotation></semantics></math>. This set, which we refer to as the <em class="ltx_emph ltx_font_italic" id="S2.SS2.p1.11.1">identified set of parameters</em>, or simply <em class="ltx_emph ltx_font_italic" id="S2.SS2.p1.11.2">identified set</em>, will be our main object of interest.</p> </div> <div class="ltx_para" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.1">To study and eventually estimate this identified set it is useful to first ask the question how we can mathematically formalize the concept of parameters being observationally equivalent to <math alttext="\beta_{\text{true}}" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.1"><semantics id="S2.SS2.p2.1.m1.1a"><msub id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml"><mi id="S2.SS2.p2.1.m1.1.1.2" xref="S2.SS2.p2.1.m1.1.1.2.cmml">β</mi><mtext id="S2.SS2.p2.1.m1.1.1.3" xref="S2.SS2.p2.1.m1.1.1.3a.cmml">true</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.1b"><apply id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.1.m1.1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1">subscript</csymbol><ci id="S2.SS2.p2.1.m1.1.1.2.cmml" xref="S2.SS2.p2.1.m1.1.1.2">𝛽</ci><ci id="S2.SS2.p2.1.m1.1.1.3a.cmml" xref="S2.SS2.p2.1.m1.1.1.3"><mtext id="S2.SS2.p2.1.m1.1.1.3.cmml" mathsize="70%" xref="S2.SS2.p2.1.m1.1.1.3">true</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">\beta_{\text{true}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.1d">italic_β start_POSTSUBSCRIPT true end_POSTSUBSCRIPT</annotation></semantics></math>. The answer is provided by <cite class="ltx_cite ltx_citemacro_cite">Peterson, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib26" title="">1976</a>)</cite>, who derives the bounds</p> <table class="ltx_equation ltx_eqn_table" id="S2.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathbb{P}(Y\leq t,\Delta=1|X=x)\leq\Lambda(x^{\top}\beta)\leq\mathbb{P}(Y\leq t% |X=x)," class="ltx_Math" display="block" id="S2.E2.m1.1"><semantics id="S2.E2.m1.1a"><mrow id="S2.E2.m1.1.1.1" xref="S2.E2.m1.1.1.1.1.cmml"><mrow id="S2.E2.m1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.cmml"><mrow id="S2.E2.m1.1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.1.cmml"><mi id="S2.E2.m1.1.1.1.1.1.3" xref="S2.E2.m1.1.1.1.1.1.3.cmml">ℙ</mi><mo id="S2.E2.m1.1.1.1.1.1.2" xref="S2.E2.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E2.m1.1.1.1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.1.cmml"><mo 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id="S2.E2.m1.1.1.1.1.3.1.1.1.6.cmml" xref="S2.E2.m1.1.1.1.1.3.1.1.1.6">𝑥</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.1c">\mathbb{P}(Y\leq t,\Delta=1|X=x)\leq\Lambda(x^{\top}\beta)\leq\mathbb{P}(Y\leq t% |X=x),</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.1d">blackboard_P ( italic_Y ≤ italic_t , roman_Δ = 1 | italic_X = italic_x ) ≤ roman_Λ ( italic_x start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT italic_β ) ≤ blackboard_P ( italic_Y ≤ italic_t | italic_X = italic_x ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p2.5">based on which we define</p> <table class="ltx_equation ltx_eqn_table" id="S2.E3"> <tbody><tr class="ltx_equation ltx_eqn_row 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models"><span class="ltx_text ltx_ref_tag">2</span></a></mtext><mtext id="S2.E3.m1.1.1.1.1.2.2.2.3.3e.cmml" xref="S2.E3.m1.1.1.1.1.2.2.2.3.3">)</mtext></mrow></ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.1c">\mathcal{B}^{*}_{I}=\{\beta\in\mathcal{B}\mid\forall x\in\mathcal{X}:\beta% \text{ satisfies \eqref{eq: Peterson bounds}}\}.</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.1d">caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT = { italic_β ∈ caligraphic_B ∣ ∀ italic_x ∈ caligraphic_X : italic_β satisfies ( ) } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p2.4">Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E3" title="In 2.2 Methodology ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">3</span></a>) precisely characterizes the parameters that are observationally equivalent to <math alttext="\beta_{\text{true}}" class="ltx_Math" display="inline" id="S2.SS2.p2.2.m1.1"><semantics id="S2.SS2.p2.2.m1.1a"><msub id="S2.SS2.p2.2.m1.1.1" xref="S2.SS2.p2.2.m1.1.1.cmml"><mi id="S2.SS2.p2.2.m1.1.1.2" xref="S2.SS2.p2.2.m1.1.1.2.cmml">β</mi><mtext id="S2.SS2.p2.2.m1.1.1.3" xref="S2.SS2.p2.2.m1.1.1.3a.cmml">true</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.2.m1.1b"><apply id="S2.SS2.p2.2.m1.1.1.cmml" xref="S2.SS2.p2.2.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.2.m1.1.1.1.cmml" xref="S2.SS2.p2.2.m1.1.1">subscript</csymbol><ci id="S2.SS2.p2.2.m1.1.1.2.cmml" xref="S2.SS2.p2.2.m1.1.1.2">𝛽</ci><ci id="S2.SS2.p2.2.m1.1.1.3a.cmml" xref="S2.SS2.p2.2.m1.1.1.3"><mtext id="S2.SS2.p2.2.m1.1.1.3.cmml" mathsize="70%" xref="S2.SS2.p2.2.m1.1.1.3">true</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.2.m1.1c">\beta_{\text{true}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.2.m1.1d">italic_β start_POSTSUBSCRIPT true end_POSTSUBSCRIPT</annotation></semantics></math>: all parameter vectors in <math alttext="\mathcal{B}_{I}^{*}" class="ltx_Math" display="inline" id="S2.SS2.p2.3.m2.1"><semantics id="S2.SS2.p2.3.m2.1a"><msubsup id="S2.SS2.p2.3.m2.1.1" xref="S2.SS2.p2.3.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.3.m2.1.1.2.2" xref="S2.SS2.p2.3.m2.1.1.2.2.cmml">ℬ</mi><mi id="S2.SS2.p2.3.m2.1.1.2.3" xref="S2.SS2.p2.3.m2.1.1.2.3.cmml">I</mi><mo id="S2.SS2.p2.3.m2.1.1.3" xref="S2.SS2.p2.3.m2.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.3.m2.1b"><apply id="S2.SS2.p2.3.m2.1.1.cmml" xref="S2.SS2.p2.3.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.3.m2.1.1.1.cmml" xref="S2.SS2.p2.3.m2.1.1">superscript</csymbol><apply id="S2.SS2.p2.3.m2.1.1.2.cmml" xref="S2.SS2.p2.3.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.3.m2.1.1.2.1.cmml" xref="S2.SS2.p2.3.m2.1.1">subscript</csymbol><ci id="S2.SS2.p2.3.m2.1.1.2.2.cmml" xref="S2.SS2.p2.3.m2.1.1.2.2">ℬ</ci><ci id="S2.SS2.p2.3.m2.1.1.2.3.cmml" xref="S2.SS2.p2.3.m2.1.1.2.3">𝐼</ci></apply><times id="S2.SS2.p2.3.m2.1.1.3.cmml" xref="S2.SS2.p2.3.m2.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.3.m2.1c">\mathcal{B}_{I}^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.3.m2.1d">caligraphic_B start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> must lead to a model that is consistent with the bounds (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E2" title="In 2.2 Methodology ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">2</span></a>). Moreover, the bounds in (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E2" title="In 2.2 Methodology ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">2</span></a>) are sharp. 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id="S2.SS2.p2.4.m3.1c">\mathbb{P}(T\leq t|X=x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m3.1d">blackboard_P ( italic_T ≤ italic_t | italic_X = italic_x )</annotation></semantics></math> that are based on the observed data alone.</p> </div> <div class="ltx_para" id="S2.SS2.p3"> <p class="ltx_p" id="S2.SS2.p3.4">With the formal characterization of the identified set (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E3" title="In 2.2 Methodology ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">3</span></a>) in hand, the idea behind estimating <math alttext="\mathcal{B}^{*}_{I}" class="ltx_Math" display="inline" id="S2.SS2.p3.1.m1.1"><semantics id="S2.SS2.p3.1.m1.1a"><msubsup id="S2.SS2.p3.1.m1.1.1" xref="S2.SS2.p3.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.1.m1.1.1.2.2" xref="S2.SS2.p3.1.m1.1.1.2.2.cmml">ℬ</mi><mi id="S2.SS2.p3.1.m1.1.1.3" xref="S2.SS2.p3.1.m1.1.1.3.cmml">I</mi><mo id="S2.SS2.p3.1.m1.1.1.2.3" xref="S2.SS2.p3.1.m1.1.1.2.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.1.m1.1b"><apply id="S2.SS2.p3.1.m1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.1.m1.1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1">subscript</csymbol><apply id="S2.SS2.p3.1.m1.1.1.2.cmml" xref="S2.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.1.m1.1.1.2.1.cmml" xref="S2.SS2.p3.1.m1.1.1">superscript</csymbol><ci id="S2.SS2.p3.1.m1.1.1.2.2.cmml" xref="S2.SS2.p3.1.m1.1.1.2.2">ℬ</ci><times id="S2.SS2.p3.1.m1.1.1.2.3.cmml" xref="S2.SS2.p3.1.m1.1.1.2.3"></times></apply><ci id="S2.SS2.p3.1.m1.1.1.3.cmml" xref="S2.SS2.p3.1.m1.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.1.m1.1c">\mathcal{B}^{*}_{I}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.1.m1.1d">caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT</annotation></semantics></math> is straightforward: we could consider a test for checking whether the model with a certain parameter vector <math alttext="\beta" class="ltx_Math" display="inline" id="S2.SS2.p3.2.m2.1"><semantics id="S2.SS2.p3.2.m2.1a"><mi id="S2.SS2.p3.2.m2.1.1" xref="S2.SS2.p3.2.m2.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.2.m2.1b"><ci id="S2.SS2.p3.2.m2.1.1.cmml" xref="S2.SS2.p3.2.m2.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.2.m2.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.2.m2.1d">italic_β</annotation></semantics></math> satisfies (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E2" title="In 2.2 Methodology ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">2</span></a>) and apply this test over the entire parameter space <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS2.p3.3.m3.1"><semantics id="S2.SS2.p3.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.3.m3.1.1" xref="S2.SS2.p3.3.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.3.m3.1b"><ci id="S2.SS2.p3.3.m3.1.1.cmml" xref="S2.SS2.p3.3.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.3.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.3.m3.1d">caligraphic_B</annotation></semantics></math>, collecting all values that are not rejected. The set of non-rejected values is then an estimator for <math alttext="\mathcal{B}^{*}_{I}" class="ltx_Math" display="inline" id="S2.SS2.p3.4.m4.1"><semantics id="S2.SS2.p3.4.m4.1a"><msubsup id="S2.SS2.p3.4.m4.1.1" xref="S2.SS2.p3.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.4.m4.1.1.2.2" xref="S2.SS2.p3.4.m4.1.1.2.2.cmml">ℬ</mi><mi id="S2.SS2.p3.4.m4.1.1.3" xref="S2.SS2.p3.4.m4.1.1.3.cmml">I</mi><mo id="S2.SS2.p3.4.m4.1.1.2.3" xref="S2.SS2.p3.4.m4.1.1.2.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.4.m4.1b"><apply id="S2.SS2.p3.4.m4.1.1.cmml" xref="S2.SS2.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.4.m4.1.1.1.cmml" xref="S2.SS2.p3.4.m4.1.1">subscript</csymbol><apply id="S2.SS2.p3.4.m4.1.1.2.cmml" xref="S2.SS2.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.4.m4.1.1.2.1.cmml" xref="S2.SS2.p3.4.m4.1.1">superscript</csymbol><ci id="S2.SS2.p3.4.m4.1.1.2.2.cmml" xref="S2.SS2.p3.4.m4.1.1.2.2">ℬ</ci><times id="S2.SS2.p3.4.m4.1.1.2.3.cmml" xref="S2.SS2.p3.4.m4.1.1.2.3"></times></apply><ci id="S2.SS2.p3.4.m4.1.1.3.cmml" xref="S2.SS2.p3.4.m4.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.4.m4.1c">\mathcal{B}^{*}_{I}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.4.m4.1d">caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT</annotation></semantics></math>. In the literature, such an approach is referred to as <em class="ltx_emph ltx_font_italic" id="S2.SS2.p3.4.1">test inversion</em>.</p> </div> <div class="ltx_para" id="S2.SS2.p4"> <p class="ltx_p" id="S2.SS2.p4.11">Typically, however, researchers are only interested in one or a few elements of the vector <math alttext="\beta_{\text{true}}" class="ltx_Math" display="inline" id="S2.SS2.p4.1.m1.1"><semantics id="S2.SS2.p4.1.m1.1a"><msub id="S2.SS2.p4.1.m1.1.1" xref="S2.SS2.p4.1.m1.1.1.cmml"><mi id="S2.SS2.p4.1.m1.1.1.2" xref="S2.SS2.p4.1.m1.1.1.2.cmml">β</mi><mtext id="S2.SS2.p4.1.m1.1.1.3" xref="S2.SS2.p4.1.m1.1.1.3a.cmml">true</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.1.m1.1b"><apply id="S2.SS2.p4.1.m1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.1.m1.1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1">subscript</csymbol><ci id="S2.SS2.p4.1.m1.1.1.2.cmml" xref="S2.SS2.p4.1.m1.1.1.2">𝛽</ci><ci id="S2.SS2.p4.1.m1.1.1.3a.cmml" xref="S2.SS2.p4.1.m1.1.1.3"><mtext id="S2.SS2.p4.1.m1.1.1.3.cmml" mathsize="70%" xref="S2.SS2.p4.1.m1.1.1.3">true</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.1.m1.1c">\beta_{\text{true}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.1.m1.1d">italic_β start_POSTSUBSCRIPT true end_POSTSUBSCRIPT</annotation></semantics></math>. Suppose that one of the elements of interest has index <math alttext="k" class="ltx_Math" display="inline" id="S2.SS2.p4.2.m2.1"><semantics id="S2.SS2.p4.2.m2.1a"><mi id="S2.SS2.p4.2.m2.1.1" xref="S2.SS2.p4.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.2.m2.1b"><ci id="S2.SS2.p4.2.m2.1.1.cmml" xref="S2.SS2.p4.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.2.m2.1d">italic_k</annotation></semantics></math>, then this means that interest is only in the projection of <math alttext="\mathcal{B}^{*}_{I}" class="ltx_Math" display="inline" id="S2.SS2.p4.3.m3.1"><semantics id="S2.SS2.p4.3.m3.1a"><msubsup id="S2.SS2.p4.3.m3.1.1" xref="S2.SS2.p4.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p4.3.m3.1.1.2.2" xref="S2.SS2.p4.3.m3.1.1.2.2.cmml">ℬ</mi><mi id="S2.SS2.p4.3.m3.1.1.3" xref="S2.SS2.p4.3.m3.1.1.3.cmml">I</mi><mo id="S2.SS2.p4.3.m3.1.1.2.3" xref="S2.SS2.p4.3.m3.1.1.2.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.3.m3.1b"><apply id="S2.SS2.p4.3.m3.1.1.cmml" xref="S2.SS2.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.3.m3.1.1.1.cmml" xref="S2.SS2.p4.3.m3.1.1">subscript</csymbol><apply id="S2.SS2.p4.3.m3.1.1.2.cmml" xref="S2.SS2.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.3.m3.1.1.2.1.cmml" xref="S2.SS2.p4.3.m3.1.1">superscript</csymbol><ci id="S2.SS2.p4.3.m3.1.1.2.2.cmml" xref="S2.SS2.p4.3.m3.1.1.2.2">ℬ</ci><times id="S2.SS2.p4.3.m3.1.1.2.3.cmml" xref="S2.SS2.p4.3.m3.1.1.2.3"></times></apply><ci id="S2.SS2.p4.3.m3.1.1.3.cmml" xref="S2.SS2.p4.3.m3.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.3.m3.1c">\mathcal{B}^{*}_{I}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.3.m3.1d">caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT</annotation></semantics></math> onto the <math alttext="k" class="ltx_Math" display="inline" id="S2.SS2.p4.4.m4.1"><semantics id="S2.SS2.p4.4.m4.1a"><mi id="S2.SS2.p4.4.m4.1.1" xref="S2.SS2.p4.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.4.m4.1b"><ci id="S2.SS2.p4.4.m4.1.1.cmml" xref="S2.SS2.p4.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.4.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.4.m4.1d">italic_k</annotation></semantics></math>-th coordinate axis, denoted as <math alttext="\mathcal{B}^{*}_{I,k}" class="ltx_Math" display="inline" id="S2.SS2.p4.5.m5.2"><semantics id="S2.SS2.p4.5.m5.2a"><msubsup id="S2.SS2.p4.5.m5.2.3" xref="S2.SS2.p4.5.m5.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p4.5.m5.2.3.2.2" xref="S2.SS2.p4.5.m5.2.3.2.2.cmml">ℬ</mi><mrow id="S2.SS2.p4.5.m5.2.2.2.4" xref="S2.SS2.p4.5.m5.2.2.2.3.cmml"><mi id="S2.SS2.p4.5.m5.1.1.1.1" xref="S2.SS2.p4.5.m5.1.1.1.1.cmml">I</mi><mo id="S2.SS2.p4.5.m5.2.2.2.4.1" xref="S2.SS2.p4.5.m5.2.2.2.3.cmml">,</mo><mi id="S2.SS2.p4.5.m5.2.2.2.2" xref="S2.SS2.p4.5.m5.2.2.2.2.cmml">k</mi></mrow><mo id="S2.SS2.p4.5.m5.2.3.2.3" xref="S2.SS2.p4.5.m5.2.3.2.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.5.m5.2b"><apply id="S2.SS2.p4.5.m5.2.3.cmml" xref="S2.SS2.p4.5.m5.2.3"><csymbol cd="ambiguous" id="S2.SS2.p4.5.m5.2.3.1.cmml" xref="S2.SS2.p4.5.m5.2.3">subscript</csymbol><apply id="S2.SS2.p4.5.m5.2.3.2.cmml" xref="S2.SS2.p4.5.m5.2.3"><csymbol cd="ambiguous" id="S2.SS2.p4.5.m5.2.3.2.1.cmml" xref="S2.SS2.p4.5.m5.2.3">superscript</csymbol><ci id="S2.SS2.p4.5.m5.2.3.2.2.cmml" xref="S2.SS2.p4.5.m5.2.3.2.2">ℬ</ci><times id="S2.SS2.p4.5.m5.2.3.2.3.cmml" xref="S2.SS2.p4.5.m5.2.3.2.3"></times></apply><list id="S2.SS2.p4.5.m5.2.2.2.3.cmml" xref="S2.SS2.p4.5.m5.2.2.2.4"><ci id="S2.SS2.p4.5.m5.1.1.1.1.cmml" xref="S2.SS2.p4.5.m5.1.1.1.1">𝐼</ci><ci id="S2.SS2.p4.5.m5.2.2.2.2.cmml" xref="S2.SS2.p4.5.m5.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.5.m5.2c">\mathcal{B}^{*}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.5.m5.2d">caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. In the literature, studying <math alttext="\mathcal{B}^{*}_{I,k}" class="ltx_Math" display="inline" id="S2.SS2.p4.6.m6.2"><semantics id="S2.SS2.p4.6.m6.2a"><msubsup id="S2.SS2.p4.6.m6.2.3" xref="S2.SS2.p4.6.m6.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p4.6.m6.2.3.2.2" xref="S2.SS2.p4.6.m6.2.3.2.2.cmml">ℬ</mi><mrow id="S2.SS2.p4.6.m6.2.2.2.4" xref="S2.SS2.p4.6.m6.2.2.2.3.cmml"><mi id="S2.SS2.p4.6.m6.1.1.1.1" xref="S2.SS2.p4.6.m6.1.1.1.1.cmml">I</mi><mo id="S2.SS2.p4.6.m6.2.2.2.4.1" xref="S2.SS2.p4.6.m6.2.2.2.3.cmml">,</mo><mi id="S2.SS2.p4.6.m6.2.2.2.2" xref="S2.SS2.p4.6.m6.2.2.2.2.cmml">k</mi></mrow><mo id="S2.SS2.p4.6.m6.2.3.2.3" xref="S2.SS2.p4.6.m6.2.3.2.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.6.m6.2b"><apply id="S2.SS2.p4.6.m6.2.3.cmml" xref="S2.SS2.p4.6.m6.2.3"><csymbol cd="ambiguous" id="S2.SS2.p4.6.m6.2.3.1.cmml" xref="S2.SS2.p4.6.m6.2.3">subscript</csymbol><apply id="S2.SS2.p4.6.m6.2.3.2.cmml" xref="S2.SS2.p4.6.m6.2.3"><csymbol cd="ambiguous" id="S2.SS2.p4.6.m6.2.3.2.1.cmml" xref="S2.SS2.p4.6.m6.2.3">superscript</csymbol><ci id="S2.SS2.p4.6.m6.2.3.2.2.cmml" xref="S2.SS2.p4.6.m6.2.3.2.2">ℬ</ci><times id="S2.SS2.p4.6.m6.2.3.2.3.cmml" xref="S2.SS2.p4.6.m6.2.3.2.3"></times></apply><list id="S2.SS2.p4.6.m6.2.2.2.3.cmml" xref="S2.SS2.p4.6.m6.2.2.2.4"><ci id="S2.SS2.p4.6.m6.1.1.1.1.cmml" xref="S2.SS2.p4.6.m6.1.1.1.1">𝐼</ci><ci id="S2.SS2.p4.6.m6.2.2.2.2.cmml" xref="S2.SS2.p4.6.m6.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.6.m6.2c">\mathcal{B}^{*}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.6.m6.2d">caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> directly is often referred to as <em class="ltx_emph ltx_font_italic" id="S2.SS2.p4.11.1">subvector inference</em> (as opposed to <em class="ltx_emph ltx_font_italic" id="S2.SS2.p4.11.2">full vector inference</em>, where one studies <math alttext="\mathcal{B}_{I}^{*}" class="ltx_Math" display="inline" id="S2.SS2.p4.7.m7.1"><semantics id="S2.SS2.p4.7.m7.1a"><msubsup id="S2.SS2.p4.7.m7.1.1" xref="S2.SS2.p4.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p4.7.m7.1.1.2.2" xref="S2.SS2.p4.7.m7.1.1.2.2.cmml">ℬ</mi><mi id="S2.SS2.p4.7.m7.1.1.2.3" xref="S2.SS2.p4.7.m7.1.1.2.3.cmml">I</mi><mo id="S2.SS2.p4.7.m7.1.1.3" xref="S2.SS2.p4.7.m7.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.7.m7.1b"><apply id="S2.SS2.p4.7.m7.1.1.cmml" xref="S2.SS2.p4.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.7.m7.1.1.1.cmml" xref="S2.SS2.p4.7.m7.1.1">superscript</csymbol><apply id="S2.SS2.p4.7.m7.1.1.2.cmml" xref="S2.SS2.p4.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.7.m7.1.1.2.1.cmml" xref="S2.SS2.p4.7.m7.1.1">subscript</csymbol><ci id="S2.SS2.p4.7.m7.1.1.2.2.cmml" xref="S2.SS2.p4.7.m7.1.1.2.2">ℬ</ci><ci id="S2.SS2.p4.7.m7.1.1.2.3.cmml" xref="S2.SS2.p4.7.m7.1.1.2.3">𝐼</ci></apply><times id="S2.SS2.p4.7.m7.1.1.3.cmml" xref="S2.SS2.p4.7.m7.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.7.m7.1c">\mathcal{B}_{I}^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.7.m7.1d">caligraphic_B start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>). In this paper, we will also focus on the subvector problem, and, analogously to what is often done in the full vector case, use test inversion for the estimation. Note that multiple elements of <math alttext="\beta_{\text{true}}" class="ltx_Math" display="inline" id="S2.SS2.p4.8.m8.1"><semantics id="S2.SS2.p4.8.m8.1a"><msub id="S2.SS2.p4.8.m8.1.1" xref="S2.SS2.p4.8.m8.1.1.cmml"><mi id="S2.SS2.p4.8.m8.1.1.2" xref="S2.SS2.p4.8.m8.1.1.2.cmml">β</mi><mtext id="S2.SS2.p4.8.m8.1.1.3" xref="S2.SS2.p4.8.m8.1.1.3a.cmml">true</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.8.m8.1b"><apply id="S2.SS2.p4.8.m8.1.1.cmml" xref="S2.SS2.p4.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.8.m8.1.1.1.cmml" xref="S2.SS2.p4.8.m8.1.1">subscript</csymbol><ci id="S2.SS2.p4.8.m8.1.1.2.cmml" xref="S2.SS2.p4.8.m8.1.1.2">𝛽</ci><ci id="S2.SS2.p4.8.m8.1.1.3a.cmml" xref="S2.SS2.p4.8.m8.1.1.3"><mtext id="S2.SS2.p4.8.m8.1.1.3.cmml" mathsize="70%" xref="S2.SS2.p4.8.m8.1.1.3">true</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.8.m8.1c">\beta_{\text{true}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.8.m8.1d">italic_β start_POSTSUBSCRIPT true end_POSTSUBSCRIPT</annotation></semantics></math> might be of interest simultaneously when a categorical covariate enters the model through several dummy variables. To keep the exposition clear, however, we will restrict to the case where interest is only in the <math alttext="k" class="ltx_Math" display="inline" id="S2.SS2.p4.9.m9.1"><semantics id="S2.SS2.p4.9.m9.1a"><mi id="S2.SS2.p4.9.m9.1.1" xref="S2.SS2.p4.9.m9.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.9.m9.1b"><ci id="S2.SS2.p4.9.m9.1.1.cmml" xref="S2.SS2.p4.9.m9.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.9.m9.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.9.m9.1d">italic_k</annotation></semantics></math>-th element. All results can be generalized by replacing <math alttext="\beta_{k}" class="ltx_Math" display="inline" id="S2.SS2.p4.10.m10.1"><semantics id="S2.SS2.p4.10.m10.1a"><msub id="S2.SS2.p4.10.m10.1.1" xref="S2.SS2.p4.10.m10.1.1.cmml"><mi id="S2.SS2.p4.10.m10.1.1.2" xref="S2.SS2.p4.10.m10.1.1.2.cmml">β</mi><mi id="S2.SS2.p4.10.m10.1.1.3" xref="S2.SS2.p4.10.m10.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.10.m10.1b"><apply id="S2.SS2.p4.10.m10.1.1.cmml" xref="S2.SS2.p4.10.m10.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.10.m10.1.1.1.cmml" xref="S2.SS2.p4.10.m10.1.1">subscript</csymbol><ci id="S2.SS2.p4.10.m10.1.1.2.cmml" xref="S2.SS2.p4.10.m10.1.1.2">𝛽</ci><ci id="S2.SS2.p4.10.m10.1.1.3.cmml" xref="S2.SS2.p4.10.m10.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.10.m10.1c">\beta_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.10.m10.1d">italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> with a general subvector of dimension smaller than <math alttext="d+1" class="ltx_Math" display="inline" id="S2.SS2.p4.11.m11.1"><semantics id="S2.SS2.p4.11.m11.1a"><mrow id="S2.SS2.p4.11.m11.1.1" xref="S2.SS2.p4.11.m11.1.1.cmml"><mi id="S2.SS2.p4.11.m11.1.1.2" xref="S2.SS2.p4.11.m11.1.1.2.cmml">d</mi><mo id="S2.SS2.p4.11.m11.1.1.1" xref="S2.SS2.p4.11.m11.1.1.1.cmml">+</mo><mn id="S2.SS2.p4.11.m11.1.1.3" xref="S2.SS2.p4.11.m11.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.11.m11.1b"><apply id="S2.SS2.p4.11.m11.1.1.cmml" xref="S2.SS2.p4.11.m11.1.1"><plus id="S2.SS2.p4.11.m11.1.1.1.cmml" xref="S2.SS2.p4.11.m11.1.1.1"></plus><ci id="S2.SS2.p4.11.m11.1.1.2.cmml" xref="S2.SS2.p4.11.m11.1.1.2">𝑑</ci><cn id="S2.SS2.p4.11.m11.1.1.3.cmml" type="integer" xref="S2.SS2.p4.11.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.11.m11.1c">d+1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.11.m11.1d">italic_d + 1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS2.p5"> <p class="ltx_p" id="S2.SS2.p5.10">Concretely, we define</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S5.EGx1"> <tbody id="S2.Ex2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{B}^{*}_{I,k}" class="ltx_Math" display="inline" id="S2.Ex2.m1.2"><semantics id="S2.Ex2.m1.2a"><msubsup id="S2.Ex2.m1.2.3" xref="S2.Ex2.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex2.m1.2.3.2.2" xref="S2.Ex2.m1.2.3.2.2.cmml">ℬ</mi><mrow id="S2.Ex2.m1.2.2.2.4" xref="S2.Ex2.m1.2.2.2.3.cmml"><mi id="S2.Ex2.m1.1.1.1.1" xref="S2.Ex2.m1.1.1.1.1.cmml">I</mi><mo id="S2.Ex2.m1.2.2.2.4.1" xref="S2.Ex2.m1.2.2.2.3.cmml">,</mo><mi id="S2.Ex2.m1.2.2.2.2" xref="S2.Ex2.m1.2.2.2.2.cmml">k</mi></mrow><mo id="S2.Ex2.m1.2.3.2.3" xref="S2.Ex2.m1.2.3.2.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.Ex2.m1.2b"><apply id="S2.Ex2.m1.2.3.cmml" xref="S2.Ex2.m1.2.3"><csymbol cd="ambiguous" id="S2.Ex2.m1.2.3.1.cmml" xref="S2.Ex2.m1.2.3">subscript</csymbol><apply id="S2.Ex2.m1.2.3.2.cmml" xref="S2.Ex2.m1.2.3"><csymbol cd="ambiguous" id="S2.Ex2.m1.2.3.2.1.cmml" xref="S2.Ex2.m1.2.3">superscript</csymbol><ci id="S2.Ex2.m1.2.3.2.2.cmml" xref="S2.Ex2.m1.2.3.2.2">ℬ</ci><times id="S2.Ex2.m1.2.3.2.3.cmml" xref="S2.Ex2.m1.2.3.2.3"></times></apply><list id="S2.Ex2.m1.2.2.2.3.cmml" xref="S2.Ex2.m1.2.2.2.4"><ci id="S2.Ex2.m1.1.1.1.1.cmml" xref="S2.Ex2.m1.1.1.1.1">𝐼</ci><ci id="S2.Ex2.m1.2.2.2.2.cmml" xref="S2.Ex2.m1.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m1.2c">\displaystyle\mathcal{B}^{*}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m1.2d">caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\{r\in\mathcal{B}_{k}\mid\exists\beta\in\mathcal{B}_{I}^{*}:% \beta_{k}=r\}" class="ltx_Math" display="inline" id="S2.Ex2.m2.2"><semantics id="S2.Ex2.m2.2a"><mrow id="S2.Ex2.m2.2.2" xref="S2.Ex2.m2.2.2.cmml"><mi id="S2.Ex2.m2.2.2.4" xref="S2.Ex2.m2.2.2.4.cmml"></mi><mo id="S2.Ex2.m2.2.2.3" xref="S2.Ex2.m2.2.2.3.cmml">=</mo><mrow id="S2.Ex2.m2.2.2.2.2" xref="S2.Ex2.m2.2.2.2.3.cmml"><mo id="S2.Ex2.m2.2.2.2.2.3" stretchy="false" xref="S2.Ex2.m2.2.2.2.3.1.cmml">{</mo><mrow id="S2.Ex2.m2.1.1.1.1.1" xref="S2.Ex2.m2.1.1.1.1.1.cmml"><mi id="S2.Ex2.m2.1.1.1.1.1.2" xref="S2.Ex2.m2.1.1.1.1.1.2.cmml">r</mi><mo id="S2.Ex2.m2.1.1.1.1.1.1" xref="S2.Ex2.m2.1.1.1.1.1.1.cmml">∈</mo><msub id="S2.Ex2.m2.1.1.1.1.1.3" xref="S2.Ex2.m2.1.1.1.1.1.3.cmml"><mi 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xref="S2.Ex2.m2.2.2.2.2.2.2.3.2.3.cmml">I</mi><mo id="S2.Ex2.m2.2.2.2.2.2.2.3.3" xref="S2.Ex2.m2.2.2.2.2.2.2.3.3.cmml">∗</mo></msubsup></mrow><mo id="S2.Ex2.m2.2.2.2.2.2.1" lspace="0.278em" rspace="0.278em" xref="S2.Ex2.m2.2.2.2.2.2.1.cmml">:</mo><mrow id="S2.Ex2.m2.2.2.2.2.2.3" xref="S2.Ex2.m2.2.2.2.2.2.3.cmml"><msub id="S2.Ex2.m2.2.2.2.2.2.3.2" xref="S2.Ex2.m2.2.2.2.2.2.3.2.cmml"><mi id="S2.Ex2.m2.2.2.2.2.2.3.2.2" xref="S2.Ex2.m2.2.2.2.2.2.3.2.2.cmml">β</mi><mi id="S2.Ex2.m2.2.2.2.2.2.3.2.3" xref="S2.Ex2.m2.2.2.2.2.2.3.2.3.cmml">k</mi></msub><mo id="S2.Ex2.m2.2.2.2.2.2.3.1" xref="S2.Ex2.m2.2.2.2.2.2.3.1.cmml">=</mo><mi id="S2.Ex2.m2.2.2.2.2.2.3.3" xref="S2.Ex2.m2.2.2.2.2.2.3.3.cmml">r</mi></mrow></mrow><mo id="S2.Ex2.m2.2.2.2.2.5" stretchy="false" xref="S2.Ex2.m2.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex2.m2.2b"><apply id="S2.Ex2.m2.2.2.cmml" xref="S2.Ex2.m2.2.2"><eq id="S2.Ex2.m2.2.2.3.cmml" xref="S2.Ex2.m2.2.2.3"></eq><csymbol 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xref="S2.Ex2.m2.2.2.2.2.2.3"><eq id="S2.Ex2.m2.2.2.2.2.2.3.1.cmml" xref="S2.Ex2.m2.2.2.2.2.2.3.1"></eq><apply id="S2.Ex2.m2.2.2.2.2.2.3.2.cmml" xref="S2.Ex2.m2.2.2.2.2.2.3.2"><csymbol cd="ambiguous" id="S2.Ex2.m2.2.2.2.2.2.3.2.1.cmml" xref="S2.Ex2.m2.2.2.2.2.2.3.2">subscript</csymbol><ci id="S2.Ex2.m2.2.2.2.2.2.3.2.2.cmml" xref="S2.Ex2.m2.2.2.2.2.2.3.2.2">𝛽</ci><ci id="S2.Ex2.m2.2.2.2.2.2.3.2.3.cmml" xref="S2.Ex2.m2.2.2.2.2.2.3.2.3">𝑘</ci></apply><ci id="S2.Ex2.m2.2.2.2.2.2.3.3.cmml" xref="S2.Ex2.m2.2.2.2.2.2.3.3">𝑟</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m2.2c">\displaystyle=\{r\in\mathcal{B}_{k}\mid\exists\beta\in\mathcal{B}_{I}^{*}:% \beta_{k}=r\}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m2.2d">= { italic_r ∈ caligraphic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∣ ∃ italic_β ∈ caligraphic_B start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_r }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.E4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\{r\in\mathcal{B}_{k}\mid\exists\beta\in\mathcal{B}:\beta_{k}=r% \text{ and }\forall x\in\mathcal{X}:\beta\text{ satisfies \eqref{eq: Peterson % bounds}}\}," class="ltx_Math" display="inline" id="S2.E4.m1.1"><semantics id="S2.E4.m1.1a"><mrow id="S2.E4.m1.1.1.1" xref="S2.E4.m1.1.1.1.1.cmml"><mrow id="S2.E4.m1.1.1.1.1" xref="S2.E4.m1.1.1.1.1.cmml"><mi id="S2.E4.m1.1.1.1.1.4" xref="S2.E4.m1.1.1.1.1.4.cmml"></mi><mo id="S2.E4.m1.1.1.1.1.3" xref="S2.E4.m1.1.1.1.1.3.cmml">=</mo><mrow id="S2.E4.m1.1.1.1.1.2.2" xref="S2.E4.m1.1.1.1.1.2.3.cmml"><mo id="S2.E4.m1.1.1.1.1.2.2.3" 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href="https://arxiv.org/html/2503.11210v2#S2.E4.m1.1.1.1.1.2.2.2.4.cmml" id="S2.E4.m1.1.1.1.1.2.2.2d.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2"></share><apply id="S2.E4.m1.1.1.1.1.2.2.2.6.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2.6"><times id="S2.E4.m1.1.1.1.1.2.2.2.6.1.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2.6.1"></times><ci id="S2.E4.m1.1.1.1.1.2.2.2.6.2.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2.6.2">𝛽</ci><ci id="S2.E4.m1.1.1.1.1.2.2.2.6.3f.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2.6.3"><mrow id="S2.E4.m1.1.1.1.1.2.2.2.6.3.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2.6.3"><mtext id="S2.E4.m1.1.1.1.1.2.2.2.6.3a.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2.6.3"> satisfies (</mtext><mtext id="S2.E4.m1.1.1.1.1.2.2.2.6.3b.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2.6.3"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E2" title="In 2.2 Methodology ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">2</span></a></mtext><mtext id="S2.E4.m1.1.1.1.1.2.2.2.6.3e.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2.6.3">)</mtext></mrow></ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.m1.1c">\displaystyle=\{r\in\mathcal{B}_{k}\mid\exists\beta\in\mathcal{B}:\beta_{k}=r% \text{ and }\forall x\in\mathcal{X}:\beta\text{ satisfies \eqref{eq: Peterson % bounds}}\},</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.1d">= { italic_r ∈ caligraphic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∣ ∃ italic_β ∈ caligraphic_B : italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_r and ∀ italic_x ∈ caligraphic_X : italic_β satisfies ( ) } ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p5.9">where <math alttext="\mathcal{B}_{k}" class="ltx_Math" display="inline" id="S2.SS2.p5.1.m1.1"><semantics id="S2.SS2.p5.1.m1.1a"><msub id="S2.SS2.p5.1.m1.1.1" xref="S2.SS2.p5.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p5.1.m1.1.1.2" xref="S2.SS2.p5.1.m1.1.1.2.cmml">ℬ</mi><mi id="S2.SS2.p5.1.m1.1.1.3" xref="S2.SS2.p5.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.1.m1.1b"><apply id="S2.SS2.p5.1.m1.1.1.cmml" xref="S2.SS2.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.1.m1.1.1.1.cmml" xref="S2.SS2.p5.1.m1.1.1">subscript</csymbol><ci id="S2.SS2.p5.1.m1.1.1.2.cmml" xref="S2.SS2.p5.1.m1.1.1.2">ℬ</ci><ci id="S2.SS2.p5.1.m1.1.1.3.cmml" xref="S2.SS2.p5.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.1.m1.1c">\mathcal{B}_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.1.m1.1d">caligraphic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> denotes the projection of <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.SS2.p5.2.m2.1"><semantics id="S2.SS2.p5.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p5.2.m2.1.1" xref="S2.SS2.p5.2.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.2.m2.1b"><ci id="S2.SS2.p5.2.m2.1.1.cmml" xref="S2.SS2.p5.2.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.2.m2.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.2.m2.1d">caligraphic_B</annotation></semantics></math> onto the <math alttext="k" class="ltx_Math" display="inline" id="S2.SS2.p5.3.m3.1"><semantics id="S2.SS2.p5.3.m3.1a"><mi id="S2.SS2.p5.3.m3.1.1" xref="S2.SS2.p5.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.3.m3.1b"><ci id="S2.SS2.p5.3.m3.1.1.cmml" xref="S2.SS2.p5.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.3.m3.1d">italic_k</annotation></semantics></math>-th coordinate. When testing the condition of the set defined in (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E4" title="In 2.2 Methodology ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">4</span></a>) for the test inversion approach, one difficulty to note is that as soon as a continuous covariate enters the model, we require a continuum of restrictions to be tested: (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E2" title="In 2.2 Methodology ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">2</span></a>) has to hold for each value of <math alttext="x\in\mathcal{X}" class="ltx_Math" display="inline" id="S2.SS2.p5.4.m4.1"><semantics id="S2.SS2.p5.4.m4.1a"><mrow id="S2.SS2.p5.4.m4.1.1" xref="S2.SS2.p5.4.m4.1.1.cmml"><mi id="S2.SS2.p5.4.m4.1.1.2" xref="S2.SS2.p5.4.m4.1.1.2.cmml">x</mi><mo id="S2.SS2.p5.4.m4.1.1.1" xref="S2.SS2.p5.4.m4.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p5.4.m4.1.1.3" xref="S2.SS2.p5.4.m4.1.1.3.cmml">𝒳</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.4.m4.1b"><apply id="S2.SS2.p5.4.m4.1.1.cmml" xref="S2.SS2.p5.4.m4.1.1"><in id="S2.SS2.p5.4.m4.1.1.1.cmml" xref="S2.SS2.p5.4.m4.1.1.1"></in><ci id="S2.SS2.p5.4.m4.1.1.2.cmml" xref="S2.SS2.p5.4.m4.1.1.2">𝑥</ci><ci id="S2.SS2.p5.4.m4.1.1.3.cmml" xref="S2.SS2.p5.4.m4.1.1.3">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.4.m4.1c">x\in\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.4.m4.1d">italic_x ∈ caligraphic_X</annotation></semantics></math>. To our knowledge, subvector inference tests that can handle this case (i.e. testing conditional moment restrictions) do not exist in the literature yet. Therefore, following <cite class="ltx_cite ltx_citemacro_cite">Andrews and Shi, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib1" title="">2013</a>)</cite>, we discretize this continuum of restrictions making use of instrumental functions. In this way, we obtain a condition that is similar to (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E4" title="In 2.2 Methodology ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">4</span></a>), only now <math alttext="\beta" class="ltx_Math" display="inline" id="S2.SS2.p5.5.m5.1"><semantics id="S2.SS2.p5.5.m5.1a"><mi id="S2.SS2.p5.5.m5.1.1" xref="S2.SS2.p5.5.m5.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.5.m5.1b"><ci id="S2.SS2.p5.5.m5.1.1.cmml" xref="S2.SS2.p5.5.m5.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.5.m5.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.5.m5.1d">italic_β</annotation></semantics></math> has to satisfy finitely many restrictions. The cost of taking this approach is that the new condition will be slightly weaker than the original one, and hence the set <math alttext="\mathcal{B}_{I,k}" class="ltx_Math" display="inline" id="S2.SS2.p5.6.m6.2"><semantics id="S2.SS2.p5.6.m6.2a"><msub id="S2.SS2.p5.6.m6.2.3" xref="S2.SS2.p5.6.m6.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p5.6.m6.2.3.2" xref="S2.SS2.p5.6.m6.2.3.2.cmml">ℬ</mi><mrow id="S2.SS2.p5.6.m6.2.2.2.4" xref="S2.SS2.p5.6.m6.2.2.2.3.cmml"><mi id="S2.SS2.p5.6.m6.1.1.1.1" xref="S2.SS2.p5.6.m6.1.1.1.1.cmml">I</mi><mo id="S2.SS2.p5.6.m6.2.2.2.4.1" xref="S2.SS2.p5.6.m6.2.2.2.3.cmml">,</mo><mi id="S2.SS2.p5.6.m6.2.2.2.2" xref="S2.SS2.p5.6.m6.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.6.m6.2b"><apply id="S2.SS2.p5.6.m6.2.3.cmml" xref="S2.SS2.p5.6.m6.2.3"><csymbol cd="ambiguous" id="S2.SS2.p5.6.m6.2.3.1.cmml" xref="S2.SS2.p5.6.m6.2.3">subscript</csymbol><ci id="S2.SS2.p5.6.m6.2.3.2.cmml" xref="S2.SS2.p5.6.m6.2.3.2">ℬ</ci><list id="S2.SS2.p5.6.m6.2.2.2.3.cmml" xref="S2.SS2.p5.6.m6.2.2.2.4"><ci id="S2.SS2.p5.6.m6.1.1.1.1.cmml" xref="S2.SS2.p5.6.m6.1.1.1.1">𝐼</ci><ci id="S2.SS2.p5.6.m6.2.2.2.2.cmml" xref="S2.SS2.p5.6.m6.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.6.m6.2c">\mathcal{B}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.6.m6.2d">caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> corresponding to this new condition will be a superset of <math alttext="\mathcal{B}_{I,k}^{*}" class="ltx_Math" display="inline" id="S2.SS2.p5.7.m7.2"><semantics id="S2.SS2.p5.7.m7.2a"><msubsup id="S2.SS2.p5.7.m7.2.3" xref="S2.SS2.p5.7.m7.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p5.7.m7.2.3.2.2" xref="S2.SS2.p5.7.m7.2.3.2.2.cmml">ℬ</mi><mrow id="S2.SS2.p5.7.m7.2.2.2.4" xref="S2.SS2.p5.7.m7.2.2.2.3.cmml"><mi id="S2.SS2.p5.7.m7.1.1.1.1" xref="S2.SS2.p5.7.m7.1.1.1.1.cmml">I</mi><mo id="S2.SS2.p5.7.m7.2.2.2.4.1" xref="S2.SS2.p5.7.m7.2.2.2.3.cmml">,</mo><mi id="S2.SS2.p5.7.m7.2.2.2.2" xref="S2.SS2.p5.7.m7.2.2.2.2.cmml">k</mi></mrow><mo id="S2.SS2.p5.7.m7.2.3.3" xref="S2.SS2.p5.7.m7.2.3.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.7.m7.2b"><apply id="S2.SS2.p5.7.m7.2.3.cmml" xref="S2.SS2.p5.7.m7.2.3"><csymbol cd="ambiguous" id="S2.SS2.p5.7.m7.2.3.1.cmml" xref="S2.SS2.p5.7.m7.2.3">superscript</csymbol><apply id="S2.SS2.p5.7.m7.2.3.2.cmml" xref="S2.SS2.p5.7.m7.2.3"><csymbol cd="ambiguous" id="S2.SS2.p5.7.m7.2.3.2.1.cmml" xref="S2.SS2.p5.7.m7.2.3">subscript</csymbol><ci id="S2.SS2.p5.7.m7.2.3.2.2.cmml" xref="S2.SS2.p5.7.m7.2.3.2.2">ℬ</ci><list id="S2.SS2.p5.7.m7.2.2.2.3.cmml" xref="S2.SS2.p5.7.m7.2.2.2.4"><ci id="S2.SS2.p5.7.m7.1.1.1.1.cmml" xref="S2.SS2.p5.7.m7.1.1.1.1">𝐼</ci><ci id="S2.SS2.p5.7.m7.2.2.2.2.cmml" xref="S2.SS2.p5.7.m7.2.2.2.2">𝑘</ci></list></apply><times id="S2.SS2.p5.7.m7.2.3.3.cmml" xref="S2.SS2.p5.7.m7.2.3.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.7.m7.2c">\mathcal{B}_{I,k}^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.7.m7.2d">caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>. As a consequence, when regarding an eventual estimator for <math alttext="\mathcal{B}_{I,k}" class="ltx_Math" display="inline" id="S2.SS2.p5.8.m8.2"><semantics id="S2.SS2.p5.8.m8.2a"><msub id="S2.SS2.p5.8.m8.2.3" xref="S2.SS2.p5.8.m8.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p5.8.m8.2.3.2" xref="S2.SS2.p5.8.m8.2.3.2.cmml">ℬ</mi><mrow id="S2.SS2.p5.8.m8.2.2.2.4" xref="S2.SS2.p5.8.m8.2.2.2.3.cmml"><mi id="S2.SS2.p5.8.m8.1.1.1.1" xref="S2.SS2.p5.8.m8.1.1.1.1.cmml">I</mi><mo id="S2.SS2.p5.8.m8.2.2.2.4.1" xref="S2.SS2.p5.8.m8.2.2.2.3.cmml">,</mo><mi id="S2.SS2.p5.8.m8.2.2.2.2" xref="S2.SS2.p5.8.m8.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.8.m8.2b"><apply id="S2.SS2.p5.8.m8.2.3.cmml" xref="S2.SS2.p5.8.m8.2.3"><csymbol cd="ambiguous" id="S2.SS2.p5.8.m8.2.3.1.cmml" xref="S2.SS2.p5.8.m8.2.3">subscript</csymbol><ci id="S2.SS2.p5.8.m8.2.3.2.cmml" xref="S2.SS2.p5.8.m8.2.3.2">ℬ</ci><list id="S2.SS2.p5.8.m8.2.2.2.3.cmml" xref="S2.SS2.p5.8.m8.2.2.2.4"><ci id="S2.SS2.p5.8.m8.1.1.1.1.cmml" xref="S2.SS2.p5.8.m8.1.1.1.1">𝐼</ci><ci id="S2.SS2.p5.8.m8.2.2.2.2.cmml" xref="S2.SS2.p5.8.m8.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.8.m8.2c">\mathcal{B}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.8.m8.2d">caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> as an estimator for <math alttext="\mathcal{B}_{I,k}^{*}" class="ltx_Math" display="inline" id="S2.SS2.p5.9.m9.2"><semantics id="S2.SS2.p5.9.m9.2a"><msubsup id="S2.SS2.p5.9.m9.2.3" xref="S2.SS2.p5.9.m9.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p5.9.m9.2.3.2.2" xref="S2.SS2.p5.9.m9.2.3.2.2.cmml">ℬ</mi><mrow id="S2.SS2.p5.9.m9.2.2.2.4" xref="S2.SS2.p5.9.m9.2.2.2.3.cmml"><mi id="S2.SS2.p5.9.m9.1.1.1.1" xref="S2.SS2.p5.9.m9.1.1.1.1.cmml">I</mi><mo id="S2.SS2.p5.9.m9.2.2.2.4.1" xref="S2.SS2.p5.9.m9.2.2.2.3.cmml">,</mo><mi id="S2.SS2.p5.9.m9.2.2.2.2" xref="S2.SS2.p5.9.m9.2.2.2.2.cmml">k</mi></mrow><mo id="S2.SS2.p5.9.m9.2.3.3" xref="S2.SS2.p5.9.m9.2.3.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.9.m9.2b"><apply id="S2.SS2.p5.9.m9.2.3.cmml" xref="S2.SS2.p5.9.m9.2.3"><csymbol cd="ambiguous" id="S2.SS2.p5.9.m9.2.3.1.cmml" xref="S2.SS2.p5.9.m9.2.3">superscript</csymbol><apply id="S2.SS2.p5.9.m9.2.3.2.cmml" xref="S2.SS2.p5.9.m9.2.3"><csymbol cd="ambiguous" id="S2.SS2.p5.9.m9.2.3.2.1.cmml" xref="S2.SS2.p5.9.m9.2.3">subscript</csymbol><ci id="S2.SS2.p5.9.m9.2.3.2.2.cmml" xref="S2.SS2.p5.9.m9.2.3.2.2">ℬ</ci><list id="S2.SS2.p5.9.m9.2.2.2.3.cmml" xref="S2.SS2.p5.9.m9.2.2.2.4"><ci id="S2.SS2.p5.9.m9.1.1.1.1.cmml" xref="S2.SS2.p5.9.m9.1.1.1.1">𝐼</ci><ci id="S2.SS2.p5.9.m9.2.2.2.2.cmml" xref="S2.SS2.p5.9.m9.2.2.2.2">𝑘</ci></list></apply><times id="S2.SS2.p5.9.m9.2.3.3.cmml" xref="S2.SS2.p5.9.m9.2.3.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.9.m9.2c">\mathcal{B}_{I,k}^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.9.m9.2d">caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, it will be slightly conservative.</p> </div> <div class="ltx_para" id="S2.SS2.p6"> <p class="ltx_p" id="S2.SS2.p6.1">In the following, we first elaborate on how the new condition is constructed and formally define <math alttext="\mathcal{B}_{I,k}" class="ltx_Math" display="inline" id="S2.SS2.p6.1.m1.2"><semantics id="S2.SS2.p6.1.m1.2a"><msub id="S2.SS2.p6.1.m1.2.3" xref="S2.SS2.p6.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p6.1.m1.2.3.2" xref="S2.SS2.p6.1.m1.2.3.2.cmml">ℬ</mi><mrow id="S2.SS2.p6.1.m1.2.2.2.4" xref="S2.SS2.p6.1.m1.2.2.2.3.cmml"><mi id="S2.SS2.p6.1.m1.1.1.1.1" xref="S2.SS2.p6.1.m1.1.1.1.1.cmml">I</mi><mo id="S2.SS2.p6.1.m1.2.2.2.4.1" xref="S2.SS2.p6.1.m1.2.2.2.3.cmml">,</mo><mi id="S2.SS2.p6.1.m1.2.2.2.2" xref="S2.SS2.p6.1.m1.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.1.m1.2b"><apply id="S2.SS2.p6.1.m1.2.3.cmml" xref="S2.SS2.p6.1.m1.2.3"><csymbol cd="ambiguous" id="S2.SS2.p6.1.m1.2.3.1.cmml" xref="S2.SS2.p6.1.m1.2.3">subscript</csymbol><ci id="S2.SS2.p6.1.m1.2.3.2.cmml" xref="S2.SS2.p6.1.m1.2.3.2">ℬ</ci><list id="S2.SS2.p6.1.m1.2.2.2.3.cmml" xref="S2.SS2.p6.1.m1.2.2.2.4"><ci id="S2.SS2.p6.1.m1.1.1.1.1.cmml" xref="S2.SS2.p6.1.m1.1.1.1.1">𝐼</ci><ci id="S2.SS2.p6.1.m1.2.2.2.2.cmml" xref="S2.SS2.p6.1.m1.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.1.m1.2c">\mathcal{B}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.1.m1.2d">caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. Next, a test for this modified condition is obtained. Lastly, we give an overview of the entire estimation procedure.</p> </div> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.3 </span>Unconditional moment restrictions</h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.18">To solve the aforementioned challenge posed by continuous covariates entering the model, we first transform the possible continuum of restrictions in (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E4" title="In 2.2 Methodology ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">4</span></a>) to a finite number of restrictions. To this end, we start by rewriting Peterson’s bounds in Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E2" title="In 2.2 Methodology ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">2</span></a>) in the form of two conditional moment inequalities:</p> <table class="ltx_equation ltx_eqn_table" id="S2.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\forall x\in\mathcal{X}:\begin{cases}\mathbb{E}[\mathbbm{1}(Y\leq t)-\Lambda(X% ^{\top}\beta)|X=x]\geq 0\\ \mathbb{E}[\Lambda(X^{\top}\beta)-\mathbbm{1}(Y\leq t,\Delta=1)|X=x]\geq 0.% \end{cases}" class="ltx_Math" display="block" id="S2.E5.m1.2"><semantics id="S2.E5.m1.2a"><mrow id="S2.E5.m1.2.3" xref="S2.E5.m1.2.3.cmml"><mrow id="S2.E5.m1.2.3.2" xref="S2.E5.m1.2.3.2.cmml"><mrow 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xref="S2.E5.m1.2.2.2.2.1.1.1.1.1.1.1.1.2.2.1.1.1.1.2.2.2">Δ</ci><cn id="S2.E5.m1.2.2.2.2.1.1.1.1.1.1.1.1.2.2.1.1.1.1.2.2.3.cmml" type="integer" xref="S2.E5.m1.2.2.2.2.1.1.1.1.1.1.1.1.2.2.1.1.1.1.2.2.3">1</cn></apply></apply></apply><ci id="S2.E5.m1.2.2.2.2.1.1.1.1.1.1.1.1.2.2.3.cmml" xref="S2.E5.m1.2.2.2.2.1.1.1.1.1.1.1.1.2.2.3">𝑋</ci></apply></apply><ci id="S2.E5.m1.2.2.2.2.1.1.1.1.1.1.1.1.4.cmml" xref="S2.E5.m1.2.2.2.2.1.1.1.1.1.1.1.1.4">𝑥</ci></apply></apply></apply><cn id="S2.E5.m1.2.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S2.E5.m1.2.2.2.2.1.1.1.1.3">0</cn></apply><ci id="S2.E5.m1.2.3.3.1.5a.cmml" xref="S2.E5.m1.2.2"><mtext class="ltx_mathvariant_italic" id="S2.E5.m1.2.3.3.1.5.cmml" xref="S2.E5.m1.2.2.3">otherwise</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E5.m1.2c">\forall x\in\mathcal{X}:\begin{cases}\mathbb{E}[\mathbbm{1}(Y\leq t)-\Lambda(X% ^{\top}\beta)|X=x]\geq 0\\ \mathbb{E}[\Lambda(X^{\top}\beta)-\mathbbm{1}(Y\leq t,\Delta=1)|X=x]\geq 0.% \end{cases}</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m1.2d">∀ italic_x ∈ caligraphic_X : { start_ROW start_CELL blackboard_E [ blackboard_1 ( italic_Y ≤ italic_t ) - roman_Λ ( italic_X start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT italic_β ) | italic_X = italic_x ] ≥ 0 end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL blackboard_E [ roman_Λ ( italic_X start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT italic_β ) - blackboard_1 ( italic_Y ≤ italic_t , roman_Δ = 1 ) | italic_X = italic_x ] ≥ 0 . end_CELL start_CELL end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p1.2">Next, these conditional moment restrictions are transformed to unconditional ones making use of a class <math alttext="\mathcal{G}=\{g_{j},j=1,\dots,J\}" class="ltx_math_unparsed" display="inline" id="S2.SS3.p1.1.m1.1"><semantics id="S2.SS3.p1.1.m1.1a"><mrow id="S2.SS3.p1.1.m1.1b"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.1.m1.1.2">𝒢</mi><mo id="S2.SS3.p1.1.m1.1.3">=</mo><mrow id="S2.SS3.p1.1.m1.1.4"><mo id="S2.SS3.p1.1.m1.1.4.1" stretchy="false">{</mo><msub id="S2.SS3.p1.1.m1.1.4.2"><mi id="S2.SS3.p1.1.m1.1.4.2.2">g</mi><mi id="S2.SS3.p1.1.m1.1.4.2.3">j</mi></msub><mo id="S2.SS3.p1.1.m1.1.4.3">,</mo><mi id="S2.SS3.p1.1.m1.1.1">j</mi><mo id="S2.SS3.p1.1.m1.1.4.4">=</mo><mn id="S2.SS3.p1.1.m1.1.4.5">1</mn><mo id="S2.SS3.p1.1.m1.1.4.6">,</mo><mi id="S2.SS3.p1.1.m1.1.4.7" mathvariant="normal">…</mi><mo id="S2.SS3.p1.1.m1.1.4.8">,</mo><mi id="S2.SS3.p1.1.m1.1.4.9">J</mi><mo id="S2.SS3.p1.1.m1.1.4.10" stretchy="false">}</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.SS3.p1.1.m1.1c">\mathcal{G}=\{g_{j},j=1,\dots,J\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.1.m1.1d">caligraphic_G = { italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_j = 1 , … , italic_J }</annotation></semantics></math> of <em class="ltx_emph ltx_font_italic" id="S2.SS3.p1.2.1">instrumental functions</em> <cite class="ltx_cite ltx_citemacro_citep">(Andrews and Shi,, <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib1" title="">2013</a>)</cite>, where <math alttext="J" class="ltx_Math" display="inline" id="S2.SS3.p1.2.m2.1"><semantics id="S2.SS3.p1.2.m2.1a"><mi id="S2.SS3.p1.2.m2.1.1" xref="S2.SS3.p1.2.m2.1.1.cmml">J</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.2.m2.1b"><ci id="S2.SS3.p1.2.m2.1.1.cmml" xref="S2.SS3.p1.2.m2.1.1">𝐽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.2.m2.1c">J</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.2.m2.1d">italic_J</annotation></semantics></math> denotes the number of instrumental functions considered:</p> <table class="ltx_equation ltx_eqn_table" id="S2.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\forall j\in\{1,\dots,J\}:\begin{cases}m_{j,1}(W,\beta)=\mathbb{E}[(\mathbbm{1% }(Y\leq t)-\Lambda(X^{\top}\beta))g_{j}(X)]\geq 0\\ m_{j,2}(W,\beta)=\mathbb{E}[(\Lambda(X^{\top}\beta)-\mathbbm{1}(Y\leq t,\Delta% =1))g_{j}(X)]\geq 0.\end{cases}" class="ltx_Math" display="block" id="S2.E6.m1.5"><semantics id="S2.E6.m1.5a"><mrow id="S2.E6.m1.5.6" xref="S2.E6.m1.5.6.cmml"><mrow id="S2.E6.m1.5.6.2" xref="S2.E6.m1.5.6.2.cmml"><mrow id="S2.E6.m1.5.6.2.2" xref="S2.E6.m1.5.6.2.2.cmml"><mo id="S2.E6.m1.5.6.2.2.1" rspace="0.167em" xref="S2.E6.m1.5.6.2.2.1.cmml">∀</mo><mi id="S2.E6.m1.5.6.2.2.2" xref="S2.E6.m1.5.6.2.2.2.cmml">j</mi></mrow><mo id="S2.E6.m1.5.6.2.1" xref="S2.E6.m1.5.6.2.1.cmml">∈</mo><mrow id="S2.E6.m1.5.6.2.3.2" xref="S2.E6.m1.5.6.2.3.1.cmml"><mo id="S2.E6.m1.5.6.2.3.2.1" stretchy="false" xref="S2.E6.m1.5.6.2.3.1.cmml">{</mo><mn id="S2.E6.m1.3.3" xref="S2.E6.m1.3.3.cmml">1</mn><mo id="S2.E6.m1.5.6.2.3.2.2" xref="S2.E6.m1.5.6.2.3.1.cmml">,</mo><mi id="S2.E6.m1.4.4" mathvariant="normal" xref="S2.E6.m1.4.4.cmml">…</mi><mo id="S2.E6.m1.5.6.2.3.2.3" xref="S2.E6.m1.5.6.2.3.1.cmml">,</mo><mi id="S2.E6.m1.5.5" xref="S2.E6.m1.5.5.cmml">J</mi><mo id="S2.E6.m1.5.6.2.3.2.4" rspace="0.278em" stretchy="false" xref="S2.E6.m1.5.6.2.3.1.cmml">}</mo></mrow></mrow><mo id="S2.E6.m1.5.6.1" rspace="0.278em" xref="S2.E6.m1.5.6.1.cmml">:</mo><mrow id="S2.E6.m1.2.2" xref="S2.E6.m1.5.6.3.1.cmml"><mo id="S2.E6.m1.2.2.3" xref="S2.E6.m1.5.6.3.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" id="S2.E6.m1.2.2.2" rowspacing="0pt" xref="S2.E6.m1.5.6.3.1.cmml"><mtr id="S2.E6.m1.2.2.2a" xref="S2.E6.m1.5.6.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S2.E6.m1.2.2.2b" 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xref="S2.E6.m1.1.1.1.1.1.1.6.1.1.1.1.1.1.2.1.1.1.cmml">(</mo><mrow id="S2.E6.m1.1.1.1.1.1.1.6.1.1.1.1.1.1.2.1.1.1" xref="S2.E6.m1.1.1.1.1.1.1.6.1.1.1.1.1.1.2.1.1.1.cmml"><msup id="S2.E6.m1.1.1.1.1.1.1.6.1.1.1.1.1.1.2.1.1.1.2" xref="S2.E6.m1.1.1.1.1.1.1.6.1.1.1.1.1.1.2.1.1.1.2.cmml"><mi id="S2.E6.m1.1.1.1.1.1.1.6.1.1.1.1.1.1.2.1.1.1.2.2" xref="S2.E6.m1.1.1.1.1.1.1.6.1.1.1.1.1.1.2.1.1.1.2.2.cmml">X</mi><mo id="S2.E6.m1.1.1.1.1.1.1.6.1.1.1.1.1.1.2.1.1.1.2.3" xref="S2.E6.m1.1.1.1.1.1.1.6.1.1.1.1.1.1.2.1.1.1.2.3.cmml">⊤</mo></msup><mo id="S2.E6.m1.1.1.1.1.1.1.6.1.1.1.1.1.1.2.1.1.1.1" xref="S2.E6.m1.1.1.1.1.1.1.6.1.1.1.1.1.1.2.1.1.1.1.cmml">⁢</mo><mi id="S2.E6.m1.1.1.1.1.1.1.6.1.1.1.1.1.1.2.1.1.1.3" xref="S2.E6.m1.1.1.1.1.1.1.6.1.1.1.1.1.1.2.1.1.1.3.cmml">β</mi></mrow><mo id="S2.E6.m1.1.1.1.1.1.1.6.1.1.1.1.1.1.2.1.1.3" stretchy="false" xref="S2.E6.m1.1.1.1.1.1.1.6.1.1.1.1.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E6.m1.1.1.1.1.1.1.6.1.1.1.1.1.3" stretchy="false" 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xref="S2.E6.m1.1.1.1.1.1.1.6.1.2.1.cmml">]</mo></mrow></mrow><mo id="S2.E6.m1.1.1.1.1.1.1.10" xref="S2.E6.m1.1.1.1.1.1.1.10.cmml">≥</mo><mn id="S2.E6.m1.1.1.1.1.1.1.11" xref="S2.E6.m1.1.1.1.1.1.1.11.cmml">0</mn></mrow></mtd><mtd id="S2.E6.m1.2.2.2c" xref="S2.E6.m1.5.6.3.1.1.cmml"></mtd></mtr><mtr id="S2.E6.m1.2.2.2d" xref="S2.E6.m1.5.6.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S2.E6.m1.2.2.2e" xref="S2.E6.m1.5.6.3.1.cmml"><mrow id="S2.E6.m1.2.2.2.2.1.1.6" xref="S2.E6.m1.2.2.2.2.1.1.6.1.cmml"><mrow id="S2.E6.m1.2.2.2.2.1.1.6.1" xref="S2.E6.m1.2.2.2.2.1.1.6.1.cmml"><mrow id="S2.E6.m1.2.2.2.2.1.1.6.1.3" xref="S2.E6.m1.2.2.2.2.1.1.6.1.3.cmml"><msub id="S2.E6.m1.2.2.2.2.1.1.6.1.3.2" xref="S2.E6.m1.2.2.2.2.1.1.6.1.3.2.cmml"><mi id="S2.E6.m1.2.2.2.2.1.1.6.1.3.2.2" xref="S2.E6.m1.2.2.2.2.1.1.6.1.3.2.2.cmml">m</mi><mrow id="S2.E6.m1.2.2.2.2.1.1.2.2.4" xref="S2.E6.m1.2.2.2.2.1.1.2.2.3.cmml"><mi id="S2.E6.m1.2.2.2.2.1.1.1.1.1" xref="S2.E6.m1.2.2.2.2.1.1.1.1.1.cmml">j</mi><mo id="S2.E6.m1.2.2.2.2.1.1.2.2.4.1" xref="S2.E6.m1.2.2.2.2.1.1.2.2.3.cmml">,</mo><mn id="S2.E6.m1.2.2.2.2.1.1.2.2.2" xref="S2.E6.m1.2.2.2.2.1.1.2.2.2.cmml">2</mn></mrow></msub><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.3.1" xref="S2.E6.m1.2.2.2.2.1.1.6.1.3.1.cmml">⁢</mo><mrow id="S2.E6.m1.2.2.2.2.1.1.6.1.3.3.2" xref="S2.E6.m1.2.2.2.2.1.1.6.1.3.3.1.cmml"><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.3.3.2.1" stretchy="false" xref="S2.E6.m1.2.2.2.2.1.1.6.1.3.3.1.cmml">(</mo><mi id="S2.E6.m1.2.2.2.2.1.1.3" xref="S2.E6.m1.2.2.2.2.1.1.3.cmml">W</mi><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.3.3.2.2" xref="S2.E6.m1.2.2.2.2.1.1.6.1.3.3.1.cmml">,</mo><mi id="S2.E6.m1.2.2.2.2.1.1.4" xref="S2.E6.m1.2.2.2.2.1.1.4.cmml">β</mi><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.3.3.2.3" stretchy="false" xref="S2.E6.m1.2.2.2.2.1.1.6.1.3.3.1.cmml">)</mo></mrow></mrow><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.4" xref="S2.E6.m1.2.2.2.2.1.1.6.1.4.cmml">=</mo><mrow id="S2.E6.m1.2.2.2.2.1.1.6.1.1" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.cmml"><mi id="S2.E6.m1.2.2.2.2.1.1.6.1.1.3" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.3.cmml">𝔼</mi><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.1.2" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.2.cmml">⁢</mo><mrow id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.2.cmml"><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.2" stretchy="false" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.2.1.cmml">[</mo><mrow id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.cmml"><mrow id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.cmml"><mrow id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.3" mathvariant="normal" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.3.cmml">Λ</mi><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.2" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><msup id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1.1.2.2" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1.1.2.2.cmml">X</mi><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1.1.2.3" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1.1.2.3.cmml">⊤</mo></msup><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">β</mi></mrow><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.3" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.3.cmml">−</mo><mrow id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.cmml"><mn id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.3" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.3.cmml">𝟙</mn><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.2" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.2.cmml">⁢</mo><mrow id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.cmml"><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.2" stretchy="false" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.cmml">(</mo><mrow id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.2" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.3.cmml"><mrow id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.1.1" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.1.1.cmml"><mi id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.1.1.2" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.1.1.2.cmml">Y</mi><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1.cmml">≤</mo><mi id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.1.1.3" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.1.1.3.cmml">t</mi></mrow><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.2.3" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.3a.cmml">,</mo><mrow id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.2.2" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.2.2.cmml"><mi id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.2.2.2" mathvariant="normal" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.2.2.2.cmml">Δ</mi><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.2.2.1" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.2.2.1.cmml">=</mo><mn id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.2.2.3" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.1.2.2.3.cmml">1</mn></mrow></mrow><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.1.1.3" stretchy="false" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.2" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.2.cmml">⁢</mo><msub id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.3" xref="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.3.cmml"><mi id="S2.E6.m1.2.2.2.2.1.1.6.1.1.1.1.1.3.2" 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xref="S2.E6.m1.2.2.3">otherwise</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m1.5c">\forall j\in\{1,\dots,J\}:\begin{cases}m_{j,1}(W,\beta)=\mathbb{E}[(\mathbbm{1% }(Y\leq t)-\Lambda(X^{\top}\beta))g_{j}(X)]\geq 0\\ m_{j,2}(W,\beta)=\mathbb{E}[(\Lambda(X^{\top}\beta)-\mathbbm{1}(Y\leq t,\Delta% =1))g_{j}(X)]\geq 0.\end{cases}</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m1.5d">∀ italic_j ∈ { 1 , … , italic_J } : { start_ROW start_CELL italic_m start_POSTSUBSCRIPT italic_j , 1 end_POSTSUBSCRIPT ( italic_W , italic_β ) = blackboard_E [ ( blackboard_1 ( italic_Y ≤ italic_t ) - roman_Λ ( italic_X start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT italic_β ) ) italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_X ) ] ≥ 0 end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_m start_POSTSUBSCRIPT italic_j , 2 end_POSTSUBSCRIPT ( italic_W , italic_β ) = blackboard_E [ ( roman_Λ ( italic_X start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT italic_β ) - blackboard_1 ( italic_Y ≤ italic_t , roman_Δ = 1 ) ) italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_X ) ] ≥ 0 . end_CELL start_CELL end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p1.15">Intuitively, the idea behind this transformation can be explained as follows. Expression (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E5" title="In 2.3 Unconditional moment restrictions ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">5</span></a>) defines two conditional moment inequalities, which can be viewed as two inequalities that should hold for each <math alttext="x\in\mathcal{X}" class="ltx_Math" display="inline" id="S2.SS3.p1.3.m1.1"><semantics id="S2.SS3.p1.3.m1.1a"><mrow id="S2.SS3.p1.3.m1.1.1" xref="S2.SS3.p1.3.m1.1.1.cmml"><mi id="S2.SS3.p1.3.m1.1.1.2" xref="S2.SS3.p1.3.m1.1.1.2.cmml">x</mi><mo id="S2.SS3.p1.3.m1.1.1.1" xref="S2.SS3.p1.3.m1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.3.m1.1.1.3" xref="S2.SS3.p1.3.m1.1.1.3.cmml">𝒳</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.3.m1.1b"><apply id="S2.SS3.p1.3.m1.1.1.cmml" xref="S2.SS3.p1.3.m1.1.1"><in id="S2.SS3.p1.3.m1.1.1.1.cmml" xref="S2.SS3.p1.3.m1.1.1.1"></in><ci id="S2.SS3.p1.3.m1.1.1.2.cmml" xref="S2.SS3.p1.3.m1.1.1.2">𝑥</ci><ci id="S2.SS3.p1.3.m1.1.1.3.cmml" xref="S2.SS3.p1.3.m1.1.1.3">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.3.m1.1c">x\in\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.3.m1.1d">italic_x ∈ caligraphic_X</annotation></semantics></math>. As already mentioned, this is problematic when <math alttext="\mathcal{X}" class="ltx_Math" display="inline" id="S2.SS3.p1.4.m2.1"><semantics id="S2.SS3.p1.4.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.4.m2.1.1" xref="S2.SS3.p1.4.m2.1.1.cmml">𝒳</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.4.m2.1b"><ci id="S2.SS3.p1.4.m2.1.1.cmml" xref="S2.SS3.p1.4.m2.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.4.m2.1c">\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.4.m2.1d">caligraphic_X</annotation></semantics></math> is an infinite set. The class <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S2.SS3.p1.5.m3.1"><semantics id="S2.SS3.p1.5.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.5.m3.1.1" xref="S2.SS3.p1.5.m3.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.5.m3.1b"><ci id="S2.SS3.p1.5.m3.1.1.cmml" xref="S2.SS3.p1.5.m3.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.5.m3.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.5.m3.1d">caligraphic_G</annotation></semantics></math> contains functions <math alttext="g:\mathcal{X}\to\mathbb{R}_{\geq 0}" class="ltx_Math" display="inline" id="S2.SS3.p1.6.m4.1"><semantics id="S2.SS3.p1.6.m4.1a"><mrow id="S2.SS3.p1.6.m4.1.1" xref="S2.SS3.p1.6.m4.1.1.cmml"><mi id="S2.SS3.p1.6.m4.1.1.2" xref="S2.SS3.p1.6.m4.1.1.2.cmml">g</mi><mo id="S2.SS3.p1.6.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS3.p1.6.m4.1.1.1.cmml">:</mo><mrow id="S2.SS3.p1.6.m4.1.1.3" xref="S2.SS3.p1.6.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.6.m4.1.1.3.2" xref="S2.SS3.p1.6.m4.1.1.3.2.cmml">𝒳</mi><mo id="S2.SS3.p1.6.m4.1.1.3.1" stretchy="false" xref="S2.SS3.p1.6.m4.1.1.3.1.cmml">→</mo><msub id="S2.SS3.p1.6.m4.1.1.3.3" xref="S2.SS3.p1.6.m4.1.1.3.3.cmml"><mi id="S2.SS3.p1.6.m4.1.1.3.3.2" xref="S2.SS3.p1.6.m4.1.1.3.3.2.cmml">ℝ</mi><mrow id="S2.SS3.p1.6.m4.1.1.3.3.3" xref="S2.SS3.p1.6.m4.1.1.3.3.3.cmml"><mi id="S2.SS3.p1.6.m4.1.1.3.3.3.2" xref="S2.SS3.p1.6.m4.1.1.3.3.3.2.cmml"></mi><mo id="S2.SS3.p1.6.m4.1.1.3.3.3.1" xref="S2.SS3.p1.6.m4.1.1.3.3.3.1.cmml">≥</mo><mn id="S2.SS3.p1.6.m4.1.1.3.3.3.3" xref="S2.SS3.p1.6.m4.1.1.3.3.3.3.cmml">0</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.6.m4.1b"><apply id="S2.SS3.p1.6.m4.1.1.cmml" xref="S2.SS3.p1.6.m4.1.1"><ci id="S2.SS3.p1.6.m4.1.1.1.cmml" xref="S2.SS3.p1.6.m4.1.1.1">:</ci><ci id="S2.SS3.p1.6.m4.1.1.2.cmml" xref="S2.SS3.p1.6.m4.1.1.2">𝑔</ci><apply id="S2.SS3.p1.6.m4.1.1.3.cmml" xref="S2.SS3.p1.6.m4.1.1.3"><ci id="S2.SS3.p1.6.m4.1.1.3.1.cmml" xref="S2.SS3.p1.6.m4.1.1.3.1">→</ci><ci id="S2.SS3.p1.6.m4.1.1.3.2.cmml" xref="S2.SS3.p1.6.m4.1.1.3.2">𝒳</ci><apply id="S2.SS3.p1.6.m4.1.1.3.3.cmml" xref="S2.SS3.p1.6.m4.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS3.p1.6.m4.1.1.3.3.1.cmml" xref="S2.SS3.p1.6.m4.1.1.3.3">subscript</csymbol><ci id="S2.SS3.p1.6.m4.1.1.3.3.2.cmml" xref="S2.SS3.p1.6.m4.1.1.3.3.2">ℝ</ci><apply id="S2.SS3.p1.6.m4.1.1.3.3.3.cmml" xref="S2.SS3.p1.6.m4.1.1.3.3.3"><geq id="S2.SS3.p1.6.m4.1.1.3.3.3.1.cmml" xref="S2.SS3.p1.6.m4.1.1.3.3.3.1"></geq><csymbol cd="latexml" id="S2.SS3.p1.6.m4.1.1.3.3.3.2.cmml" xref="S2.SS3.p1.6.m4.1.1.3.3.3.2">absent</csymbol><cn id="S2.SS3.p1.6.m4.1.1.3.3.3.3.cmml" type="integer" xref="S2.SS3.p1.6.m4.1.1.3.3.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.6.m4.1c">g:\mathcal{X}\to\mathbb{R}_{\geq 0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.6.m4.1d">italic_g : caligraphic_X → blackboard_R start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT</annotation></semantics></math> that are only non-zero on <math alttext="\mathcal{X}_{g}" class="ltx_Math" display="inline" id="S2.SS3.p1.7.m5.1"><semantics id="S2.SS3.p1.7.m5.1a"><msub id="S2.SS3.p1.7.m5.1.1" xref="S2.SS3.p1.7.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.7.m5.1.1.2" xref="S2.SS3.p1.7.m5.1.1.2.cmml">𝒳</mi><mi id="S2.SS3.p1.7.m5.1.1.3" xref="S2.SS3.p1.7.m5.1.1.3.cmml">g</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.7.m5.1b"><apply id="S2.SS3.p1.7.m5.1.1.cmml" xref="S2.SS3.p1.7.m5.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.7.m5.1.1.1.cmml" xref="S2.SS3.p1.7.m5.1.1">subscript</csymbol><ci id="S2.SS3.p1.7.m5.1.1.2.cmml" xref="S2.SS3.p1.7.m5.1.1.2">𝒳</ci><ci id="S2.SS3.p1.7.m5.1.1.3.cmml" xref="S2.SS3.p1.7.m5.1.1.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.7.m5.1c">\mathcal{X}_{g}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.7.m5.1d">caligraphic_X start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="\mathcal{X}_{g}" class="ltx_Math" display="inline" id="S2.SS3.p1.8.m6.1"><semantics id="S2.SS3.p1.8.m6.1a"><msub id="S2.SS3.p1.8.m6.1.1" xref="S2.SS3.p1.8.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.8.m6.1.1.2" xref="S2.SS3.p1.8.m6.1.1.2.cmml">𝒳</mi><mi id="S2.SS3.p1.8.m6.1.1.3" xref="S2.SS3.p1.8.m6.1.1.3.cmml">g</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.8.m6.1b"><apply id="S2.SS3.p1.8.m6.1.1.cmml" xref="S2.SS3.p1.8.m6.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.8.m6.1.1.1.cmml" xref="S2.SS3.p1.8.m6.1.1">subscript</csymbol><ci id="S2.SS3.p1.8.m6.1.1.2.cmml" xref="S2.SS3.p1.8.m6.1.1.2">𝒳</ci><ci id="S2.SS3.p1.8.m6.1.1.3.cmml" xref="S2.SS3.p1.8.m6.1.1.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.8.m6.1c">\mathcal{X}_{g}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.8.m6.1d">caligraphic_X start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT</annotation></semantics></math> is a small region inside <math alttext="\mathcal{X}" class="ltx_Math" display="inline" id="S2.SS3.p1.9.m7.1"><semantics id="S2.SS3.p1.9.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.9.m7.1.1" xref="S2.SS3.p1.9.m7.1.1.cmml">𝒳</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.9.m7.1b"><ci id="S2.SS3.p1.9.m7.1.1.cmml" xref="S2.SS3.p1.9.m7.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.9.m7.1c">\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.9.m7.1d">caligraphic_X</annotation></semantics></math> such that <math alttext="\mathcal{X}\subseteq\bigcup_{g\in\mathcal{G}}\mathcal{X}_{g}" class="ltx_Math" display="inline" id="S2.SS3.p1.10.m8.1"><semantics id="S2.SS3.p1.10.m8.1a"><mrow id="S2.SS3.p1.10.m8.1.1" xref="S2.SS3.p1.10.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.10.m8.1.1.2" xref="S2.SS3.p1.10.m8.1.1.2.cmml">𝒳</mi><mo id="S2.SS3.p1.10.m8.1.1.1" rspace="0.111em" xref="S2.SS3.p1.10.m8.1.1.1.cmml">⊆</mo><mrow id="S2.SS3.p1.10.m8.1.1.3" xref="S2.SS3.p1.10.m8.1.1.3.cmml"><msub id="S2.SS3.p1.10.m8.1.1.3.1" xref="S2.SS3.p1.10.m8.1.1.3.1.cmml"><mo id="S2.SS3.p1.10.m8.1.1.3.1.2" xref="S2.SS3.p1.10.m8.1.1.3.1.2.cmml">⋃</mo><mrow id="S2.SS3.p1.10.m8.1.1.3.1.3" xref="S2.SS3.p1.10.m8.1.1.3.1.3.cmml"><mi id="S2.SS3.p1.10.m8.1.1.3.1.3.2" xref="S2.SS3.p1.10.m8.1.1.3.1.3.2.cmml">g</mi><mo id="S2.SS3.p1.10.m8.1.1.3.1.3.1" xref="S2.SS3.p1.10.m8.1.1.3.1.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.10.m8.1.1.3.1.3.3" xref="S2.SS3.p1.10.m8.1.1.3.1.3.3.cmml">𝒢</mi></mrow></msub><msub id="S2.SS3.p1.10.m8.1.1.3.2" xref="S2.SS3.p1.10.m8.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.10.m8.1.1.3.2.2" xref="S2.SS3.p1.10.m8.1.1.3.2.2.cmml">𝒳</mi><mi id="S2.SS3.p1.10.m8.1.1.3.2.3" xref="S2.SS3.p1.10.m8.1.1.3.2.3.cmml">g</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.10.m8.1b"><apply id="S2.SS3.p1.10.m8.1.1.cmml" xref="S2.SS3.p1.10.m8.1.1"><subset id="S2.SS3.p1.10.m8.1.1.1.cmml" xref="S2.SS3.p1.10.m8.1.1.1"></subset><ci id="S2.SS3.p1.10.m8.1.1.2.cmml" xref="S2.SS3.p1.10.m8.1.1.2">𝒳</ci><apply id="S2.SS3.p1.10.m8.1.1.3.cmml" xref="S2.SS3.p1.10.m8.1.1.3"><apply id="S2.SS3.p1.10.m8.1.1.3.1.cmml" xref="S2.SS3.p1.10.m8.1.1.3.1"><csymbol cd="ambiguous" id="S2.SS3.p1.10.m8.1.1.3.1.1.cmml" xref="S2.SS3.p1.10.m8.1.1.3.1">subscript</csymbol><union id="S2.SS3.p1.10.m8.1.1.3.1.2.cmml" xref="S2.SS3.p1.10.m8.1.1.3.1.2"></union><apply id="S2.SS3.p1.10.m8.1.1.3.1.3.cmml" xref="S2.SS3.p1.10.m8.1.1.3.1.3"><in id="S2.SS3.p1.10.m8.1.1.3.1.3.1.cmml" xref="S2.SS3.p1.10.m8.1.1.3.1.3.1"></in><ci id="S2.SS3.p1.10.m8.1.1.3.1.3.2.cmml" xref="S2.SS3.p1.10.m8.1.1.3.1.3.2">𝑔</ci><ci id="S2.SS3.p1.10.m8.1.1.3.1.3.3.cmml" xref="S2.SS3.p1.10.m8.1.1.3.1.3.3">𝒢</ci></apply></apply><apply id="S2.SS3.p1.10.m8.1.1.3.2.cmml" xref="S2.SS3.p1.10.m8.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS3.p1.10.m8.1.1.3.2.1.cmml" xref="S2.SS3.p1.10.m8.1.1.3.2">subscript</csymbol><ci id="S2.SS3.p1.10.m8.1.1.3.2.2.cmml" xref="S2.SS3.p1.10.m8.1.1.3.2.2">𝒳</ci><ci id="S2.SS3.p1.10.m8.1.1.3.2.3.cmml" xref="S2.SS3.p1.10.m8.1.1.3.2.3">𝑔</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.10.m8.1c">\mathcal{X}\subseteq\bigcup_{g\in\mathcal{G}}\mathcal{X}_{g}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.10.m8.1d">caligraphic_X ⊆ ⋃ start_POSTSUBSCRIPT italic_g ∈ caligraphic_G end_POSTSUBSCRIPT caligraphic_X start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT</annotation></semantics></math>. The transformed unconditional moment restrictions thus only require that the inequalities are expected to hold on each of the regions <math alttext="\mathcal{X}_{g}" class="ltx_Math" display="inline" id="S2.SS3.p1.11.m9.1"><semantics id="S2.SS3.p1.11.m9.1a"><msub id="S2.SS3.p1.11.m9.1.1" xref="S2.SS3.p1.11.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.11.m9.1.1.2" xref="S2.SS3.p1.11.m9.1.1.2.cmml">𝒳</mi><mi id="S2.SS3.p1.11.m9.1.1.3" xref="S2.SS3.p1.11.m9.1.1.3.cmml">g</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.11.m9.1b"><apply id="S2.SS3.p1.11.m9.1.1.cmml" xref="S2.SS3.p1.11.m9.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.11.m9.1.1.1.cmml" xref="S2.SS3.p1.11.m9.1.1">subscript</csymbol><ci id="S2.SS3.p1.11.m9.1.1.2.cmml" xref="S2.SS3.p1.11.m9.1.1.2">𝒳</ci><ci id="S2.SS3.p1.11.m9.1.1.3.cmml" xref="S2.SS3.p1.11.m9.1.1.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.11.m9.1c">\mathcal{X}_{g}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.11.m9.1d">caligraphic_X start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT</annotation></semantics></math> (instead of for all <math alttext="x\in\mathcal{X}" class="ltx_Math" display="inline" id="S2.SS3.p1.12.m10.1"><semantics id="S2.SS3.p1.12.m10.1a"><mrow id="S2.SS3.p1.12.m10.1.1" xref="S2.SS3.p1.12.m10.1.1.cmml"><mi id="S2.SS3.p1.12.m10.1.1.2" xref="S2.SS3.p1.12.m10.1.1.2.cmml">x</mi><mo id="S2.SS3.p1.12.m10.1.1.1" xref="S2.SS3.p1.12.m10.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.12.m10.1.1.3" xref="S2.SS3.p1.12.m10.1.1.3.cmml">𝒳</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.12.m10.1b"><apply id="S2.SS3.p1.12.m10.1.1.cmml" xref="S2.SS3.p1.12.m10.1.1"><in id="S2.SS3.p1.12.m10.1.1.1.cmml" xref="S2.SS3.p1.12.m10.1.1.1"></in><ci id="S2.SS3.p1.12.m10.1.1.2.cmml" xref="S2.SS3.p1.12.m10.1.1.2">𝑥</ci><ci id="S2.SS3.p1.12.m10.1.1.3.cmml" xref="S2.SS3.p1.12.m10.1.1.3">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.12.m10.1c">x\in\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.12.m10.1d">italic_x ∈ caligraphic_X</annotation></semantics></math>), essentially discretizing the possible continuum of restrictions in (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E5" title="In 2.3 Unconditional moment restrictions ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">5</span></a>). Also note that if <math alttext="X" class="ltx_Math" display="inline" id="S2.SS3.p1.13.m11.1"><semantics id="S2.SS3.p1.13.m11.1a"><mi id="S2.SS3.p1.13.m11.1.1" xref="S2.SS3.p1.13.m11.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.13.m11.1b"><ci id="S2.SS3.p1.13.m11.1.1.cmml" xref="S2.SS3.p1.13.m11.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.13.m11.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.13.m11.1d">italic_X</annotation></semantics></math> contains only discrete elements, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E5" title="In 2.3 Unconditional moment restrictions ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">5</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E6" title="In 2.3 Unconditional moment restrictions ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">6</span></a>) are equivalent when each instrumental function in <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S2.SS3.p1.14.m12.1"><semantics id="S2.SS3.p1.14.m12.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.14.m12.1.1" xref="S2.SS3.p1.14.m12.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.14.m12.1b"><ci id="S2.SS3.p1.14.m12.1.1.cmml" xref="S2.SS3.p1.14.m12.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.14.m12.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.14.m12.1d">caligraphic_G</annotation></semantics></math> is only non-zero for one level of <math alttext="X" class="ltx_Math" display="inline" id="S2.SS3.p1.15.m13.1"><semantics id="S2.SS3.p1.15.m13.1a"><mi id="S2.SS3.p1.15.m13.1.1" xref="S2.SS3.p1.15.m13.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.15.m13.1b"><ci id="S2.SS3.p1.15.m13.1.1.cmml" xref="S2.SS3.p1.15.m13.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.15.m13.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.15.m13.1d">italic_X</annotation></semantics></math>. We define</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S5.EGx2"> <tbody id="S2.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{B}_{I}" class="ltx_Math" display="inline" id="S2.E7.m1.1"><semantics id="S2.E7.m1.1a"><msub id="S2.E7.m1.1.1" xref="S2.E7.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E7.m1.1.1.2" xref="S2.E7.m1.1.1.2.cmml">ℬ</mi><mi id="S2.E7.m1.1.1.3" xref="S2.E7.m1.1.1.3.cmml">I</mi></msub><annotation-xml encoding="MathML-Content" id="S2.E7.m1.1b"><apply id="S2.E7.m1.1.1.cmml" xref="S2.E7.m1.1.1"><csymbol cd="ambiguous" id="S2.E7.m1.1.1.1.cmml" xref="S2.E7.m1.1.1">subscript</csymbol><ci id="S2.E7.m1.1.1.2.cmml" xref="S2.E7.m1.1.1.2">ℬ</ci><ci id="S2.E7.m1.1.1.3.cmml" xref="S2.E7.m1.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E7.m1.1c">\displaystyle\mathcal{B}_{I}</annotation><annotation encoding="application/x-llamapun" id="S2.E7.m1.1d">caligraphic_B start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\{\beta\in\mathcal{B}\mid\beta\text{ satisfies \eqref{eq: % unconditional moment inequalities}}\}," class="ltx_Math" display="inline" id="S2.E7.m2.1"><semantics id="S2.E7.m2.1a"><mrow id="S2.E7.m2.1.1.1" xref="S2.E7.m2.1.1.1.1.cmml"><mrow id="S2.E7.m2.1.1.1.1" xref="S2.E7.m2.1.1.1.1.cmml"><mi id="S2.E7.m2.1.1.1.1.4" xref="S2.E7.m2.1.1.1.1.4.cmml"></mi><mo id="S2.E7.m2.1.1.1.1.3" xref="S2.E7.m2.1.1.1.1.3.cmml">=</mo><mrow id="S2.E7.m2.1.1.1.1.2.2" xref="S2.E7.m2.1.1.1.1.2.3.cmml"><mo id="S2.E7.m2.1.1.1.1.2.2.3" stretchy="false" xref="S2.E7.m2.1.1.1.1.2.3.1.cmml">{</mo><mrow id="S2.E7.m2.1.1.1.1.1.1.1" xref="S2.E7.m2.1.1.1.1.1.1.1.cmml"><mi id="S2.E7.m2.1.1.1.1.1.1.1.2" xref="S2.E7.m2.1.1.1.1.1.1.1.2.cmml">β</mi><mo id="S2.E7.m2.1.1.1.1.1.1.1.1" xref="S2.E7.m2.1.1.1.1.1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.E7.m2.1.1.1.1.1.1.1.3" xref="S2.E7.m2.1.1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo fence="true" id="S2.E7.m2.1.1.1.1.2.2.4" lspace="0em" rspace="0em" xref="S2.E7.m2.1.1.1.1.2.3.1.cmml">∣</mo><mrow id="S2.E7.m2.1.1.1.1.2.2.2" xref="S2.E7.m2.1.1.1.1.2.2.2.cmml"><mi id="S2.E7.m2.1.1.1.1.2.2.2.2" xref="S2.E7.m2.1.1.1.1.2.2.2.2.cmml">β</mi><mo id="S2.E7.m2.1.1.1.1.2.2.2.1" xref="S2.E7.m2.1.1.1.1.2.2.2.1.cmml">⁢</mo><mrow id="S2.E7.m2.1.1.1.1.2.2.2.3" xref="S2.E7.m2.1.1.1.1.2.2.2.3f.cmml"><mtext id="S2.E7.m2.1.1.1.1.2.2.2.3a" xref="S2.E7.m2.1.1.1.1.2.2.2.3f.cmml"> satisfies (</mtext><mtext id="S2.E7.m2.1.1.1.1.2.2.2.3b" xref="S2.E7.m2.1.1.1.1.2.2.2.3f.cmml"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E6" title="In 2.3 Unconditional moment restrictions ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">6</span></a></mtext><mtext id="S2.E7.m2.1.1.1.1.2.2.2.3e" xref="S2.E7.m2.1.1.1.1.2.2.2.3f.cmml">)</mtext></mrow></mrow><mo id="S2.E7.m2.1.1.1.1.2.2.5" stretchy="false" xref="S2.E7.m2.1.1.1.1.2.3.1.cmml">}</mo></mrow></mrow><mo id="S2.E7.m2.1.1.1.2" xref="S2.E7.m2.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E7.m2.1b"><apply id="S2.E7.m2.1.1.1.1.cmml" xref="S2.E7.m2.1.1.1"><eq id="S2.E7.m2.1.1.1.1.3.cmml" xref="S2.E7.m2.1.1.1.1.3"></eq><csymbol cd="latexml" id="S2.E7.m2.1.1.1.1.4.cmml" xref="S2.E7.m2.1.1.1.1.4">absent</csymbol><apply id="S2.E7.m2.1.1.1.1.2.3.cmml" xref="S2.E7.m2.1.1.1.1.2.2"><csymbol cd="latexml" id="S2.E7.m2.1.1.1.1.2.3.1.cmml" xref="S2.E7.m2.1.1.1.1.2.2.3">conditional-set</csymbol><apply id="S2.E7.m2.1.1.1.1.1.1.1.cmml" xref="S2.E7.m2.1.1.1.1.1.1.1"><in id="S2.E7.m2.1.1.1.1.1.1.1.1.cmml" xref="S2.E7.m2.1.1.1.1.1.1.1.1"></in><ci id="S2.E7.m2.1.1.1.1.1.1.1.2.cmml" xref="S2.E7.m2.1.1.1.1.1.1.1.2">𝛽</ci><ci id="S2.E7.m2.1.1.1.1.1.1.1.3.cmml" xref="S2.E7.m2.1.1.1.1.1.1.1.3">ℬ</ci></apply><apply id="S2.E7.m2.1.1.1.1.2.2.2.cmml" xref="S2.E7.m2.1.1.1.1.2.2.2"><times id="S2.E7.m2.1.1.1.1.2.2.2.1.cmml" xref="S2.E7.m2.1.1.1.1.2.2.2.1"></times><ci id="S2.E7.m2.1.1.1.1.2.2.2.2.cmml" xref="S2.E7.m2.1.1.1.1.2.2.2.2">𝛽</ci><ci id="S2.E7.m2.1.1.1.1.2.2.2.3f.cmml" xref="S2.E7.m2.1.1.1.1.2.2.2.3"><mrow id="S2.E7.m2.1.1.1.1.2.2.2.3.cmml" xref="S2.E7.m2.1.1.1.1.2.2.2.3"><mtext id="S2.E7.m2.1.1.1.1.2.2.2.3a.cmml" xref="S2.E7.m2.1.1.1.1.2.2.2.3"> satisfies (</mtext><mtext id="S2.E7.m2.1.1.1.1.2.2.2.3b.cmml" xref="S2.E7.m2.1.1.1.1.2.2.2.3"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E6" title="In 2.3 Unconditional moment restrictions ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">6</span></a></mtext><mtext id="S2.E7.m2.1.1.1.1.2.2.2.3e.cmml" xref="S2.E7.m2.1.1.1.1.2.2.2.3">)</mtext></mrow></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E7.m2.1c">\displaystyle=\{\beta\in\mathcal{B}\mid\beta\text{ satisfies \eqref{eq: % unconditional moment inequalities}}\},</annotation><annotation encoding="application/x-llamapun" id="S2.E7.m2.1d">= { italic_β ∈ caligraphic_B ∣ italic_β satisfies ( ) } ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> <tbody id="S2.E8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{B}_{I,k}" class="ltx_Math" display="inline" id="S2.E8.m1.2"><semantics id="S2.E8.m1.2a"><msub id="S2.E8.m1.2.3" xref="S2.E8.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E8.m1.2.3.2" xref="S2.E8.m1.2.3.2.cmml">ℬ</mi><mrow id="S2.E8.m1.2.2.2.4" xref="S2.E8.m1.2.2.2.3.cmml"><mi id="S2.E8.m1.1.1.1.1" xref="S2.E8.m1.1.1.1.1.cmml">I</mi><mo id="S2.E8.m1.2.2.2.4.1" xref="S2.E8.m1.2.2.2.3.cmml">,</mo><mi id="S2.E8.m1.2.2.2.2" xref="S2.E8.m1.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.E8.m1.2b"><apply id="S2.E8.m1.2.3.cmml" xref="S2.E8.m1.2.3"><csymbol cd="ambiguous" id="S2.E8.m1.2.3.1.cmml" xref="S2.E8.m1.2.3">subscript</csymbol><ci id="S2.E8.m1.2.3.2.cmml" xref="S2.E8.m1.2.3.2">ℬ</ci><list id="S2.E8.m1.2.2.2.3.cmml" xref="S2.E8.m1.2.2.2.4"><ci id="S2.E8.m1.1.1.1.1.cmml" xref="S2.E8.m1.1.1.1.1">𝐼</ci><ci id="S2.E8.m1.2.2.2.2.cmml" xref="S2.E8.m1.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.m1.2c">\displaystyle\mathcal{B}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S2.E8.m1.2d">caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\{r\in\mathcal{B}_{k}\mid\exists\beta\in\mathcal{B}:\beta_{k}=r% \text{ and }\beta\text{ satisfies \eqref{eq: unconditional moment inequalities% }}\}." class="ltx_Math" display="inline" id="S2.E8.m2.1"><semantics id="S2.E8.m2.1a"><mrow id="S2.E8.m2.1.1.1" xref="S2.E8.m2.1.1.1.1.cmml"><mrow id="S2.E8.m2.1.1.1.1" xref="S2.E8.m2.1.1.1.1.cmml"><mi id="S2.E8.m2.1.1.1.1.4" xref="S2.E8.m2.1.1.1.1.4.cmml"></mi><mo id="S2.E8.m2.1.1.1.1.3" xref="S2.E8.m2.1.1.1.1.3.cmml">=</mo><mrow id="S2.E8.m2.1.1.1.1.2.2" xref="S2.E8.m2.1.1.1.1.2.3.cmml"><mo id="S2.E8.m2.1.1.1.1.2.2.3" stretchy="false" xref="S2.E8.m2.1.1.1.1.2.3.1.cmml">{</mo><mrow id="S2.E8.m2.1.1.1.1.1.1.1" xref="S2.E8.m2.1.1.1.1.1.1.1.cmml"><mi id="S2.E8.m2.1.1.1.1.1.1.1.2" xref="S2.E8.m2.1.1.1.1.1.1.1.2.cmml">r</mi><mo id="S2.E8.m2.1.1.1.1.1.1.1.1" xref="S2.E8.m2.1.1.1.1.1.1.1.1.cmml">∈</mo><msub id="S2.E8.m2.1.1.1.1.1.1.1.3" xref="S2.E8.m2.1.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E8.m2.1.1.1.1.1.1.1.3.2" xref="S2.E8.m2.1.1.1.1.1.1.1.3.2.cmml">ℬ</mi><mi id="S2.E8.m2.1.1.1.1.1.1.1.3.3" xref="S2.E8.m2.1.1.1.1.1.1.1.3.3.cmml">k</mi></msub></mrow><mo fence="true" id="S2.E8.m2.1.1.1.1.2.2.4" lspace="0em" rspace="0.167em" xref="S2.E8.m2.1.1.1.1.2.3.1.cmml">∣</mo><mrow id="S2.E8.m2.1.1.1.1.2.2.2" xref="S2.E8.m2.1.1.1.1.2.2.2.cmml"><mrow id="S2.E8.m2.1.1.1.1.2.2.2.2" xref="S2.E8.m2.1.1.1.1.2.2.2.2.cmml"><mrow id="S2.E8.m2.1.1.1.1.2.2.2.2.2" xref="S2.E8.m2.1.1.1.1.2.2.2.2.2.cmml"><mo id="S2.E8.m2.1.1.1.1.2.2.2.2.2.1" rspace="0.167em" xref="S2.E8.m2.1.1.1.1.2.2.2.2.2.1.cmml">∃</mo><mi id="S2.E8.m2.1.1.1.1.2.2.2.2.2.2" xref="S2.E8.m2.1.1.1.1.2.2.2.2.2.2.cmml">β</mi></mrow><mo id="S2.E8.m2.1.1.1.1.2.2.2.2.1" xref="S2.E8.m2.1.1.1.1.2.2.2.2.1.cmml">∈</mo><mi 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id="S2.E8.m2.1.1.1.1.2.2.2.3.3.5.cmml" xref="S2.E8.m2.1.1.1.1.2.2.2.3.3.5"><mtext id="S2.E8.m2.1.1.1.1.2.2.2.3.3.5a.cmml" xref="S2.E8.m2.1.1.1.1.2.2.2.3.3.5"> satisfies (</mtext><mtext id="S2.E8.m2.1.1.1.1.2.2.2.3.3.5b.cmml" xref="S2.E8.m2.1.1.1.1.2.2.2.3.3.5"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E6" title="In 2.3 Unconditional moment restrictions ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">6</span></a></mtext><mtext id="S2.E8.m2.1.1.1.1.2.2.2.3.3.5e.cmml" xref="S2.E8.m2.1.1.1.1.2.2.2.3.3.5">)</mtext></mrow></ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.m2.1c">\displaystyle=\{r\in\mathcal{B}_{k}\mid\exists\beta\in\mathcal{B}:\beta_{k}=r% \text{ and }\beta\text{ satisfies \eqref{eq: unconditional moment inequalities% }}\}.</annotation><annotation encoding="application/x-llamapun" id="S2.E8.m2.1d">= { italic_r ∈ caligraphic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∣ ∃ italic_β ∈ caligraphic_B : italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_r and italic_β satisfies ( ) } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p1.17">In the process of transforming (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E5" title="In 2.3 Unconditional moment restrictions ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">5</span></a>) to (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E6" title="In 2.3 Unconditional moment restrictions ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">6</span></a>) in the presence of continuous covariates, some information is lost. As a consequence, the condition in Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E8" title="In 2.3 Unconditional moment restrictions ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">8</span></a>) is weaker than the one in (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E4" title="In 2.2 Methodology ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">4</span></a>) and therefore <math alttext="\mathcal{B}_{I,k}" class="ltx_Math" display="inline" id="S2.SS3.p1.16.m1.2"><semantics id="S2.SS3.p1.16.m1.2a"><msub id="S2.SS3.p1.16.m1.2.3" xref="S2.SS3.p1.16.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.16.m1.2.3.2" xref="S2.SS3.p1.16.m1.2.3.2.cmml">ℬ</mi><mrow id="S2.SS3.p1.16.m1.2.2.2.4" xref="S2.SS3.p1.16.m1.2.2.2.3.cmml"><mi id="S2.SS3.p1.16.m1.1.1.1.1" xref="S2.SS3.p1.16.m1.1.1.1.1.cmml">I</mi><mo id="S2.SS3.p1.16.m1.2.2.2.4.1" xref="S2.SS3.p1.16.m1.2.2.2.3.cmml">,</mo><mi id="S2.SS3.p1.16.m1.2.2.2.2" xref="S2.SS3.p1.16.m1.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.16.m1.2b"><apply id="S2.SS3.p1.16.m1.2.3.cmml" xref="S2.SS3.p1.16.m1.2.3"><csymbol cd="ambiguous" id="S2.SS3.p1.16.m1.2.3.1.cmml" xref="S2.SS3.p1.16.m1.2.3">subscript</csymbol><ci id="S2.SS3.p1.16.m1.2.3.2.cmml" xref="S2.SS3.p1.16.m1.2.3.2">ℬ</ci><list id="S2.SS3.p1.16.m1.2.2.2.3.cmml" xref="S2.SS3.p1.16.m1.2.2.2.4"><ci id="S2.SS3.p1.16.m1.1.1.1.1.cmml" xref="S2.SS3.p1.16.m1.1.1.1.1">𝐼</ci><ci id="S2.SS3.p1.16.m1.2.2.2.2.cmml" xref="S2.SS3.p1.16.m1.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.16.m1.2c">\mathcal{B}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.16.m1.2d">caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> will be a superset of <math alttext="\mathcal{B}_{I,k}^{*}" class="ltx_Math" display="inline" id="S2.SS3.p1.17.m2.2"><semantics id="S2.SS3.p1.17.m2.2a"><msubsup id="S2.SS3.p1.17.m2.2.3" xref="S2.SS3.p1.17.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.17.m2.2.3.2.2" xref="S2.SS3.p1.17.m2.2.3.2.2.cmml">ℬ</mi><mrow id="S2.SS3.p1.17.m2.2.2.2.4" xref="S2.SS3.p1.17.m2.2.2.2.3.cmml"><mi id="S2.SS3.p1.17.m2.1.1.1.1" xref="S2.SS3.p1.17.m2.1.1.1.1.cmml">I</mi><mo id="S2.SS3.p1.17.m2.2.2.2.4.1" xref="S2.SS3.p1.17.m2.2.2.2.3.cmml">,</mo><mi id="S2.SS3.p1.17.m2.2.2.2.2" xref="S2.SS3.p1.17.m2.2.2.2.2.cmml">k</mi></mrow><mo id="S2.SS3.p1.17.m2.2.3.3" xref="S2.SS3.p1.17.m2.2.3.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.17.m2.2b"><apply id="S2.SS3.p1.17.m2.2.3.cmml" xref="S2.SS3.p1.17.m2.2.3"><csymbol cd="ambiguous" id="S2.SS3.p1.17.m2.2.3.1.cmml" xref="S2.SS3.p1.17.m2.2.3">superscript</csymbol><apply id="S2.SS3.p1.17.m2.2.3.2.cmml" xref="S2.SS3.p1.17.m2.2.3"><csymbol cd="ambiguous" id="S2.SS3.p1.17.m2.2.3.2.1.cmml" xref="S2.SS3.p1.17.m2.2.3">subscript</csymbol><ci id="S2.SS3.p1.17.m2.2.3.2.2.cmml" xref="S2.SS3.p1.17.m2.2.3.2.2">ℬ</ci><list id="S2.SS3.p1.17.m2.2.2.2.3.cmml" xref="S2.SS3.p1.17.m2.2.2.2.4"><ci id="S2.SS3.p1.17.m2.1.1.1.1.cmml" xref="S2.SS3.p1.17.m2.1.1.1.1">𝐼</ci><ci id="S2.SS3.p1.17.m2.2.2.2.2.cmml" xref="S2.SS3.p1.17.m2.2.2.2.2">𝑘</ci></list></apply><times id="S2.SS3.p1.17.m2.2.3.3.cmml" xref="S2.SS3.p1.17.m2.2.3.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.17.m2.2c">\mathcal{B}_{I,k}^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.17.m2.2d">caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>. The simulations show, however, that the information loss is limited, and data applications illustrate that inference based on the unconditional moment restrictions can still be informative. For a more detailed explanation of instrumental functions, we refer to Section 3.3 of <cite class="ltx_cite ltx_citemacro_cite">Andrews and Shi, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib1" title="">2013</a>)</cite>.</p> </div> </section> <section class="ltx_subsection" id="S2.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.4 </span>Testing procedure</h3> <div class="ltx_para" id="S2.SS4.p1"> <p class="ltx_p" id="S2.SS4.p1.1">To estimate <math alttext="\mathcal{B}_{I,k}" class="ltx_Math" display="inline" id="S2.SS4.p1.1.m1.2"><semantics id="S2.SS4.p1.1.m1.2a"><msub id="S2.SS4.p1.1.m1.2.3" xref="S2.SS4.p1.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.p1.1.m1.2.3.2" xref="S2.SS4.p1.1.m1.2.3.2.cmml">ℬ</mi><mrow id="S2.SS4.p1.1.m1.2.2.2.4" xref="S2.SS4.p1.1.m1.2.2.2.3.cmml"><mi id="S2.SS4.p1.1.m1.1.1.1.1" xref="S2.SS4.p1.1.m1.1.1.1.1.cmml">I</mi><mo id="S2.SS4.p1.1.m1.2.2.2.4.1" xref="S2.SS4.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="S2.SS4.p1.1.m1.2.2.2.2" xref="S2.SS4.p1.1.m1.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.1.m1.2b"><apply id="S2.SS4.p1.1.m1.2.3.cmml" xref="S2.SS4.p1.1.m1.2.3"><csymbol cd="ambiguous" id="S2.SS4.p1.1.m1.2.3.1.cmml" xref="S2.SS4.p1.1.m1.2.3">subscript</csymbol><ci id="S2.SS4.p1.1.m1.2.3.2.cmml" xref="S2.SS4.p1.1.m1.2.3.2">ℬ</ci><list id="S2.SS4.p1.1.m1.2.2.2.3.cmml" xref="S2.SS4.p1.1.m1.2.2.2.4"><ci id="S2.SS4.p1.1.m1.1.1.1.1.cmml" xref="S2.SS4.p1.1.m1.1.1.1.1">𝐼</ci><ci id="S2.SS4.p1.1.m1.2.2.2.2.cmml" xref="S2.SS4.p1.1.m1.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.1.m1.2c">\mathcal{B}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.1.m1.2d">caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> defined in (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E8" title="In 2.3 Unconditional moment restrictions ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">8</span></a>) by means of test inversion we require a test for the hypothesis</p> <table class="ltx_equation ltx_eqn_table" id="S2.E9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{H}_{0}(r):\exists\beta\in\mathcal{B}:\beta_{k}=r\text{ and }\beta% \text{ satisfies \eqref{eq: unconditional moment inequalities}}." class="ltx_Math" display="block" id="S2.E9.m1.2"><semantics id="S2.E9.m1.2a"><mrow id="S2.E9.m1.2.2.1" xref="S2.E9.m1.2.2.1.1.cmml"><mrow id="S2.E9.m1.2.2.1.1" xref="S2.E9.m1.2.2.1.1.cmml"><mrow id="S2.E9.m1.2.2.1.1.2" xref="S2.E9.m1.2.2.1.1.2.cmml"><msub id="S2.E9.m1.2.2.1.1.2.2" xref="S2.E9.m1.2.2.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E9.m1.2.2.1.1.2.2.2" xref="S2.E9.m1.2.2.1.1.2.2.2.cmml">ℋ</mi><mn id="S2.E9.m1.2.2.1.1.2.2.3" xref="S2.E9.m1.2.2.1.1.2.2.3.cmml">0</mn></msub><mo id="S2.E9.m1.2.2.1.1.2.1" xref="S2.E9.m1.2.2.1.1.2.1.cmml">⁢</mo><mrow id="S2.E9.m1.2.2.1.1.2.3.2" xref="S2.E9.m1.2.2.1.1.2.cmml"><mo id="S2.E9.m1.2.2.1.1.2.3.2.1" stretchy="false" xref="S2.E9.m1.2.2.1.1.2.cmml">(</mo><mi id="S2.E9.m1.1.1" xref="S2.E9.m1.1.1.cmml">r</mi><mo id="S2.E9.m1.2.2.1.1.2.3.2.2" rspace="0.278em" stretchy="false" xref="S2.E9.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E9.m1.2.2.1.1.3" rspace="0.278em" xref="S2.E9.m1.2.2.1.1.3.cmml">:</mo><mrow id="S2.E9.m1.2.2.1.1.4" xref="S2.E9.m1.2.2.1.1.4.cmml"><mrow id="S2.E9.m1.2.2.1.1.4.2" xref="S2.E9.m1.2.2.1.1.4.2.cmml"><mo id="S2.E9.m1.2.2.1.1.4.2.1" rspace="0.167em" xref="S2.E9.m1.2.2.1.1.4.2.1.cmml">∃</mo><mi id="S2.E9.m1.2.2.1.1.4.2.2" xref="S2.E9.m1.2.2.1.1.4.2.2.cmml">β</mi></mrow><mo id="S2.E9.m1.2.2.1.1.4.1" xref="S2.E9.m1.2.2.1.1.4.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.E9.m1.2.2.1.1.4.3" xref="S2.E9.m1.2.2.1.1.4.3.cmml">ℬ</mi></mrow><mo id="S2.E9.m1.2.2.1.1.5" lspace="0.278em" rspace="0.278em" xref="S2.E9.m1.2.2.1.1.5.cmml">:</mo><mrow id="S2.E9.m1.2.2.1.1.6" xref="S2.E9.m1.2.2.1.1.6.cmml"><msub id="S2.E9.m1.2.2.1.1.6.2" xref="S2.E9.m1.2.2.1.1.6.2.cmml"><mi id="S2.E9.m1.2.2.1.1.6.2.2" xref="S2.E9.m1.2.2.1.1.6.2.2.cmml">β</mi><mi id="S2.E9.m1.2.2.1.1.6.2.3" xref="S2.E9.m1.2.2.1.1.6.2.3.cmml">k</mi></msub><mo id="S2.E9.m1.2.2.1.1.6.1" xref="S2.E9.m1.2.2.1.1.6.1.cmml">=</mo><mrow id="S2.E9.m1.2.2.1.1.6.3" xref="S2.E9.m1.2.2.1.1.6.3.cmml"><mi id="S2.E9.m1.2.2.1.1.6.3.2" xref="S2.E9.m1.2.2.1.1.6.3.2.cmml">r</mi><mo id="S2.E9.m1.2.2.1.1.6.3.1" xref="S2.E9.m1.2.2.1.1.6.3.1.cmml">⁢</mo><mtext id="S2.E9.m1.2.2.1.1.6.3.3" xref="S2.E9.m1.2.2.1.1.6.3.3a.cmml"> and </mtext><mo id="S2.E9.m1.2.2.1.1.6.3.1a" xref="S2.E9.m1.2.2.1.1.6.3.1.cmml">⁢</mo><mi id="S2.E9.m1.2.2.1.1.6.3.4" xref="S2.E9.m1.2.2.1.1.6.3.4.cmml">β</mi><mo id="S2.E9.m1.2.2.1.1.6.3.1b" xref="S2.E9.m1.2.2.1.1.6.3.1.cmml">⁢</mo><mrow id="S2.E9.m1.2.2.1.1.6.3.5" xref="S2.E9.m1.2.2.1.1.6.3.5f.cmml"><mtext id="S2.E9.m1.2.2.1.1.6.3.5a" xref="S2.E9.m1.2.2.1.1.6.3.5f.cmml"> satisfies (</mtext><mtext id="S2.E9.m1.2.2.1.1.6.3.5b" xref="S2.E9.m1.2.2.1.1.6.3.5f.cmml"><a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E6" title="In 2.3 Unconditional moment restrictions ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">6</span></a></mtext><mtext id="S2.E9.m1.2.2.1.1.6.3.5e" xref="S2.E9.m1.2.2.1.1.6.3.5f.cmml">)</mtext></mrow></mrow></mrow></mrow><mo id="S2.E9.m1.2.2.1.2" lspace="0em" xref="S2.E9.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E9.m1.2b"><apply id="S2.E9.m1.2.2.1.1.cmml" xref="S2.E9.m1.2.2.1"><and id="S2.E9.m1.2.2.1.1a.cmml" xref="S2.E9.m1.2.2.1"></and><apply id="S2.E9.m1.2.2.1.1b.cmml" xref="S2.E9.m1.2.2.1"><ci id="S2.E9.m1.2.2.1.1.3.cmml" xref="S2.E9.m1.2.2.1.1.3">:</ci><apply id="S2.E9.m1.2.2.1.1.2.cmml" xref="S2.E9.m1.2.2.1.1.2"><times id="S2.E9.m1.2.2.1.1.2.1.cmml" xref="S2.E9.m1.2.2.1.1.2.1"></times><apply id="S2.E9.m1.2.2.1.1.2.2.cmml" xref="S2.E9.m1.2.2.1.1.2.2"><csymbol cd="ambiguous" id="S2.E9.m1.2.2.1.1.2.2.1.cmml" xref="S2.E9.m1.2.2.1.1.2.2">subscript</csymbol><ci id="S2.E9.m1.2.2.1.1.2.2.2.cmml" xref="S2.E9.m1.2.2.1.1.2.2.2">ℋ</ci><cn id="S2.E9.m1.2.2.1.1.2.2.3.cmml" type="integer" xref="S2.E9.m1.2.2.1.1.2.2.3">0</cn></apply><ci id="S2.E9.m1.1.1.cmml" xref="S2.E9.m1.1.1">𝑟</ci></apply><apply id="S2.E9.m1.2.2.1.1.4.cmml" xref="S2.E9.m1.2.2.1.1.4"><in id="S2.E9.m1.2.2.1.1.4.1.cmml" xref="S2.E9.m1.2.2.1.1.4.1"></in><apply id="S2.E9.m1.2.2.1.1.4.2.cmml" xref="S2.E9.m1.2.2.1.1.4.2"><exists id="S2.E9.m1.2.2.1.1.4.2.1.cmml" xref="S2.E9.m1.2.2.1.1.4.2.1"></exists><ci id="S2.E9.m1.2.2.1.1.4.2.2.cmml" 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xref="S2.E9.m1.2.2.1.1.6.3.5">)</mtext></mrow></ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E9.m1.2c">\mathcal{H}_{0}(r):\exists\beta\in\mathcal{B}:\beta_{k}=r\text{ and }\beta% \text{ satisfies \eqref{eq: unconditional moment inequalities}}.</annotation><annotation encoding="application/x-llamapun" id="S2.E9.m1.2d">caligraphic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_r ) : ∃ italic_β ∈ caligraphic_B : italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_r and italic_β satisfies ( ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.p1.10">Such a test is described in <cite class="ltx_cite ltx_citemacro_cite">Bei, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib2" title="">2024</a>)</cite>, who uses the statistic <math alttext="T_{n}(r)=\inf_{\beta\in\mathcal{B}(r)}S(\sqrt{n}\bar{m}(\beta),\hat{\sigma}(% \beta))" class="ltx_Math" display="inline" id="S2.SS4.p1.2.m1.6"><semantics id="S2.SS4.p1.2.m1.6a"><mrow id="S2.SS4.p1.2.m1.6.6" xref="S2.SS4.p1.2.m1.6.6.cmml"><mrow id="S2.SS4.p1.2.m1.6.6.4" xref="S2.SS4.p1.2.m1.6.6.4.cmml"><msub id="S2.SS4.p1.2.m1.6.6.4.2" xref="S2.SS4.p1.2.m1.6.6.4.2.cmml"><mi id="S2.SS4.p1.2.m1.6.6.4.2.2" xref="S2.SS4.p1.2.m1.6.6.4.2.2.cmml">T</mi><mi id="S2.SS4.p1.2.m1.6.6.4.2.3" xref="S2.SS4.p1.2.m1.6.6.4.2.3.cmml">n</mi></msub><mo id="S2.SS4.p1.2.m1.6.6.4.1" xref="S2.SS4.p1.2.m1.6.6.4.1.cmml">⁢</mo><mrow id="S2.SS4.p1.2.m1.6.6.4.3.2" xref="S2.SS4.p1.2.m1.6.6.4.cmml"><mo id="S2.SS4.p1.2.m1.6.6.4.3.2.1" stretchy="false" xref="S2.SS4.p1.2.m1.6.6.4.cmml">(</mo><mi id="S2.SS4.p1.2.m1.2.2" xref="S2.SS4.p1.2.m1.2.2.cmml">r</mi><mo id="S2.SS4.p1.2.m1.6.6.4.3.2.2" 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xref="S2.SS4.p1.2.m1.6.6.2.2.2.2.2.2.1">^</ci><ci id="S2.SS4.p1.2.m1.6.6.2.2.2.2.2.2.2.cmml" xref="S2.SS4.p1.2.m1.6.6.2.2.2.2.2.2.2">𝜎</ci></apply><ci id="S2.SS4.p1.2.m1.4.4.cmml" xref="S2.SS4.p1.2.m1.4.4">𝛽</ci></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.2.m1.6c">T_{n}(r)=\inf_{\beta\in\mathcal{B}(r)}S(\sqrt{n}\bar{m}(\beta),\hat{\sigma}(% \beta))</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.2.m1.6d">italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_r ) = roman_inf start_POSTSUBSCRIPT italic_β ∈ caligraphic_B ( italic_r ) end_POSTSUBSCRIPT italic_S ( square-root start_ARG italic_n end_ARG over¯ start_ARG italic_m end_ARG ( italic_β ) , over^ start_ARG italic_σ end_ARG ( italic_β ) )</annotation></semantics></math>, where <math alttext="S(\sqrt{n}\bar{m}(\beta),\hat{\sigma}(\beta))=\sum_{q=1}^{2}\sum_{j=1}^{J}\max% \{-\sqrt{n}\bar{m}_{j,q}(\beta)/\hat{\sigma}_{j,q}(\beta),0\}^{2}" class="ltx_math_unparsed" display="inline" id="S2.SS4.p1.3.m2.10"><semantics id="S2.SS4.p1.3.m2.10a"><mrow id="S2.SS4.p1.3.m2.10b"><mi id="S2.SS4.p1.3.m2.10.11">S</mi><mrow id="S2.SS4.p1.3.m2.10.12"><mo id="S2.SS4.p1.3.m2.10.12.1" stretchy="false">(</mo><msqrt id="S2.SS4.p1.3.m2.10.12.2"><mi id="S2.SS4.p1.3.m2.10.12.2.2">n</mi></msqrt><mover accent="true" id="S2.SS4.p1.3.m2.10.12.3"><mi id="S2.SS4.p1.3.m2.10.12.3.2">m</mi><mo id="S2.SS4.p1.3.m2.10.12.3.1">¯</mo></mover><mrow id="S2.SS4.p1.3.m2.10.12.4"><mo id="S2.SS4.p1.3.m2.10.12.4.1" stretchy="false">(</mo><mi id="S2.SS4.p1.3.m2.5.5">β</mi><mo id="S2.SS4.p1.3.m2.10.12.4.2" stretchy="false">)</mo></mrow><mo id="S2.SS4.p1.3.m2.10.12.5">,</mo><mover accent="true" id="S2.SS4.p1.3.m2.10.12.6"><mi id="S2.SS4.p1.3.m2.10.12.6.2">σ</mi><mo id="S2.SS4.p1.3.m2.10.12.6.1">^</mo></mover><mrow id="S2.SS4.p1.3.m2.10.12.7"><mo id="S2.SS4.p1.3.m2.10.12.7.1" stretchy="false">(</mo><mi id="S2.SS4.p1.3.m2.6.6">β</mi><mo id="S2.SS4.p1.3.m2.10.12.7.2" stretchy="false">)</mo></mrow><mo id="S2.SS4.p1.3.m2.10.12.8" stretchy="false">)</mo></mrow><mo id="S2.SS4.p1.3.m2.10.13" rspace="0.111em">=</mo><msubsup id="S2.SS4.p1.3.m2.10.14"><mo id="S2.SS4.p1.3.m2.10.14.2.2" rspace="0em">∑</mo><mrow id="S2.SS4.p1.3.m2.10.14.2.3"><mi id="S2.SS4.p1.3.m2.10.14.2.3.2">q</mi><mo id="S2.SS4.p1.3.m2.10.14.2.3.1">=</mo><mn id="S2.SS4.p1.3.m2.10.14.2.3.3">1</mn></mrow><mn id="S2.SS4.p1.3.m2.10.14.3">2</mn></msubsup><msubsup id="S2.SS4.p1.3.m2.10.15"><mo id="S2.SS4.p1.3.m2.10.15.2.2">∑</mo><mrow id="S2.SS4.p1.3.m2.10.15.2.3"><mi id="S2.SS4.p1.3.m2.10.15.2.3.2">j</mi><mo id="S2.SS4.p1.3.m2.10.15.2.3.1">=</mo><mn id="S2.SS4.p1.3.m2.10.15.2.3.3">1</mn></mrow><mi id="S2.SS4.p1.3.m2.10.15.3">J</mi></msubsup><mi id="S2.SS4.p1.3.m2.9.9">max</mi><msup id="S2.SS4.p1.3.m2.10.16"><mrow id="S2.SS4.p1.3.m2.10.16.2"><mo id="S2.SS4.p1.3.m2.10.16.2.1" stretchy="false">{</mo><mo id="S2.SS4.p1.3.m2.10.16.2.2" lspace="0em">−</mo><msqrt id="S2.SS4.p1.3.m2.10.16.2.3"><mi id="S2.SS4.p1.3.m2.10.16.2.3.2">n</mi></msqrt><msub id="S2.SS4.p1.3.m2.10.16.2.4"><mover accent="true" id="S2.SS4.p1.3.m2.10.16.2.4.2"><mi id="S2.SS4.p1.3.m2.10.16.2.4.2.2">m</mi><mo id="S2.SS4.p1.3.m2.10.16.2.4.2.1">¯</mo></mover><mrow id="S2.SS4.p1.3.m2.2.2.2.4"><mi id="S2.SS4.p1.3.m2.1.1.1.1">j</mi><mo id="S2.SS4.p1.3.m2.2.2.2.4.1">,</mo><mi id="S2.SS4.p1.3.m2.2.2.2.2">q</mi></mrow></msub><mrow id="S2.SS4.p1.3.m2.10.16.2.5"><mo id="S2.SS4.p1.3.m2.10.16.2.5.1" stretchy="false">(</mo><mi id="S2.SS4.p1.3.m2.7.7">β</mi><mo id="S2.SS4.p1.3.m2.10.16.2.5.2" stretchy="false">)</mo></mrow><mo id="S2.SS4.p1.3.m2.10.16.2.6">/</mo><msub id="S2.SS4.p1.3.m2.10.16.2.7"><mover accent="true" id="S2.SS4.p1.3.m2.10.16.2.7.2"><mi id="S2.SS4.p1.3.m2.10.16.2.7.2.2">σ</mi><mo id="S2.SS4.p1.3.m2.10.16.2.7.2.1">^</mo></mover><mrow id="S2.SS4.p1.3.m2.4.4.2.4"><mi id="S2.SS4.p1.3.m2.3.3.1.1">j</mi><mo id="S2.SS4.p1.3.m2.4.4.2.4.1">,</mo><mi id="S2.SS4.p1.3.m2.4.4.2.2">q</mi></mrow></msub><mrow id="S2.SS4.p1.3.m2.10.16.2.8"><mo id="S2.SS4.p1.3.m2.10.16.2.8.1" stretchy="false">(</mo><mi id="S2.SS4.p1.3.m2.8.8">β</mi><mo id="S2.SS4.p1.3.m2.10.16.2.8.2" stretchy="false">)</mo></mrow><mo id="S2.SS4.p1.3.m2.10.16.2.9">,</mo><mn id="S2.SS4.p1.3.m2.10.10">0</mn><mo id="S2.SS4.p1.3.m2.10.16.2.10" stretchy="false">}</mo></mrow><mn id="S2.SS4.p1.3.m2.10.16.3">2</mn></msup></mrow><annotation encoding="application/x-tex" id="S2.SS4.p1.3.m2.10c">S(\sqrt{n}\bar{m}(\beta),\hat{\sigma}(\beta))=\sum_{q=1}^{2}\sum_{j=1}^{J}\max% \{-\sqrt{n}\bar{m}_{j,q}(\beta)/\hat{\sigma}_{j,q}(\beta),0\}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.3.m2.10d">italic_S ( square-root start_ARG italic_n end_ARG over¯ start_ARG italic_m end_ARG ( italic_β ) , over^ start_ARG italic_σ end_ARG ( italic_β ) ) = ∑ start_POSTSUBSCRIPT italic_q = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_J end_POSTSUPERSCRIPT roman_max { - square-root start_ARG italic_n end_ARG over¯ start_ARG italic_m end_ARG start_POSTSUBSCRIPT italic_j , italic_q end_POSTSUBSCRIPT ( italic_β ) / over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT italic_j , italic_q end_POSTSUBSCRIPT ( italic_β ) , 0 } start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. In this notation, we define the vector-valued function <math alttext="m=(m_{1,1},\dots,m_{J,1},m_{1,2}\dots,m_{J,2})" class="ltx_Math" display="inline" id="S2.SS4.p1.4.m3.13"><semantics id="S2.SS4.p1.4.m3.13a"><mrow id="S2.SS4.p1.4.m3.13.13" xref="S2.SS4.p1.4.m3.13.13.cmml"><mi id="S2.SS4.p1.4.m3.13.13.6" xref="S2.SS4.p1.4.m3.13.13.6.cmml">m</mi><mo id="S2.SS4.p1.4.m3.13.13.5" xref="S2.SS4.p1.4.m3.13.13.5.cmml">=</mo><mrow id="S2.SS4.p1.4.m3.13.13.4.4" xref="S2.SS4.p1.4.m3.13.13.4.5.cmml"><mo id="S2.SS4.p1.4.m3.13.13.4.4.5" stretchy="false" xref="S2.SS4.p1.4.m3.13.13.4.5.cmml">(</mo><msub id="S2.SS4.p1.4.m3.10.10.1.1.1" xref="S2.SS4.p1.4.m3.10.10.1.1.1.cmml"><mi id="S2.SS4.p1.4.m3.10.10.1.1.1.2" xref="S2.SS4.p1.4.m3.10.10.1.1.1.2.cmml">m</mi><mrow id="S2.SS4.p1.4.m3.2.2.2.4" xref="S2.SS4.p1.4.m3.2.2.2.3.cmml"><mn id="S2.SS4.p1.4.m3.1.1.1.1" xref="S2.SS4.p1.4.m3.1.1.1.1.cmml">1</mn><mo id="S2.SS4.p1.4.m3.2.2.2.4.1" xref="S2.SS4.p1.4.m3.2.2.2.3.cmml">,</mo><mn id="S2.SS4.p1.4.m3.2.2.2.2" xref="S2.SS4.p1.4.m3.2.2.2.2.cmml">1</mn></mrow></msub><mo id="S2.SS4.p1.4.m3.13.13.4.4.6" xref="S2.SS4.p1.4.m3.13.13.4.5.cmml">,</mo><mi id="S2.SS4.p1.4.m3.9.9" mathvariant="normal" xref="S2.SS4.p1.4.m3.9.9.cmml">…</mi><mo id="S2.SS4.p1.4.m3.13.13.4.4.7" xref="S2.SS4.p1.4.m3.13.13.4.5.cmml">,</mo><msub id="S2.SS4.p1.4.m3.11.11.2.2.2" xref="S2.SS4.p1.4.m3.11.11.2.2.2.cmml"><mi id="S2.SS4.p1.4.m3.11.11.2.2.2.2" xref="S2.SS4.p1.4.m3.11.11.2.2.2.2.cmml">m</mi><mrow id="S2.SS4.p1.4.m3.4.4.2.4" xref="S2.SS4.p1.4.m3.4.4.2.3.cmml"><mi id="S2.SS4.p1.4.m3.3.3.1.1" xref="S2.SS4.p1.4.m3.3.3.1.1.cmml">J</mi><mo id="S2.SS4.p1.4.m3.4.4.2.4.1" xref="S2.SS4.p1.4.m3.4.4.2.3.cmml">,</mo><mn id="S2.SS4.p1.4.m3.4.4.2.2" xref="S2.SS4.p1.4.m3.4.4.2.2.cmml">1</mn></mrow></msub><mo id="S2.SS4.p1.4.m3.13.13.4.4.8" xref="S2.SS4.p1.4.m3.13.13.4.5.cmml">,</mo><mrow id="S2.SS4.p1.4.m3.12.12.3.3.3" xref="S2.SS4.p1.4.m3.12.12.3.3.3.cmml"><msub id="S2.SS4.p1.4.m3.12.12.3.3.3.2" xref="S2.SS4.p1.4.m3.12.12.3.3.3.2.cmml"><mi id="S2.SS4.p1.4.m3.12.12.3.3.3.2.2" xref="S2.SS4.p1.4.m3.12.12.3.3.3.2.2.cmml">m</mi><mrow id="S2.SS4.p1.4.m3.6.6.2.4" xref="S2.SS4.p1.4.m3.6.6.2.3.cmml"><mn id="S2.SS4.p1.4.m3.5.5.1.1" xref="S2.SS4.p1.4.m3.5.5.1.1.cmml">1</mn><mo id="S2.SS4.p1.4.m3.6.6.2.4.1" xref="S2.SS4.p1.4.m3.6.6.2.3.cmml">,</mo><mn id="S2.SS4.p1.4.m3.6.6.2.2" xref="S2.SS4.p1.4.m3.6.6.2.2.cmml">2</mn></mrow></msub><mo id="S2.SS4.p1.4.m3.12.12.3.3.3.1" xref="S2.SS4.p1.4.m3.12.12.3.3.3.1.cmml">⁢</mo><mi id="S2.SS4.p1.4.m3.12.12.3.3.3.3" mathvariant="normal" xref="S2.SS4.p1.4.m3.12.12.3.3.3.3.cmml">…</mi></mrow><mo id="S2.SS4.p1.4.m3.13.13.4.4.9" xref="S2.SS4.p1.4.m3.13.13.4.5.cmml">,</mo><msub id="S2.SS4.p1.4.m3.13.13.4.4.4" xref="S2.SS4.p1.4.m3.13.13.4.4.4.cmml"><mi id="S2.SS4.p1.4.m3.13.13.4.4.4.2" xref="S2.SS4.p1.4.m3.13.13.4.4.4.2.cmml">m</mi><mrow id="S2.SS4.p1.4.m3.8.8.2.4" xref="S2.SS4.p1.4.m3.8.8.2.3.cmml"><mi id="S2.SS4.p1.4.m3.7.7.1.1" xref="S2.SS4.p1.4.m3.7.7.1.1.cmml">J</mi><mo id="S2.SS4.p1.4.m3.8.8.2.4.1" xref="S2.SS4.p1.4.m3.8.8.2.3.cmml">,</mo><mn id="S2.SS4.p1.4.m3.8.8.2.2" xref="S2.SS4.p1.4.m3.8.8.2.2.cmml">2</mn></mrow></msub><mo id="S2.SS4.p1.4.m3.13.13.4.4.10" stretchy="false" xref="S2.SS4.p1.4.m3.13.13.4.5.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.4.m3.13b"><apply id="S2.SS4.p1.4.m3.13.13.cmml" xref="S2.SS4.p1.4.m3.13.13"><eq id="S2.SS4.p1.4.m3.13.13.5.cmml" xref="S2.SS4.p1.4.m3.13.13.5"></eq><ci id="S2.SS4.p1.4.m3.13.13.6.cmml" xref="S2.SS4.p1.4.m3.13.13.6">𝑚</ci><vector id="S2.SS4.p1.4.m3.13.13.4.5.cmml" xref="S2.SS4.p1.4.m3.13.13.4.4"><apply id="S2.SS4.p1.4.m3.10.10.1.1.1.cmml" xref="S2.SS4.p1.4.m3.10.10.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.4.m3.10.10.1.1.1.1.cmml" xref="S2.SS4.p1.4.m3.10.10.1.1.1">subscript</csymbol><ci id="S2.SS4.p1.4.m3.10.10.1.1.1.2.cmml" xref="S2.SS4.p1.4.m3.10.10.1.1.1.2">𝑚</ci><list id="S2.SS4.p1.4.m3.2.2.2.3.cmml" xref="S2.SS4.p1.4.m3.2.2.2.4"><cn id="S2.SS4.p1.4.m3.1.1.1.1.cmml" type="integer" xref="S2.SS4.p1.4.m3.1.1.1.1">1</cn><cn id="S2.SS4.p1.4.m3.2.2.2.2.cmml" type="integer" xref="S2.SS4.p1.4.m3.2.2.2.2">1</cn></list></apply><ci id="S2.SS4.p1.4.m3.9.9.cmml" xref="S2.SS4.p1.4.m3.9.9">…</ci><apply id="S2.SS4.p1.4.m3.11.11.2.2.2.cmml" xref="S2.SS4.p1.4.m3.11.11.2.2.2"><csymbol cd="ambiguous" id="S2.SS4.p1.4.m3.11.11.2.2.2.1.cmml" xref="S2.SS4.p1.4.m3.11.11.2.2.2">subscript</csymbol><ci id="S2.SS4.p1.4.m3.11.11.2.2.2.2.cmml" xref="S2.SS4.p1.4.m3.11.11.2.2.2.2">𝑚</ci><list id="S2.SS4.p1.4.m3.4.4.2.3.cmml" xref="S2.SS4.p1.4.m3.4.4.2.4"><ci id="S2.SS4.p1.4.m3.3.3.1.1.cmml" xref="S2.SS4.p1.4.m3.3.3.1.1">𝐽</ci><cn id="S2.SS4.p1.4.m3.4.4.2.2.cmml" type="integer" xref="S2.SS4.p1.4.m3.4.4.2.2">1</cn></list></apply><apply id="S2.SS4.p1.4.m3.12.12.3.3.3.cmml" xref="S2.SS4.p1.4.m3.12.12.3.3.3"><times id="S2.SS4.p1.4.m3.12.12.3.3.3.1.cmml" xref="S2.SS4.p1.4.m3.12.12.3.3.3.1"></times><apply id="S2.SS4.p1.4.m3.12.12.3.3.3.2.cmml" xref="S2.SS4.p1.4.m3.12.12.3.3.3.2"><csymbol cd="ambiguous" id="S2.SS4.p1.4.m3.12.12.3.3.3.2.1.cmml" xref="S2.SS4.p1.4.m3.12.12.3.3.3.2">subscript</csymbol><ci id="S2.SS4.p1.4.m3.12.12.3.3.3.2.2.cmml" xref="S2.SS4.p1.4.m3.12.12.3.3.3.2.2">𝑚</ci><list id="S2.SS4.p1.4.m3.6.6.2.3.cmml" xref="S2.SS4.p1.4.m3.6.6.2.4"><cn id="S2.SS4.p1.4.m3.5.5.1.1.cmml" type="integer" xref="S2.SS4.p1.4.m3.5.5.1.1">1</cn><cn id="S2.SS4.p1.4.m3.6.6.2.2.cmml" type="integer" xref="S2.SS4.p1.4.m3.6.6.2.2">2</cn></list></apply><ci id="S2.SS4.p1.4.m3.12.12.3.3.3.3.cmml" xref="S2.SS4.p1.4.m3.12.12.3.3.3.3">…</ci></apply><apply id="S2.SS4.p1.4.m3.13.13.4.4.4.cmml" xref="S2.SS4.p1.4.m3.13.13.4.4.4"><csymbol cd="ambiguous" id="S2.SS4.p1.4.m3.13.13.4.4.4.1.cmml" xref="S2.SS4.p1.4.m3.13.13.4.4.4">subscript</csymbol><ci id="S2.SS4.p1.4.m3.13.13.4.4.4.2.cmml" xref="S2.SS4.p1.4.m3.13.13.4.4.4.2">𝑚</ci><list id="S2.SS4.p1.4.m3.8.8.2.3.cmml" xref="S2.SS4.p1.4.m3.8.8.2.4"><ci id="S2.SS4.p1.4.m3.7.7.1.1.cmml" xref="S2.SS4.p1.4.m3.7.7.1.1">𝐽</ci><cn id="S2.SS4.p1.4.m3.8.8.2.2.cmml" type="integer" xref="S2.SS4.p1.4.m3.8.8.2.2">2</cn></list></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.4.m3.13c">m=(m_{1,1},\dots,m_{J,1},m_{1,2}\dots,m_{J,2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.4.m3.13d">italic_m = ( italic_m start_POSTSUBSCRIPT 1 , 1 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_J , 1 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT … , italic_m start_POSTSUBSCRIPT italic_J , 2 end_POSTSUBSCRIPT )</annotation></semantics></math> and denote by <math alttext="\bar{m}(\beta)" class="ltx_Math" display="inline" id="S2.SS4.p1.5.m4.1"><semantics id="S2.SS4.p1.5.m4.1a"><mrow id="S2.SS4.p1.5.m4.1.2" xref="S2.SS4.p1.5.m4.1.2.cmml"><mover accent="true" id="S2.SS4.p1.5.m4.1.2.2" xref="S2.SS4.p1.5.m4.1.2.2.cmml"><mi id="S2.SS4.p1.5.m4.1.2.2.2" xref="S2.SS4.p1.5.m4.1.2.2.2.cmml">m</mi><mo id="S2.SS4.p1.5.m4.1.2.2.1" xref="S2.SS4.p1.5.m4.1.2.2.1.cmml">¯</mo></mover><mo id="S2.SS4.p1.5.m4.1.2.1" xref="S2.SS4.p1.5.m4.1.2.1.cmml">⁢</mo><mrow id="S2.SS4.p1.5.m4.1.2.3.2" xref="S2.SS4.p1.5.m4.1.2.cmml"><mo id="S2.SS4.p1.5.m4.1.2.3.2.1" stretchy="false" xref="S2.SS4.p1.5.m4.1.2.cmml">(</mo><mi id="S2.SS4.p1.5.m4.1.1" xref="S2.SS4.p1.5.m4.1.1.cmml">β</mi><mo id="S2.SS4.p1.5.m4.1.2.3.2.2" stretchy="false" xref="S2.SS4.p1.5.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.5.m4.1b"><apply id="S2.SS4.p1.5.m4.1.2.cmml" xref="S2.SS4.p1.5.m4.1.2"><times id="S2.SS4.p1.5.m4.1.2.1.cmml" xref="S2.SS4.p1.5.m4.1.2.1"></times><apply id="S2.SS4.p1.5.m4.1.2.2.cmml" xref="S2.SS4.p1.5.m4.1.2.2"><ci id="S2.SS4.p1.5.m4.1.2.2.1.cmml" xref="S2.SS4.p1.5.m4.1.2.2.1">¯</ci><ci id="S2.SS4.p1.5.m4.1.2.2.2.cmml" xref="S2.SS4.p1.5.m4.1.2.2.2">𝑚</ci></apply><ci id="S2.SS4.p1.5.m4.1.1.cmml" xref="S2.SS4.p1.5.m4.1.1">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.5.m4.1c">\bar{m}(\beta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.5.m4.1d">over¯ start_ARG italic_m end_ARG ( italic_β )</annotation></semantics></math> and <math alttext="\hat{\sigma}(\beta)" class="ltx_Math" display="inline" id="S2.SS4.p1.6.m5.1"><semantics id="S2.SS4.p1.6.m5.1a"><mrow id="S2.SS4.p1.6.m5.1.2" xref="S2.SS4.p1.6.m5.1.2.cmml"><mover accent="true" id="S2.SS4.p1.6.m5.1.2.2" xref="S2.SS4.p1.6.m5.1.2.2.cmml"><mi id="S2.SS4.p1.6.m5.1.2.2.2" xref="S2.SS4.p1.6.m5.1.2.2.2.cmml">σ</mi><mo id="S2.SS4.p1.6.m5.1.2.2.1" xref="S2.SS4.p1.6.m5.1.2.2.1.cmml">^</mo></mover><mo id="S2.SS4.p1.6.m5.1.2.1" xref="S2.SS4.p1.6.m5.1.2.1.cmml">⁢</mo><mrow id="S2.SS4.p1.6.m5.1.2.3.2" xref="S2.SS4.p1.6.m5.1.2.cmml"><mo id="S2.SS4.p1.6.m5.1.2.3.2.1" stretchy="false" xref="S2.SS4.p1.6.m5.1.2.cmml">(</mo><mi id="S2.SS4.p1.6.m5.1.1" xref="S2.SS4.p1.6.m5.1.1.cmml">β</mi><mo id="S2.SS4.p1.6.m5.1.2.3.2.2" stretchy="false" xref="S2.SS4.p1.6.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.6.m5.1b"><apply id="S2.SS4.p1.6.m5.1.2.cmml" xref="S2.SS4.p1.6.m5.1.2"><times id="S2.SS4.p1.6.m5.1.2.1.cmml" xref="S2.SS4.p1.6.m5.1.2.1"></times><apply id="S2.SS4.p1.6.m5.1.2.2.cmml" xref="S2.SS4.p1.6.m5.1.2.2"><ci id="S2.SS4.p1.6.m5.1.2.2.1.cmml" xref="S2.SS4.p1.6.m5.1.2.2.1">^</ci><ci id="S2.SS4.p1.6.m5.1.2.2.2.cmml" xref="S2.SS4.p1.6.m5.1.2.2.2">𝜎</ci></apply><ci id="S2.SS4.p1.6.m5.1.1.cmml" xref="S2.SS4.p1.6.m5.1.1">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.6.m5.1c">\hat{\sigma}(\beta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.6.m5.1d">over^ start_ARG italic_σ end_ARG ( italic_β )</annotation></semantics></math> the sample average and standard deviation of <math alttext="m(W,\beta)" class="ltx_Math" display="inline" id="S2.SS4.p1.7.m6.2"><semantics id="S2.SS4.p1.7.m6.2a"><mrow id="S2.SS4.p1.7.m6.2.3" xref="S2.SS4.p1.7.m6.2.3.cmml"><mi id="S2.SS4.p1.7.m6.2.3.2" xref="S2.SS4.p1.7.m6.2.3.2.cmml">m</mi><mo id="S2.SS4.p1.7.m6.2.3.1" xref="S2.SS4.p1.7.m6.2.3.1.cmml">⁢</mo><mrow id="S2.SS4.p1.7.m6.2.3.3.2" xref="S2.SS4.p1.7.m6.2.3.3.1.cmml"><mo id="S2.SS4.p1.7.m6.2.3.3.2.1" stretchy="false" xref="S2.SS4.p1.7.m6.2.3.3.1.cmml">(</mo><mi id="S2.SS4.p1.7.m6.1.1" xref="S2.SS4.p1.7.m6.1.1.cmml">W</mi><mo id="S2.SS4.p1.7.m6.2.3.3.2.2" xref="S2.SS4.p1.7.m6.2.3.3.1.cmml">,</mo><mi id="S2.SS4.p1.7.m6.2.2" xref="S2.SS4.p1.7.m6.2.2.cmml">β</mi><mo id="S2.SS4.p1.7.m6.2.3.3.2.3" stretchy="false" xref="S2.SS4.p1.7.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.7.m6.2b"><apply id="S2.SS4.p1.7.m6.2.3.cmml" xref="S2.SS4.p1.7.m6.2.3"><times id="S2.SS4.p1.7.m6.2.3.1.cmml" xref="S2.SS4.p1.7.m6.2.3.1"></times><ci id="S2.SS4.p1.7.m6.2.3.2.cmml" xref="S2.SS4.p1.7.m6.2.3.2">𝑚</ci><interval closure="open" id="S2.SS4.p1.7.m6.2.3.3.1.cmml" xref="S2.SS4.p1.7.m6.2.3.3.2"><ci id="S2.SS4.p1.7.m6.1.1.cmml" xref="S2.SS4.p1.7.m6.1.1">𝑊</ci><ci id="S2.SS4.p1.7.m6.2.2.cmml" xref="S2.SS4.p1.7.m6.2.2">𝛽</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.7.m6.2c">m(W,\beta)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.7.m6.2d">italic_m ( italic_W , italic_β )</annotation></semantics></math>. The set <math alttext="\mathcal{B}(r)=\{\beta\in\mathcal{B}\mid\beta_{k}=r\}" class="ltx_Math" display="inline" id="S2.SS4.p1.8.m7.3"><semantics id="S2.SS4.p1.8.m7.3a"><mrow id="S2.SS4.p1.8.m7.3.3" xref="S2.SS4.p1.8.m7.3.3.cmml"><mrow id="S2.SS4.p1.8.m7.3.3.4" xref="S2.SS4.p1.8.m7.3.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.p1.8.m7.3.3.4.2" xref="S2.SS4.p1.8.m7.3.3.4.2.cmml">ℬ</mi><mo id="S2.SS4.p1.8.m7.3.3.4.1" xref="S2.SS4.p1.8.m7.3.3.4.1.cmml">⁢</mo><mrow id="S2.SS4.p1.8.m7.3.3.4.3.2" xref="S2.SS4.p1.8.m7.3.3.4.cmml"><mo id="S2.SS4.p1.8.m7.3.3.4.3.2.1" stretchy="false" xref="S2.SS4.p1.8.m7.3.3.4.cmml">(</mo><mi id="S2.SS4.p1.8.m7.1.1" xref="S2.SS4.p1.8.m7.1.1.cmml">r</mi><mo id="S2.SS4.p1.8.m7.3.3.4.3.2.2" stretchy="false" xref="S2.SS4.p1.8.m7.3.3.4.cmml">)</mo></mrow></mrow><mo id="S2.SS4.p1.8.m7.3.3.3" xref="S2.SS4.p1.8.m7.3.3.3.cmml">=</mo><mrow id="S2.SS4.p1.8.m7.3.3.2.2" xref="S2.SS4.p1.8.m7.3.3.2.3.cmml"><mo id="S2.SS4.p1.8.m7.3.3.2.2.3" stretchy="false" xref="S2.SS4.p1.8.m7.3.3.2.3.1.cmml">{</mo><mrow id="S2.SS4.p1.8.m7.2.2.1.1.1" xref="S2.SS4.p1.8.m7.2.2.1.1.1.cmml"><mi id="S2.SS4.p1.8.m7.2.2.1.1.1.2" xref="S2.SS4.p1.8.m7.2.2.1.1.1.2.cmml">β</mi><mo id="S2.SS4.p1.8.m7.2.2.1.1.1.1" xref="S2.SS4.p1.8.m7.2.2.1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS4.p1.8.m7.2.2.1.1.1.3" xref="S2.SS4.p1.8.m7.2.2.1.1.1.3.cmml">ℬ</mi></mrow><mo fence="true" id="S2.SS4.p1.8.m7.3.3.2.2.4" lspace="0em" rspace="0em" xref="S2.SS4.p1.8.m7.3.3.2.3.1.cmml">∣</mo><mrow id="S2.SS4.p1.8.m7.3.3.2.2.2" xref="S2.SS4.p1.8.m7.3.3.2.2.2.cmml"><msub id="S2.SS4.p1.8.m7.3.3.2.2.2.2" xref="S2.SS4.p1.8.m7.3.3.2.2.2.2.cmml"><mi id="S2.SS4.p1.8.m7.3.3.2.2.2.2.2" xref="S2.SS4.p1.8.m7.3.3.2.2.2.2.2.cmml">β</mi><mi id="S2.SS4.p1.8.m7.3.3.2.2.2.2.3" xref="S2.SS4.p1.8.m7.3.3.2.2.2.2.3.cmml">k</mi></msub><mo id="S2.SS4.p1.8.m7.3.3.2.2.2.1" xref="S2.SS4.p1.8.m7.3.3.2.2.2.1.cmml">=</mo><mi id="S2.SS4.p1.8.m7.3.3.2.2.2.3" xref="S2.SS4.p1.8.m7.3.3.2.2.2.3.cmml">r</mi></mrow><mo id="S2.SS4.p1.8.m7.3.3.2.2.5" stretchy="false" xref="S2.SS4.p1.8.m7.3.3.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.8.m7.3b"><apply id="S2.SS4.p1.8.m7.3.3.cmml" xref="S2.SS4.p1.8.m7.3.3"><eq id="S2.SS4.p1.8.m7.3.3.3.cmml" xref="S2.SS4.p1.8.m7.3.3.3"></eq><apply id="S2.SS4.p1.8.m7.3.3.4.cmml" xref="S2.SS4.p1.8.m7.3.3.4"><times id="S2.SS4.p1.8.m7.3.3.4.1.cmml" xref="S2.SS4.p1.8.m7.3.3.4.1"></times><ci id="S2.SS4.p1.8.m7.3.3.4.2.cmml" xref="S2.SS4.p1.8.m7.3.3.4.2">ℬ</ci><ci id="S2.SS4.p1.8.m7.1.1.cmml" xref="S2.SS4.p1.8.m7.1.1">𝑟</ci></apply><apply id="S2.SS4.p1.8.m7.3.3.2.3.cmml" xref="S2.SS4.p1.8.m7.3.3.2.2"><csymbol cd="latexml" id="S2.SS4.p1.8.m7.3.3.2.3.1.cmml" xref="S2.SS4.p1.8.m7.3.3.2.2.3">conditional-set</csymbol><apply id="S2.SS4.p1.8.m7.2.2.1.1.1.cmml" xref="S2.SS4.p1.8.m7.2.2.1.1.1"><in id="S2.SS4.p1.8.m7.2.2.1.1.1.1.cmml" xref="S2.SS4.p1.8.m7.2.2.1.1.1.1"></in><ci id="S2.SS4.p1.8.m7.2.2.1.1.1.2.cmml" xref="S2.SS4.p1.8.m7.2.2.1.1.1.2">𝛽</ci><ci id="S2.SS4.p1.8.m7.2.2.1.1.1.3.cmml" xref="S2.SS4.p1.8.m7.2.2.1.1.1.3">ℬ</ci></apply><apply id="S2.SS4.p1.8.m7.3.3.2.2.2.cmml" xref="S2.SS4.p1.8.m7.3.3.2.2.2"><eq id="S2.SS4.p1.8.m7.3.3.2.2.2.1.cmml" xref="S2.SS4.p1.8.m7.3.3.2.2.2.1"></eq><apply id="S2.SS4.p1.8.m7.3.3.2.2.2.2.cmml" xref="S2.SS4.p1.8.m7.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS4.p1.8.m7.3.3.2.2.2.2.1.cmml" xref="S2.SS4.p1.8.m7.3.3.2.2.2.2">subscript</csymbol><ci id="S2.SS4.p1.8.m7.3.3.2.2.2.2.2.cmml" xref="S2.SS4.p1.8.m7.3.3.2.2.2.2.2">𝛽</ci><ci id="S2.SS4.p1.8.m7.3.3.2.2.2.2.3.cmml" xref="S2.SS4.p1.8.m7.3.3.2.2.2.2.3">𝑘</ci></apply><ci id="S2.SS4.p1.8.m7.3.3.2.2.2.3.cmml" xref="S2.SS4.p1.8.m7.3.3.2.2.2.3">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.8.m7.3c">\mathcal{B}(r)=\{\beta\in\mathcal{B}\mid\beta_{k}=r\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.8.m7.3d">caligraphic_B ( italic_r ) = { italic_β ∈ caligraphic_B ∣ italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_r }</annotation></semantics></math> contains all parameter vectors that have <math alttext="r" class="ltx_Math" display="inline" id="S2.SS4.p1.9.m8.1"><semantics id="S2.SS4.p1.9.m8.1a"><mi id="S2.SS4.p1.9.m8.1.1" xref="S2.SS4.p1.9.m8.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.9.m8.1b"><ci id="S2.SS4.p1.9.m8.1.1.cmml" xref="S2.SS4.p1.9.m8.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.9.m8.1c">r</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.9.m8.1d">italic_r</annotation></semantics></math> as their <math alttext="k" class="ltx_Math" display="inline" id="S2.SS4.p1.10.m9.1"><semantics id="S2.SS4.p1.10.m9.1a"><mi id="S2.SS4.p1.10.m9.1.1" xref="S2.SS4.p1.10.m9.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.10.m9.1b"><ci id="S2.SS4.p1.10.m9.1.1.cmml" xref="S2.SS4.p1.10.m9.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.10.m9.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.10.m9.1d">italic_k</annotation></semantics></math>-th entry.</p> </div> <div class="ltx_para" id="S2.SS4.p2"> <p class="ltx_p" id="S2.SS4.p2.10">Intuitively, the function <math alttext="S" class="ltx_Math" display="inline" id="S2.SS4.p2.1.m1.1"><semantics id="S2.SS4.p2.1.m1.1a"><mi id="S2.SS4.p2.1.m1.1.1" xref="S2.SS4.p2.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.1.m1.1b"><ci id="S2.SS4.p2.1.m1.1.1.cmml" xref="S2.SS4.p2.1.m1.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.1.m1.1d">italic_S</annotation></semantics></math> can be viewed as a measure how much the sample moment restrictions are violated at a given <math alttext="\beta" class="ltx_Math" display="inline" id="S2.SS4.p2.2.m2.1"><semantics id="S2.SS4.p2.2.m2.1a"><mi id="S2.SS4.p2.2.m2.1.1" xref="S2.SS4.p2.2.m2.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.2.m2.1b"><ci id="S2.SS4.p2.2.m2.1.1.cmml" xref="S2.SS4.p2.2.m2.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.2.m2.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.2.m2.1d">italic_β</annotation></semantics></math>. The test statistic <math alttext="T_{n}(r)" class="ltx_Math" display="inline" id="S2.SS4.p2.3.m3.1"><semantics id="S2.SS4.p2.3.m3.1a"><mrow id="S2.SS4.p2.3.m3.1.2" xref="S2.SS4.p2.3.m3.1.2.cmml"><msub id="S2.SS4.p2.3.m3.1.2.2" xref="S2.SS4.p2.3.m3.1.2.2.cmml"><mi id="S2.SS4.p2.3.m3.1.2.2.2" xref="S2.SS4.p2.3.m3.1.2.2.2.cmml">T</mi><mi id="S2.SS4.p2.3.m3.1.2.2.3" xref="S2.SS4.p2.3.m3.1.2.2.3.cmml">n</mi></msub><mo id="S2.SS4.p2.3.m3.1.2.1" xref="S2.SS4.p2.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS4.p2.3.m3.1.2.3.2" xref="S2.SS4.p2.3.m3.1.2.cmml"><mo id="S2.SS4.p2.3.m3.1.2.3.2.1" stretchy="false" xref="S2.SS4.p2.3.m3.1.2.cmml">(</mo><mi id="S2.SS4.p2.3.m3.1.1" xref="S2.SS4.p2.3.m3.1.1.cmml">r</mi><mo id="S2.SS4.p2.3.m3.1.2.3.2.2" stretchy="false" xref="S2.SS4.p2.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.3.m3.1b"><apply id="S2.SS4.p2.3.m3.1.2.cmml" xref="S2.SS4.p2.3.m3.1.2"><times id="S2.SS4.p2.3.m3.1.2.1.cmml" xref="S2.SS4.p2.3.m3.1.2.1"></times><apply id="S2.SS4.p2.3.m3.1.2.2.cmml" xref="S2.SS4.p2.3.m3.1.2.2"><csymbol cd="ambiguous" id="S2.SS4.p2.3.m3.1.2.2.1.cmml" xref="S2.SS4.p2.3.m3.1.2.2">subscript</csymbol><ci id="S2.SS4.p2.3.m3.1.2.2.2.cmml" xref="S2.SS4.p2.3.m3.1.2.2.2">𝑇</ci><ci id="S2.SS4.p2.3.m3.1.2.2.3.cmml" xref="S2.SS4.p2.3.m3.1.2.2.3">𝑛</ci></apply><ci id="S2.SS4.p2.3.m3.1.1.cmml" xref="S2.SS4.p2.3.m3.1.1">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.3.m3.1c">T_{n}(r)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.3.m3.1d">italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_r )</annotation></semantics></math> considers the value of <math alttext="\beta" class="ltx_Math" display="inline" id="S2.SS4.p2.4.m4.1"><semantics id="S2.SS4.p2.4.m4.1a"><mi id="S2.SS4.p2.4.m4.1.1" xref="S2.SS4.p2.4.m4.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.4.m4.1b"><ci id="S2.SS4.p2.4.m4.1.1.cmml" xref="S2.SS4.p2.4.m4.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.4.m4.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.4.m4.1d">italic_β</annotation></semantics></math> for which this violation is at a minimum and such that the <math alttext="k" class="ltx_Math" display="inline" id="S2.SS4.p2.5.m5.1"><semantics id="S2.SS4.p2.5.m5.1a"><mi id="S2.SS4.p2.5.m5.1.1" xref="S2.SS4.p2.5.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.5.m5.1b"><ci id="S2.SS4.p2.5.m5.1.1.cmml" xref="S2.SS4.p2.5.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.5.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.5.m5.1d">italic_k</annotation></semantics></math>-th component of <math alttext="\beta" class="ltx_Math" display="inline" id="S2.SS4.p2.6.m6.1"><semantics id="S2.SS4.p2.6.m6.1a"><mi id="S2.SS4.p2.6.m6.1.1" xref="S2.SS4.p2.6.m6.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.6.m6.1b"><ci id="S2.SS4.p2.6.m6.1.1.cmml" xref="S2.SS4.p2.6.m6.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.6.m6.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.6.m6.1d">italic_β</annotation></semantics></math> is equal to <math alttext="r" class="ltx_Math" display="inline" id="S2.SS4.p2.7.m7.1"><semantics id="S2.SS4.p2.7.m7.1a"><mi id="S2.SS4.p2.7.m7.1.1" xref="S2.SS4.p2.7.m7.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.7.m7.1b"><ci id="S2.SS4.p2.7.m7.1.1.cmml" xref="S2.SS4.p2.7.m7.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.7.m7.1c">r</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.7.m7.1d">italic_r</annotation></semantics></math>. If this minimal violation is too large, <math alttext="\mathcal{H}_{0}(r)" class="ltx_Math" display="inline" id="S2.SS4.p2.8.m8.1"><semantics id="S2.SS4.p2.8.m8.1a"><mrow id="S2.SS4.p2.8.m8.1.2" xref="S2.SS4.p2.8.m8.1.2.cmml"><msub id="S2.SS4.p2.8.m8.1.2.2" xref="S2.SS4.p2.8.m8.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.p2.8.m8.1.2.2.2" xref="S2.SS4.p2.8.m8.1.2.2.2.cmml">ℋ</mi><mn id="S2.SS4.p2.8.m8.1.2.2.3" xref="S2.SS4.p2.8.m8.1.2.2.3.cmml">0</mn></msub><mo id="S2.SS4.p2.8.m8.1.2.1" xref="S2.SS4.p2.8.m8.1.2.1.cmml">⁢</mo><mrow id="S2.SS4.p2.8.m8.1.2.3.2" xref="S2.SS4.p2.8.m8.1.2.cmml"><mo id="S2.SS4.p2.8.m8.1.2.3.2.1" stretchy="false" xref="S2.SS4.p2.8.m8.1.2.cmml">(</mo><mi id="S2.SS4.p2.8.m8.1.1" xref="S2.SS4.p2.8.m8.1.1.cmml">r</mi><mo id="S2.SS4.p2.8.m8.1.2.3.2.2" stretchy="false" xref="S2.SS4.p2.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.8.m8.1b"><apply id="S2.SS4.p2.8.m8.1.2.cmml" xref="S2.SS4.p2.8.m8.1.2"><times id="S2.SS4.p2.8.m8.1.2.1.cmml" xref="S2.SS4.p2.8.m8.1.2.1"></times><apply id="S2.SS4.p2.8.m8.1.2.2.cmml" xref="S2.SS4.p2.8.m8.1.2.2"><csymbol cd="ambiguous" id="S2.SS4.p2.8.m8.1.2.2.1.cmml" xref="S2.SS4.p2.8.m8.1.2.2">subscript</csymbol><ci id="S2.SS4.p2.8.m8.1.2.2.2.cmml" xref="S2.SS4.p2.8.m8.1.2.2.2">ℋ</ci><cn id="S2.SS4.p2.8.m8.1.2.2.3.cmml" type="integer" xref="S2.SS4.p2.8.m8.1.2.2.3">0</cn></apply><ci id="S2.SS4.p2.8.m8.1.1.cmml" xref="S2.SS4.p2.8.m8.1.1">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.8.m8.1c">\mathcal{H}_{0}(r)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.8.m8.1d">caligraphic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_r )</annotation></semantics></math> can be rejected. To this end, a specialized bootstrap procedure to obtain the critical value <math alttext="\gamma_{n,1-\alpha}(r)" class="ltx_Math" display="inline" id="S2.SS4.p2.9.m9.3"><semantics id="S2.SS4.p2.9.m9.3a"><mrow id="S2.SS4.p2.9.m9.3.4" xref="S2.SS4.p2.9.m9.3.4.cmml"><msub id="S2.SS4.p2.9.m9.3.4.2" xref="S2.SS4.p2.9.m9.3.4.2.cmml"><mi id="S2.SS4.p2.9.m9.3.4.2.2" xref="S2.SS4.p2.9.m9.3.4.2.2.cmml">γ</mi><mrow id="S2.SS4.p2.9.m9.2.2.2.2" xref="S2.SS4.p2.9.m9.2.2.2.3.cmml"><mi id="S2.SS4.p2.9.m9.1.1.1.1" xref="S2.SS4.p2.9.m9.1.1.1.1.cmml">n</mi><mo id="S2.SS4.p2.9.m9.2.2.2.2.2" xref="S2.SS4.p2.9.m9.2.2.2.3.cmml">,</mo><mrow id="S2.SS4.p2.9.m9.2.2.2.2.1" xref="S2.SS4.p2.9.m9.2.2.2.2.1.cmml"><mn id="S2.SS4.p2.9.m9.2.2.2.2.1.2" xref="S2.SS4.p2.9.m9.2.2.2.2.1.2.cmml">1</mn><mo id="S2.SS4.p2.9.m9.2.2.2.2.1.1" xref="S2.SS4.p2.9.m9.2.2.2.2.1.1.cmml">−</mo><mi id="S2.SS4.p2.9.m9.2.2.2.2.1.3" xref="S2.SS4.p2.9.m9.2.2.2.2.1.3.cmml">α</mi></mrow></mrow></msub><mo id="S2.SS4.p2.9.m9.3.4.1" xref="S2.SS4.p2.9.m9.3.4.1.cmml">⁢</mo><mrow id="S2.SS4.p2.9.m9.3.4.3.2" xref="S2.SS4.p2.9.m9.3.4.cmml"><mo id="S2.SS4.p2.9.m9.3.4.3.2.1" stretchy="false" xref="S2.SS4.p2.9.m9.3.4.cmml">(</mo><mi id="S2.SS4.p2.9.m9.3.3" xref="S2.SS4.p2.9.m9.3.3.cmml">r</mi><mo id="S2.SS4.p2.9.m9.3.4.3.2.2" stretchy="false" xref="S2.SS4.p2.9.m9.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.9.m9.3b"><apply id="S2.SS4.p2.9.m9.3.4.cmml" xref="S2.SS4.p2.9.m9.3.4"><times id="S2.SS4.p2.9.m9.3.4.1.cmml" xref="S2.SS4.p2.9.m9.3.4.1"></times><apply id="S2.SS4.p2.9.m9.3.4.2.cmml" xref="S2.SS4.p2.9.m9.3.4.2"><csymbol cd="ambiguous" id="S2.SS4.p2.9.m9.3.4.2.1.cmml" xref="S2.SS4.p2.9.m9.3.4.2">subscript</csymbol><ci id="S2.SS4.p2.9.m9.3.4.2.2.cmml" xref="S2.SS4.p2.9.m9.3.4.2.2">𝛾</ci><list id="S2.SS4.p2.9.m9.2.2.2.3.cmml" xref="S2.SS4.p2.9.m9.2.2.2.2"><ci id="S2.SS4.p2.9.m9.1.1.1.1.cmml" xref="S2.SS4.p2.9.m9.1.1.1.1">𝑛</ci><apply id="S2.SS4.p2.9.m9.2.2.2.2.1.cmml" xref="S2.SS4.p2.9.m9.2.2.2.2.1"><minus id="S2.SS4.p2.9.m9.2.2.2.2.1.1.cmml" xref="S2.SS4.p2.9.m9.2.2.2.2.1.1"></minus><cn id="S2.SS4.p2.9.m9.2.2.2.2.1.2.cmml" type="integer" xref="S2.SS4.p2.9.m9.2.2.2.2.1.2">1</cn><ci id="S2.SS4.p2.9.m9.2.2.2.2.1.3.cmml" xref="S2.SS4.p2.9.m9.2.2.2.2.1.3">𝛼</ci></apply></list></apply><ci id="S2.SS4.p2.9.m9.3.3.cmml" xref="S2.SS4.p2.9.m9.3.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.9.m9.3c">\gamma_{n,1-\alpha}(r)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.9.m9.3d">italic_γ start_POSTSUBSCRIPT italic_n , 1 - italic_α end_POSTSUBSCRIPT ( italic_r )</annotation></semantics></math> of <math alttext="T_{n}(r)" class="ltx_Math" display="inline" id="S2.SS4.p2.10.m10.1"><semantics id="S2.SS4.p2.10.m10.1a"><mrow id="S2.SS4.p2.10.m10.1.2" xref="S2.SS4.p2.10.m10.1.2.cmml"><msub id="S2.SS4.p2.10.m10.1.2.2" xref="S2.SS4.p2.10.m10.1.2.2.cmml"><mi id="S2.SS4.p2.10.m10.1.2.2.2" xref="S2.SS4.p2.10.m10.1.2.2.2.cmml">T</mi><mi id="S2.SS4.p2.10.m10.1.2.2.3" xref="S2.SS4.p2.10.m10.1.2.2.3.cmml">n</mi></msub><mo id="S2.SS4.p2.10.m10.1.2.1" xref="S2.SS4.p2.10.m10.1.2.1.cmml">⁢</mo><mrow id="S2.SS4.p2.10.m10.1.2.3.2" xref="S2.SS4.p2.10.m10.1.2.cmml"><mo id="S2.SS4.p2.10.m10.1.2.3.2.1" stretchy="false" xref="S2.SS4.p2.10.m10.1.2.cmml">(</mo><mi id="S2.SS4.p2.10.m10.1.1" xref="S2.SS4.p2.10.m10.1.1.cmml">r</mi><mo id="S2.SS4.p2.10.m10.1.2.3.2.2" stretchy="false" xref="S2.SS4.p2.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.10.m10.1b"><apply id="S2.SS4.p2.10.m10.1.2.cmml" xref="S2.SS4.p2.10.m10.1.2"><times id="S2.SS4.p2.10.m10.1.2.1.cmml" xref="S2.SS4.p2.10.m10.1.2.1"></times><apply id="S2.SS4.p2.10.m10.1.2.2.cmml" xref="S2.SS4.p2.10.m10.1.2.2"><csymbol cd="ambiguous" id="S2.SS4.p2.10.m10.1.2.2.1.cmml" xref="S2.SS4.p2.10.m10.1.2.2">subscript</csymbol><ci id="S2.SS4.p2.10.m10.1.2.2.2.cmml" xref="S2.SS4.p2.10.m10.1.2.2.2">𝑇</ci><ci id="S2.SS4.p2.10.m10.1.2.2.3.cmml" xref="S2.SS4.p2.10.m10.1.2.2.3">𝑛</ci></apply><ci id="S2.SS4.p2.10.m10.1.1.cmml" xref="S2.SS4.p2.10.m10.1.1">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.10.m10.1c">T_{n}(r)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.10.m10.1d">italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_r )</annotation></semantics></math> is developed. More detailed information on this test, and in particular on the bootstrap procedure, can be found in Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Testing_procedure_of_Bei</span>. We refer to <cite class="ltx_cite ltx_citemacro_cite">Bei, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib2" title="">2024</a>)</cite> for a full explanation.</p> </div> </section> <section class="ltx_subsection" id="S2.SS5"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.5 </span>Overview</h3> <div class="ltx_para" id="S2.SS5.p1"> <p class="ltx_p" id="S2.SS5.p1.6">In summary, after selecting a time point <math alttext="t" class="ltx_Math" display="inline" id="S2.SS5.p1.1.m1.1"><semantics id="S2.SS5.p1.1.m1.1a"><mi id="S2.SS5.p1.1.m1.1.1" xref="S2.SS5.p1.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.1.m1.1b"><ci id="S2.SS5.p1.1.m1.1.1.cmml" xref="S2.SS5.p1.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.1.m1.1d">italic_t</annotation></semantics></math> and a coefficient of interest <math alttext="\beta_{k}" class="ltx_Math" display="inline" id="S2.SS5.p1.2.m2.1"><semantics id="S2.SS5.p1.2.m2.1a"><msub id="S2.SS5.p1.2.m2.1.1" xref="S2.SS5.p1.2.m2.1.1.cmml"><mi id="S2.SS5.p1.2.m2.1.1.2" xref="S2.SS5.p1.2.m2.1.1.2.cmml">β</mi><mi id="S2.SS5.p1.2.m2.1.1.3" xref="S2.SS5.p1.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.2.m2.1b"><apply id="S2.SS5.p1.2.m2.1.1.cmml" xref="S2.SS5.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS5.p1.2.m2.1.1.1.cmml" xref="S2.SS5.p1.2.m2.1.1">subscript</csymbol><ci id="S2.SS5.p1.2.m2.1.1.2.cmml" xref="S2.SS5.p1.2.m2.1.1.2">𝛽</ci><ci id="S2.SS5.p1.2.m2.1.1.3.cmml" xref="S2.SS5.p1.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.2.m2.1c">\beta_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.2.m2.1d">italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, the methodology of this paper consists of two parts. First, a test for the hypothesis (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E9" title="In 2.4 Testing procedure ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">9</span></a>) is obtained. Next, this test is applied over the entire parameter subspace <math alttext="\mathcal{B}_{k}" class="ltx_Math" display="inline" id="S2.SS5.p1.3.m3.1"><semantics id="S2.SS5.p1.3.m3.1a"><msub id="S2.SS5.p1.3.m3.1.1" xref="S2.SS5.p1.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p1.3.m3.1.1.2" xref="S2.SS5.p1.3.m3.1.1.2.cmml">ℬ</mi><mi id="S2.SS5.p1.3.m3.1.1.3" xref="S2.SS5.p1.3.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.3.m3.1b"><apply id="S2.SS5.p1.3.m3.1.1.cmml" xref="S2.SS5.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS5.p1.3.m3.1.1.1.cmml" xref="S2.SS5.p1.3.m3.1.1">subscript</csymbol><ci id="S2.SS5.p1.3.m3.1.1.2.cmml" xref="S2.SS5.p1.3.m3.1.1.2">ℬ</ci><ci id="S2.SS5.p1.3.m3.1.1.3.cmml" xref="S2.SS5.p1.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.3.m3.1c">\mathcal{B}_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.3.m3.1d">caligraphic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and all non-rejected values are collected into the set <math alttext="\hat{\mathcal{B}}_{I,k}" class="ltx_Math" display="inline" id="S2.SS5.p1.4.m4.2"><semantics id="S2.SS5.p1.4.m4.2a"><msub id="S2.SS5.p1.4.m4.2.3" xref="S2.SS5.p1.4.m4.2.3.cmml"><mover accent="true" id="S2.SS5.p1.4.m4.2.3.2" xref="S2.SS5.p1.4.m4.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p1.4.m4.2.3.2.2" xref="S2.SS5.p1.4.m4.2.3.2.2.cmml">ℬ</mi><mo id="S2.SS5.p1.4.m4.2.3.2.1" xref="S2.SS5.p1.4.m4.2.3.2.1.cmml">^</mo></mover><mrow id="S2.SS5.p1.4.m4.2.2.2.4" xref="S2.SS5.p1.4.m4.2.2.2.3.cmml"><mi id="S2.SS5.p1.4.m4.1.1.1.1" xref="S2.SS5.p1.4.m4.1.1.1.1.cmml">I</mi><mo id="S2.SS5.p1.4.m4.2.2.2.4.1" xref="S2.SS5.p1.4.m4.2.2.2.3.cmml">,</mo><mi id="S2.SS5.p1.4.m4.2.2.2.2" xref="S2.SS5.p1.4.m4.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.4.m4.2b"><apply id="S2.SS5.p1.4.m4.2.3.cmml" xref="S2.SS5.p1.4.m4.2.3"><csymbol cd="ambiguous" id="S2.SS5.p1.4.m4.2.3.1.cmml" xref="S2.SS5.p1.4.m4.2.3">subscript</csymbol><apply id="S2.SS5.p1.4.m4.2.3.2.cmml" xref="S2.SS5.p1.4.m4.2.3.2"><ci id="S2.SS5.p1.4.m4.2.3.2.1.cmml" xref="S2.SS5.p1.4.m4.2.3.2.1">^</ci><ci id="S2.SS5.p1.4.m4.2.3.2.2.cmml" xref="S2.SS5.p1.4.m4.2.3.2.2">ℬ</ci></apply><list id="S2.SS5.p1.4.m4.2.2.2.3.cmml" xref="S2.SS5.p1.4.m4.2.2.2.4"><ci id="S2.SS5.p1.4.m4.1.1.1.1.cmml" xref="S2.SS5.p1.4.m4.1.1.1.1">𝐼</ci><ci id="S2.SS5.p1.4.m4.2.2.2.2.cmml" xref="S2.SS5.p1.4.m4.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.4.m4.2c">\hat{\mathcal{B}}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.4.m4.2d">over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, which is then an estimator for <math alttext="\mathcal{B}_{I,k}" class="ltx_Math" display="inline" id="S2.SS5.p1.5.m5.2"><semantics id="S2.SS5.p1.5.m5.2a"><msub id="S2.SS5.p1.5.m5.2.3" xref="S2.SS5.p1.5.m5.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p1.5.m5.2.3.2" xref="S2.SS5.p1.5.m5.2.3.2.cmml">ℬ</mi><mrow id="S2.SS5.p1.5.m5.2.2.2.4" xref="S2.SS5.p1.5.m5.2.2.2.3.cmml"><mi id="S2.SS5.p1.5.m5.1.1.1.1" xref="S2.SS5.p1.5.m5.1.1.1.1.cmml">I</mi><mo id="S2.SS5.p1.5.m5.2.2.2.4.1" xref="S2.SS5.p1.5.m5.2.2.2.3.cmml">,</mo><mi id="S2.SS5.p1.5.m5.2.2.2.2" xref="S2.SS5.p1.5.m5.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.5.m5.2b"><apply id="S2.SS5.p1.5.m5.2.3.cmml" xref="S2.SS5.p1.5.m5.2.3"><csymbol cd="ambiguous" id="S2.SS5.p1.5.m5.2.3.1.cmml" xref="S2.SS5.p1.5.m5.2.3">subscript</csymbol><ci id="S2.SS5.p1.5.m5.2.3.2.cmml" xref="S2.SS5.p1.5.m5.2.3.2">ℬ</ci><list id="S2.SS5.p1.5.m5.2.2.2.3.cmml" xref="S2.SS5.p1.5.m5.2.2.2.4"><ci id="S2.SS5.p1.5.m5.1.1.1.1.cmml" xref="S2.SS5.p1.5.m5.1.1.1.1">𝐼</ci><ci id="S2.SS5.p1.5.m5.2.2.2.2.cmml" xref="S2.SS5.p1.5.m5.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.5.m5.2c">\mathcal{B}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.5.m5.2d">caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. We refer to Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Algorithm_and_implementation_details</span> for details on how this can be done in practice via root finding algorithms. Notably, <math alttext="\hat{\mathcal{B}}_{I,k}" class="ltx_Math" display="inline" id="S2.SS5.p1.6.m6.2"><semantics id="S2.SS5.p1.6.m6.2a"><msub id="S2.SS5.p1.6.m6.2.3" xref="S2.SS5.p1.6.m6.2.3.cmml"><mover accent="true" id="S2.SS5.p1.6.m6.2.3.2" xref="S2.SS5.p1.6.m6.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p1.6.m6.2.3.2.2" xref="S2.SS5.p1.6.m6.2.3.2.2.cmml">ℬ</mi><mo id="S2.SS5.p1.6.m6.2.3.2.1" xref="S2.SS5.p1.6.m6.2.3.2.1.cmml">^</mo></mover><mrow id="S2.SS5.p1.6.m6.2.2.2.4" xref="S2.SS5.p1.6.m6.2.2.2.3.cmml"><mi id="S2.SS5.p1.6.m6.1.1.1.1" xref="S2.SS5.p1.6.m6.1.1.1.1.cmml">I</mi><mo id="S2.SS5.p1.6.m6.2.2.2.4.1" xref="S2.SS5.p1.6.m6.2.2.2.3.cmml">,</mo><mi id="S2.SS5.p1.6.m6.2.2.2.2" xref="S2.SS5.p1.6.m6.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.p1.6.m6.2b"><apply id="S2.SS5.p1.6.m6.2.3.cmml" xref="S2.SS5.p1.6.m6.2.3"><csymbol cd="ambiguous" id="S2.SS5.p1.6.m6.2.3.1.cmml" xref="S2.SS5.p1.6.m6.2.3">subscript</csymbol><apply id="S2.SS5.p1.6.m6.2.3.2.cmml" xref="S2.SS5.p1.6.m6.2.3.2"><ci id="S2.SS5.p1.6.m6.2.3.2.1.cmml" xref="S2.SS5.p1.6.m6.2.3.2.1">^</ci><ci id="S2.SS5.p1.6.m6.2.3.2.2.cmml" xref="S2.SS5.p1.6.m6.2.3.2.2">ℬ</ci></apply><list id="S2.SS5.p1.6.m6.2.2.2.3.cmml" xref="S2.SS5.p1.6.m6.2.2.2.4"><ci id="S2.SS5.p1.6.m6.1.1.1.1.cmml" xref="S2.SS5.p1.6.m6.1.1.1.1">𝐼</ci><ci id="S2.SS5.p1.6.m6.2.2.2.2.cmml" xref="S2.SS5.p1.6.m6.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p1.6.m6.2c">\hat{\mathcal{B}}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p1.6.m6.2d">over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> will, in most cases, be an interval so that it can be interpreted as a standard confidence interval (Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.SS4" title="3.4 Discussion ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">3.4</span></a>).</p> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Estimation procedure</h2> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.1 </span>Modeling assumptions and theoretical results</h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.1">Throughout the rest of this paper, we will assume that interest lies in the <math alttext="k" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><mi id="S3.SS1.p1.1.m1.1.1" xref="S3.SS1.p1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><ci id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">italic_k</annotation></semantics></math>-th element of the parameter vector. We require the following set of assumptions:</p> <ol class="ltx_enumerate" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(A1)</span> <div class="ltx_para" id="S3.I1.i1.p1"> <p class="ltx_p" id="S3.I1.i1.p1.1">The observations <math alttext="\{W_{i},i=1,\dots,n\}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.1.m1.4"><semantics id="S3.I1.i1.p1.1.m1.4a"><mrow id="S3.I1.i1.p1.1.m1.4.4.1" xref="S3.I1.i1.p1.1.m1.4.4.2.cmml"><mo id="S3.I1.i1.p1.1.m1.4.4.1.2" stretchy="false" xref="S3.I1.i1.p1.1.m1.4.4.2.cmml">{</mo><mrow id="S3.I1.i1.p1.1.m1.4.4.1.1.2" xref="S3.I1.i1.p1.1.m1.4.4.1.1.3.cmml"><mrow id="S3.I1.i1.p1.1.m1.4.4.1.1.1.1" xref="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.cmml"><mrow id="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.1" xref="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.2.cmml"><msub id="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.1.1" xref="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.1.1.cmml"><mi id="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.1.1.2" xref="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.1.1.2.cmml">W</mi><mi id="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.1.1.3" xref="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.1.2" xref="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.2.cmml">,</mo><mi id="S3.I1.i1.p1.1.m1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.cmml">i</mi></mrow><mo id="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.2" xref="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.2.cmml">=</mo><mn id="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.3" xref="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.I1.i1.p1.1.m1.4.4.1.1.2.3" xref="S3.I1.i1.p1.1.m1.4.4.1.1.3a.cmml">,</mo><mrow id="S3.I1.i1.p1.1.m1.4.4.1.1.2.2.2" xref="S3.I1.i1.p1.1.m1.4.4.1.1.2.2.1.cmml"><mi id="S3.I1.i1.p1.1.m1.2.2" mathvariant="normal" xref="S3.I1.i1.p1.1.m1.2.2.cmml">…</mi><mo id="S3.I1.i1.p1.1.m1.4.4.1.1.2.2.2.1" xref="S3.I1.i1.p1.1.m1.4.4.1.1.2.2.1.cmml">,</mo><mi id="S3.I1.i1.p1.1.m1.3.3" xref="S3.I1.i1.p1.1.m1.3.3.cmml">n</mi></mrow></mrow><mo id="S3.I1.i1.p1.1.m1.4.4.1.3" stretchy="false" xref="S3.I1.i1.p1.1.m1.4.4.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.1.m1.4b"><set id="S3.I1.i1.p1.1.m1.4.4.2.cmml" xref="S3.I1.i1.p1.1.m1.4.4.1"><apply id="S3.I1.i1.p1.1.m1.4.4.1.1.3.cmml" xref="S3.I1.i1.p1.1.m1.4.4.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.1.m1.4.4.1.1.3a.cmml" xref="S3.I1.i1.p1.1.m1.4.4.1.1.2.3">formulae-sequence</csymbol><apply id="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.4.4.1.1.1.1"><eq id="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.2.cmml" xref="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.2"></eq><list id="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.2.cmml" xref="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.1"><apply id="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.1.1.2.cmml" xref="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.1.1.2">𝑊</ci><ci id="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.1.1.3.cmml" xref="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.1.1.1.3">𝑖</ci></apply><ci id="S3.I1.i1.p1.1.m1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1">𝑖</ci></list><cn id="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.3.cmml" type="integer" xref="S3.I1.i1.p1.1.m1.4.4.1.1.1.1.3">1</cn></apply><list id="S3.I1.i1.p1.1.m1.4.4.1.1.2.2.1.cmml" xref="S3.I1.i1.p1.1.m1.4.4.1.1.2.2.2"><ci id="S3.I1.i1.p1.1.m1.2.2.cmml" xref="S3.I1.i1.p1.1.m1.2.2">…</ci><ci id="S3.I1.i1.p1.1.m1.3.3.cmml" xref="S3.I1.i1.p1.1.m1.3.3">𝑛</ci></list></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.1.m1.4c">\{W_{i},i=1,\dots,n\}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.1.m1.4d">{ italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_i = 1 , … , italic_n }</annotation></semantics></math> are independent and identically distributed.</p> </div> </li> <li class="ltx_item" id="S3.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(A2)</span> <div class="ltx_para" id="S3.I1.i2.p1"> <p class="ltx_p" id="S3.I1.i2.p1.1"><math alttext="\Lambda:\mathbb{R}\to(0,1)" class="ltx_Math" display="inline" id="S3.I1.i2.p1.1.m1.2"><semantics id="S3.I1.i2.p1.1.m1.2a"><mrow id="S3.I1.i2.p1.1.m1.2.3" xref="S3.I1.i2.p1.1.m1.2.3.cmml"><mi id="S3.I1.i2.p1.1.m1.2.3.2" mathvariant="normal" xref="S3.I1.i2.p1.1.m1.2.3.2.cmml">Λ</mi><mo id="S3.I1.i2.p1.1.m1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S3.I1.i2.p1.1.m1.2.3.1.cmml">:</mo><mrow id="S3.I1.i2.p1.1.m1.2.3.3" xref="S3.I1.i2.p1.1.m1.2.3.3.cmml"><mi id="S3.I1.i2.p1.1.m1.2.3.3.2" xref="S3.I1.i2.p1.1.m1.2.3.3.2.cmml">ℝ</mi><mo id="S3.I1.i2.p1.1.m1.2.3.3.1" stretchy="false" xref="S3.I1.i2.p1.1.m1.2.3.3.1.cmml">→</mo><mrow id="S3.I1.i2.p1.1.m1.2.3.3.3.2" xref="S3.I1.i2.p1.1.m1.2.3.3.3.1.cmml"><mo id="S3.I1.i2.p1.1.m1.2.3.3.3.2.1" stretchy="false" xref="S3.I1.i2.p1.1.m1.2.3.3.3.1.cmml">(</mo><mn id="S3.I1.i2.p1.1.m1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.cmml">0</mn><mo id="S3.I1.i2.p1.1.m1.2.3.3.3.2.2" xref="S3.I1.i2.p1.1.m1.2.3.3.3.1.cmml">,</mo><mn id="S3.I1.i2.p1.1.m1.2.2" xref="S3.I1.i2.p1.1.m1.2.2.cmml">1</mn><mo id="S3.I1.i2.p1.1.m1.2.3.3.3.2.3" stretchy="false" xref="S3.I1.i2.p1.1.m1.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.1.m1.2b"><apply id="S3.I1.i2.p1.1.m1.2.3.cmml" xref="S3.I1.i2.p1.1.m1.2.3"><ci id="S3.I1.i2.p1.1.m1.2.3.1.cmml" xref="S3.I1.i2.p1.1.m1.2.3.1">:</ci><ci id="S3.I1.i2.p1.1.m1.2.3.2.cmml" xref="S3.I1.i2.p1.1.m1.2.3.2">Λ</ci><apply id="S3.I1.i2.p1.1.m1.2.3.3.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3"><ci id="S3.I1.i2.p1.1.m1.2.3.3.1.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.1">→</ci><ci id="S3.I1.i2.p1.1.m1.2.3.3.2.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2">ℝ</ci><interval closure="open" id="S3.I1.i2.p1.1.m1.2.3.3.3.1.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.3.2"><cn id="S3.I1.i2.p1.1.m1.1.1.cmml" type="integer" xref="S3.I1.i2.p1.1.m1.1.1">0</cn><cn id="S3.I1.i2.p1.1.m1.2.2.cmml" type="integer" xref="S3.I1.i2.p1.1.m1.2.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.1.m1.2c">\Lambda:\mathbb{R}\to(0,1)</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.1.m1.2d">roman_Λ : blackboard_R → ( 0 , 1 )</annotation></semantics></math> is a strictly increasing function that is twice continuously differentiable.</p> </div> </li> <li class="ltx_item" id="S3.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(A3)</span> <div class="ltx_para" id="S3.I1.i3.p1"> <p class="ltx_p" id="S3.I1.i3.p1.4"><math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S3.I1.i3.p1.1.m1.1"><semantics id="S3.I1.i3.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.I1.i3.p1.1.m1.1.1" xref="S3.I1.i3.p1.1.m1.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.1.m1.1b"><ci id="S3.I1.i3.p1.1.m1.1.1.cmml" xref="S3.I1.i3.p1.1.m1.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.1.m1.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.1.m1.1d">caligraphic_G</annotation></semantics></math> is a class of continuous functions <math alttext="g_{j}:\mathbb{R}^{d+1}\to[0,M_{\mathcal{G}}]" class="ltx_Math" display="inline" id="S3.I1.i3.p1.2.m2.2"><semantics id="S3.I1.i3.p1.2.m2.2a"><mrow id="S3.I1.i3.p1.2.m2.2.2" xref="S3.I1.i3.p1.2.m2.2.2.cmml"><msub id="S3.I1.i3.p1.2.m2.2.2.3" xref="S3.I1.i3.p1.2.m2.2.2.3.cmml"><mi id="S3.I1.i3.p1.2.m2.2.2.3.2" xref="S3.I1.i3.p1.2.m2.2.2.3.2.cmml">g</mi><mi id="S3.I1.i3.p1.2.m2.2.2.3.3" xref="S3.I1.i3.p1.2.m2.2.2.3.3.cmml">j</mi></msub><mo id="S3.I1.i3.p1.2.m2.2.2.2" lspace="0.278em" rspace="0.278em" xref="S3.I1.i3.p1.2.m2.2.2.2.cmml">:</mo><mrow id="S3.I1.i3.p1.2.m2.2.2.1" xref="S3.I1.i3.p1.2.m2.2.2.1.cmml"><msup id="S3.I1.i3.p1.2.m2.2.2.1.3" xref="S3.I1.i3.p1.2.m2.2.2.1.3.cmml"><mi id="S3.I1.i3.p1.2.m2.2.2.1.3.2" xref="S3.I1.i3.p1.2.m2.2.2.1.3.2.cmml">ℝ</mi><mrow id="S3.I1.i3.p1.2.m2.2.2.1.3.3" xref="S3.I1.i3.p1.2.m2.2.2.1.3.3.cmml"><mi id="S3.I1.i3.p1.2.m2.2.2.1.3.3.2" xref="S3.I1.i3.p1.2.m2.2.2.1.3.3.2.cmml">d</mi><mo id="S3.I1.i3.p1.2.m2.2.2.1.3.3.1" xref="S3.I1.i3.p1.2.m2.2.2.1.3.3.1.cmml">+</mo><mn id="S3.I1.i3.p1.2.m2.2.2.1.3.3.3" xref="S3.I1.i3.p1.2.m2.2.2.1.3.3.3.cmml">1</mn></mrow></msup><mo id="S3.I1.i3.p1.2.m2.2.2.1.2" stretchy="false" xref="S3.I1.i3.p1.2.m2.2.2.1.2.cmml">→</mo><mrow id="S3.I1.i3.p1.2.m2.2.2.1.1.1" xref="S3.I1.i3.p1.2.m2.2.2.1.1.2.cmml"><mo id="S3.I1.i3.p1.2.m2.2.2.1.1.1.2" stretchy="false" xref="S3.I1.i3.p1.2.m2.2.2.1.1.2.cmml">[</mo><mn id="S3.I1.i3.p1.2.m2.1.1" xref="S3.I1.i3.p1.2.m2.1.1.cmml">0</mn><mo id="S3.I1.i3.p1.2.m2.2.2.1.1.1.3" xref="S3.I1.i3.p1.2.m2.2.2.1.1.2.cmml">,</mo><msub id="S3.I1.i3.p1.2.m2.2.2.1.1.1.1" xref="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.cmml"><mi id="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.2" xref="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.2.cmml">M</mi><mi class="ltx_font_mathcaligraphic" id="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.3" xref="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.3.cmml">𝒢</mi></msub><mo id="S3.I1.i3.p1.2.m2.2.2.1.1.1.4" stretchy="false" xref="S3.I1.i3.p1.2.m2.2.2.1.1.2.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.2.m2.2b"><apply id="S3.I1.i3.p1.2.m2.2.2.cmml" xref="S3.I1.i3.p1.2.m2.2.2"><ci id="S3.I1.i3.p1.2.m2.2.2.2.cmml" xref="S3.I1.i3.p1.2.m2.2.2.2">:</ci><apply id="S3.I1.i3.p1.2.m2.2.2.3.cmml" xref="S3.I1.i3.p1.2.m2.2.2.3"><csymbol cd="ambiguous" id="S3.I1.i3.p1.2.m2.2.2.3.1.cmml" xref="S3.I1.i3.p1.2.m2.2.2.3">subscript</csymbol><ci id="S3.I1.i3.p1.2.m2.2.2.3.2.cmml" xref="S3.I1.i3.p1.2.m2.2.2.3.2">𝑔</ci><ci id="S3.I1.i3.p1.2.m2.2.2.3.3.cmml" xref="S3.I1.i3.p1.2.m2.2.2.3.3">𝑗</ci></apply><apply id="S3.I1.i3.p1.2.m2.2.2.1.cmml" xref="S3.I1.i3.p1.2.m2.2.2.1"><ci id="S3.I1.i3.p1.2.m2.2.2.1.2.cmml" xref="S3.I1.i3.p1.2.m2.2.2.1.2">→</ci><apply id="S3.I1.i3.p1.2.m2.2.2.1.3.cmml" xref="S3.I1.i3.p1.2.m2.2.2.1.3"><csymbol cd="ambiguous" id="S3.I1.i3.p1.2.m2.2.2.1.3.1.cmml" xref="S3.I1.i3.p1.2.m2.2.2.1.3">superscript</csymbol><ci id="S3.I1.i3.p1.2.m2.2.2.1.3.2.cmml" xref="S3.I1.i3.p1.2.m2.2.2.1.3.2">ℝ</ci><apply id="S3.I1.i3.p1.2.m2.2.2.1.3.3.cmml" xref="S3.I1.i3.p1.2.m2.2.2.1.3.3"><plus id="S3.I1.i3.p1.2.m2.2.2.1.3.3.1.cmml" xref="S3.I1.i3.p1.2.m2.2.2.1.3.3.1"></plus><ci id="S3.I1.i3.p1.2.m2.2.2.1.3.3.2.cmml" xref="S3.I1.i3.p1.2.m2.2.2.1.3.3.2">𝑑</ci><cn id="S3.I1.i3.p1.2.m2.2.2.1.3.3.3.cmml" type="integer" xref="S3.I1.i3.p1.2.m2.2.2.1.3.3.3">1</cn></apply></apply><interval closure="closed" id="S3.I1.i3.p1.2.m2.2.2.1.1.2.cmml" xref="S3.I1.i3.p1.2.m2.2.2.1.1.1"><cn id="S3.I1.i3.p1.2.m2.1.1.cmml" type="integer" xref="S3.I1.i3.p1.2.m2.1.1">0</cn><apply id="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.cmml" xref="S3.I1.i3.p1.2.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.1.cmml" xref="S3.I1.i3.p1.2.m2.2.2.1.1.1.1">subscript</csymbol><ci id="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.2.cmml" xref="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.2">𝑀</ci><ci id="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.3.cmml" xref="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.3">𝒢</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.2.m2.2c">g_{j}:\mathbb{R}^{d+1}\to[0,M_{\mathcal{G}}]</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.2.m2.2d">italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT : blackboard_R start_POSTSUPERSCRIPT italic_d + 1 end_POSTSUPERSCRIPT → [ 0 , italic_M start_POSTSUBSCRIPT caligraphic_G end_POSTSUBSCRIPT ]</annotation></semantics></math>, <math alttext="j=1,\dots,J" class="ltx_Math" display="inline" id="S3.I1.i3.p1.3.m3.3"><semantics id="S3.I1.i3.p1.3.m3.3a"><mrow id="S3.I1.i3.p1.3.m3.3.4" xref="S3.I1.i3.p1.3.m3.3.4.cmml"><mi id="S3.I1.i3.p1.3.m3.3.4.2" xref="S3.I1.i3.p1.3.m3.3.4.2.cmml">j</mi><mo id="S3.I1.i3.p1.3.m3.3.4.1" xref="S3.I1.i3.p1.3.m3.3.4.1.cmml">=</mo><mrow id="S3.I1.i3.p1.3.m3.3.4.3.2" xref="S3.I1.i3.p1.3.m3.3.4.3.1.cmml"><mn id="S3.I1.i3.p1.3.m3.1.1" xref="S3.I1.i3.p1.3.m3.1.1.cmml">1</mn><mo id="S3.I1.i3.p1.3.m3.3.4.3.2.1" xref="S3.I1.i3.p1.3.m3.3.4.3.1.cmml">,</mo><mi id="S3.I1.i3.p1.3.m3.2.2" mathvariant="normal" xref="S3.I1.i3.p1.3.m3.2.2.cmml">…</mi><mo id="S3.I1.i3.p1.3.m3.3.4.3.2.2" xref="S3.I1.i3.p1.3.m3.3.4.3.1.cmml">,</mo><mi id="S3.I1.i3.p1.3.m3.3.3" xref="S3.I1.i3.p1.3.m3.3.3.cmml">J</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.3.m3.3b"><apply id="S3.I1.i3.p1.3.m3.3.4.cmml" xref="S3.I1.i3.p1.3.m3.3.4"><eq id="S3.I1.i3.p1.3.m3.3.4.1.cmml" xref="S3.I1.i3.p1.3.m3.3.4.1"></eq><ci id="S3.I1.i3.p1.3.m3.3.4.2.cmml" xref="S3.I1.i3.p1.3.m3.3.4.2">𝑗</ci><list id="S3.I1.i3.p1.3.m3.3.4.3.1.cmml" xref="S3.I1.i3.p1.3.m3.3.4.3.2"><cn id="S3.I1.i3.p1.3.m3.1.1.cmml" type="integer" xref="S3.I1.i3.p1.3.m3.1.1">1</cn><ci id="S3.I1.i3.p1.3.m3.2.2.cmml" xref="S3.I1.i3.p1.3.m3.2.2">…</ci><ci id="S3.I1.i3.p1.3.m3.3.3.cmml" xref="S3.I1.i3.p1.3.m3.3.3">𝐽</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.3.m3.3c">j=1,\dots,J</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.3.m3.3d">italic_j = 1 , … , italic_J</annotation></semantics></math>, for some <math alttext="M_{\mathcal{G}}<\infty" class="ltx_Math" display="inline" id="S3.I1.i3.p1.4.m4.1"><semantics id="S3.I1.i3.p1.4.m4.1a"><mrow id="S3.I1.i3.p1.4.m4.1.1" xref="S3.I1.i3.p1.4.m4.1.1.cmml"><msub id="S3.I1.i3.p1.4.m4.1.1.2" xref="S3.I1.i3.p1.4.m4.1.1.2.cmml"><mi id="S3.I1.i3.p1.4.m4.1.1.2.2" xref="S3.I1.i3.p1.4.m4.1.1.2.2.cmml">M</mi><mi class="ltx_font_mathcaligraphic" id="S3.I1.i3.p1.4.m4.1.1.2.3" xref="S3.I1.i3.p1.4.m4.1.1.2.3.cmml">𝒢</mi></msub><mo id="S3.I1.i3.p1.4.m4.1.1.1" xref="S3.I1.i3.p1.4.m4.1.1.1.cmml"><</mo><mi id="S3.I1.i3.p1.4.m4.1.1.3" mathvariant="normal" xref="S3.I1.i3.p1.4.m4.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.4.m4.1b"><apply id="S3.I1.i3.p1.4.m4.1.1.cmml" xref="S3.I1.i3.p1.4.m4.1.1"><lt id="S3.I1.i3.p1.4.m4.1.1.1.cmml" xref="S3.I1.i3.p1.4.m4.1.1.1"></lt><apply id="S3.I1.i3.p1.4.m4.1.1.2.cmml" xref="S3.I1.i3.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i3.p1.4.m4.1.1.2.1.cmml" xref="S3.I1.i3.p1.4.m4.1.1.2">subscript</csymbol><ci id="S3.I1.i3.p1.4.m4.1.1.2.2.cmml" xref="S3.I1.i3.p1.4.m4.1.1.2.2">𝑀</ci><ci id="S3.I1.i3.p1.4.m4.1.1.2.3.cmml" xref="S3.I1.i3.p1.4.m4.1.1.2.3">𝒢</ci></apply><infinity id="S3.I1.i3.p1.4.m4.1.1.3.cmml" xref="S3.I1.i3.p1.4.m4.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.4.m4.1c">M_{\mathcal{G}}<\infty</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.4.m4.1d">italic_M start_POSTSUBSCRIPT caligraphic_G end_POSTSUBSCRIPT < ∞</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S3.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(A4)</span> <div class="ltx_para" id="S3.I1.i4.p1"> <p class="ltx_p" id="S3.I1.i4.p1.5">Let <math alttext="\mathcal{X}_{g,j}" class="ltx_Math" display="inline" id="S3.I1.i4.p1.1.m1.2"><semantics id="S3.I1.i4.p1.1.m1.2a"><msub id="S3.I1.i4.p1.1.m1.2.3" xref="S3.I1.i4.p1.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I1.i4.p1.1.m1.2.3.2" xref="S3.I1.i4.p1.1.m1.2.3.2.cmml">𝒳</mi><mrow id="S3.I1.i4.p1.1.m1.2.2.2.4" xref="S3.I1.i4.p1.1.m1.2.2.2.3.cmml"><mi id="S3.I1.i4.p1.1.m1.1.1.1.1" xref="S3.I1.i4.p1.1.m1.1.1.1.1.cmml">g</mi><mo id="S3.I1.i4.p1.1.m1.2.2.2.4.1" xref="S3.I1.i4.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.I1.i4.p1.1.m1.2.2.2.2" xref="S3.I1.i4.p1.1.m1.2.2.2.2.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i4.p1.1.m1.2b"><apply id="S3.I1.i4.p1.1.m1.2.3.cmml" xref="S3.I1.i4.p1.1.m1.2.3"><csymbol cd="ambiguous" id="S3.I1.i4.p1.1.m1.2.3.1.cmml" xref="S3.I1.i4.p1.1.m1.2.3">subscript</csymbol><ci id="S3.I1.i4.p1.1.m1.2.3.2.cmml" xref="S3.I1.i4.p1.1.m1.2.3.2">𝒳</ci><list id="S3.I1.i4.p1.1.m1.2.2.2.3.cmml" xref="S3.I1.i4.p1.1.m1.2.2.2.4"><ci id="S3.I1.i4.p1.1.m1.1.1.1.1.cmml" xref="S3.I1.i4.p1.1.m1.1.1.1.1">𝑔</ci><ci id="S3.I1.i4.p1.1.m1.2.2.2.2.cmml" xref="S3.I1.i4.p1.1.m1.2.2.2.2">𝑗</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i4.p1.1.m1.2c">\mathcal{X}_{g,j}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i4.p1.1.m1.2d">caligraphic_X start_POSTSUBSCRIPT italic_g , italic_j end_POSTSUBSCRIPT</annotation></semantics></math> denote the support of the <math alttext="j" class="ltx_Math" display="inline" id="S3.I1.i4.p1.2.m2.1"><semantics id="S3.I1.i4.p1.2.m2.1a"><mi id="S3.I1.i4.p1.2.m2.1.1" xref="S3.I1.i4.p1.2.m2.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i4.p1.2.m2.1b"><ci id="S3.I1.i4.p1.2.m2.1.1.cmml" xref="S3.I1.i4.p1.2.m2.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i4.p1.2.m2.1c">j</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i4.p1.2.m2.1d">italic_j</annotation></semantics></math>-th instrumental function <math alttext="g_{j}(\cdot)" class="ltx_Math" display="inline" id="S3.I1.i4.p1.3.m3.1"><semantics id="S3.I1.i4.p1.3.m3.1a"><mrow id="S3.I1.i4.p1.3.m3.1.2" xref="S3.I1.i4.p1.3.m3.1.2.cmml"><msub id="S3.I1.i4.p1.3.m3.1.2.2" xref="S3.I1.i4.p1.3.m3.1.2.2.cmml"><mi id="S3.I1.i4.p1.3.m3.1.2.2.2" xref="S3.I1.i4.p1.3.m3.1.2.2.2.cmml">g</mi><mi id="S3.I1.i4.p1.3.m3.1.2.2.3" xref="S3.I1.i4.p1.3.m3.1.2.2.3.cmml">j</mi></msub><mo id="S3.I1.i4.p1.3.m3.1.2.1" xref="S3.I1.i4.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S3.I1.i4.p1.3.m3.1.2.3.2" xref="S3.I1.i4.p1.3.m3.1.2.cmml"><mo id="S3.I1.i4.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S3.I1.i4.p1.3.m3.1.2.cmml">(</mo><mo id="S3.I1.i4.p1.3.m3.1.1" lspace="0em" rspace="0em" xref="S3.I1.i4.p1.3.m3.1.1.cmml">⋅</mo><mo id="S3.I1.i4.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S3.I1.i4.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i4.p1.3.m3.1b"><apply id="S3.I1.i4.p1.3.m3.1.2.cmml" xref="S3.I1.i4.p1.3.m3.1.2"><times id="S3.I1.i4.p1.3.m3.1.2.1.cmml" xref="S3.I1.i4.p1.3.m3.1.2.1"></times><apply id="S3.I1.i4.p1.3.m3.1.2.2.cmml" xref="S3.I1.i4.p1.3.m3.1.2.2"><csymbol cd="ambiguous" id="S3.I1.i4.p1.3.m3.1.2.2.1.cmml" xref="S3.I1.i4.p1.3.m3.1.2.2">subscript</csymbol><ci id="S3.I1.i4.p1.3.m3.1.2.2.2.cmml" xref="S3.I1.i4.p1.3.m3.1.2.2.2">𝑔</ci><ci id="S3.I1.i4.p1.3.m3.1.2.2.3.cmml" xref="S3.I1.i4.p1.3.m3.1.2.2.3">𝑗</ci></apply><ci id="S3.I1.i4.p1.3.m3.1.1.cmml" xref="S3.I1.i4.p1.3.m3.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i4.p1.3.m3.1c">g_{j}(\cdot)</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i4.p1.3.m3.1d">italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( ⋅ )</annotation></semantics></math>. Then <math alttext="\mathcal{X}\cap\mathcal{X}_{g,j}\neq\emptyset" class="ltx_Math" display="inline" id="S3.I1.i4.p1.4.m4.2"><semantics id="S3.I1.i4.p1.4.m4.2a"><mrow id="S3.I1.i4.p1.4.m4.2.3" xref="S3.I1.i4.p1.4.m4.2.3.cmml"><mrow id="S3.I1.i4.p1.4.m4.2.3.2" xref="S3.I1.i4.p1.4.m4.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I1.i4.p1.4.m4.2.3.2.2" xref="S3.I1.i4.p1.4.m4.2.3.2.2.cmml">𝒳</mi><mo id="S3.I1.i4.p1.4.m4.2.3.2.1" xref="S3.I1.i4.p1.4.m4.2.3.2.1.cmml">∩</mo><msub id="S3.I1.i4.p1.4.m4.2.3.2.3" xref="S3.I1.i4.p1.4.m4.2.3.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I1.i4.p1.4.m4.2.3.2.3.2" xref="S3.I1.i4.p1.4.m4.2.3.2.3.2.cmml">𝒳</mi><mrow id="S3.I1.i4.p1.4.m4.2.2.2.4" xref="S3.I1.i4.p1.4.m4.2.2.2.3.cmml"><mi id="S3.I1.i4.p1.4.m4.1.1.1.1" xref="S3.I1.i4.p1.4.m4.1.1.1.1.cmml">g</mi><mo id="S3.I1.i4.p1.4.m4.2.2.2.4.1" xref="S3.I1.i4.p1.4.m4.2.2.2.3.cmml">,</mo><mi id="S3.I1.i4.p1.4.m4.2.2.2.2" xref="S3.I1.i4.p1.4.m4.2.2.2.2.cmml">j</mi></mrow></msub></mrow><mo id="S3.I1.i4.p1.4.m4.2.3.1" xref="S3.I1.i4.p1.4.m4.2.3.1.cmml">≠</mo><mi id="S3.I1.i4.p1.4.m4.2.3.3" mathvariant="normal" xref="S3.I1.i4.p1.4.m4.2.3.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i4.p1.4.m4.2b"><apply id="S3.I1.i4.p1.4.m4.2.3.cmml" xref="S3.I1.i4.p1.4.m4.2.3"><neq id="S3.I1.i4.p1.4.m4.2.3.1.cmml" xref="S3.I1.i4.p1.4.m4.2.3.1"></neq><apply id="S3.I1.i4.p1.4.m4.2.3.2.cmml" xref="S3.I1.i4.p1.4.m4.2.3.2"><intersect id="S3.I1.i4.p1.4.m4.2.3.2.1.cmml" xref="S3.I1.i4.p1.4.m4.2.3.2.1"></intersect><ci id="S3.I1.i4.p1.4.m4.2.3.2.2.cmml" xref="S3.I1.i4.p1.4.m4.2.3.2.2">𝒳</ci><apply id="S3.I1.i4.p1.4.m4.2.3.2.3.cmml" xref="S3.I1.i4.p1.4.m4.2.3.2.3"><csymbol cd="ambiguous" id="S3.I1.i4.p1.4.m4.2.3.2.3.1.cmml" xref="S3.I1.i4.p1.4.m4.2.3.2.3">subscript</csymbol><ci id="S3.I1.i4.p1.4.m4.2.3.2.3.2.cmml" xref="S3.I1.i4.p1.4.m4.2.3.2.3.2">𝒳</ci><list id="S3.I1.i4.p1.4.m4.2.2.2.3.cmml" xref="S3.I1.i4.p1.4.m4.2.2.2.4"><ci id="S3.I1.i4.p1.4.m4.1.1.1.1.cmml" xref="S3.I1.i4.p1.4.m4.1.1.1.1">𝑔</ci><ci id="S3.I1.i4.p1.4.m4.2.2.2.2.cmml" xref="S3.I1.i4.p1.4.m4.2.2.2.2">𝑗</ci></list></apply></apply><emptyset id="S3.I1.i4.p1.4.m4.2.3.3.cmml" xref="S3.I1.i4.p1.4.m4.2.3.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i4.p1.4.m4.2c">\mathcal{X}\cap\mathcal{X}_{g,j}\neq\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i4.p1.4.m4.2d">caligraphic_X ∩ caligraphic_X start_POSTSUBSCRIPT italic_g , italic_j end_POSTSUBSCRIPT ≠ ∅</annotation></semantics></math> and <math alttext="\mathcal{X}\subset\bigcup_{j=1}^{J}\mathcal{X}_{g,j}" class="ltx_Math" display="inline" id="S3.I1.i4.p1.5.m5.2"><semantics id="S3.I1.i4.p1.5.m5.2a"><mrow id="S3.I1.i4.p1.5.m5.2.3" xref="S3.I1.i4.p1.5.m5.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I1.i4.p1.5.m5.2.3.2" xref="S3.I1.i4.p1.5.m5.2.3.2.cmml">𝒳</mi><mo id="S3.I1.i4.p1.5.m5.2.3.1" rspace="0.111em" xref="S3.I1.i4.p1.5.m5.2.3.1.cmml">⊂</mo><mrow id="S3.I1.i4.p1.5.m5.2.3.3" xref="S3.I1.i4.p1.5.m5.2.3.3.cmml"><msubsup id="S3.I1.i4.p1.5.m5.2.3.3.1" xref="S3.I1.i4.p1.5.m5.2.3.3.1.cmml"><mo id="S3.I1.i4.p1.5.m5.2.3.3.1.2.2" xref="S3.I1.i4.p1.5.m5.2.3.3.1.2.2.cmml">⋃</mo><mrow id="S3.I1.i4.p1.5.m5.2.3.3.1.2.3" xref="S3.I1.i4.p1.5.m5.2.3.3.1.2.3.cmml"><mi id="S3.I1.i4.p1.5.m5.2.3.3.1.2.3.2" xref="S3.I1.i4.p1.5.m5.2.3.3.1.2.3.2.cmml">j</mi><mo id="S3.I1.i4.p1.5.m5.2.3.3.1.2.3.1" xref="S3.I1.i4.p1.5.m5.2.3.3.1.2.3.1.cmml">=</mo><mn id="S3.I1.i4.p1.5.m5.2.3.3.1.2.3.3" xref="S3.I1.i4.p1.5.m5.2.3.3.1.2.3.3.cmml">1</mn></mrow><mi id="S3.I1.i4.p1.5.m5.2.3.3.1.3" xref="S3.I1.i4.p1.5.m5.2.3.3.1.3.cmml">J</mi></msubsup><msub id="S3.I1.i4.p1.5.m5.2.3.3.2" xref="S3.I1.i4.p1.5.m5.2.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I1.i4.p1.5.m5.2.3.3.2.2" xref="S3.I1.i4.p1.5.m5.2.3.3.2.2.cmml">𝒳</mi><mrow id="S3.I1.i4.p1.5.m5.2.2.2.4" xref="S3.I1.i4.p1.5.m5.2.2.2.3.cmml"><mi id="S3.I1.i4.p1.5.m5.1.1.1.1" xref="S3.I1.i4.p1.5.m5.1.1.1.1.cmml">g</mi><mo id="S3.I1.i4.p1.5.m5.2.2.2.4.1" xref="S3.I1.i4.p1.5.m5.2.2.2.3.cmml">,</mo><mi id="S3.I1.i4.p1.5.m5.2.2.2.2" xref="S3.I1.i4.p1.5.m5.2.2.2.2.cmml">j</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i4.p1.5.m5.2b"><apply id="S3.I1.i4.p1.5.m5.2.3.cmml" xref="S3.I1.i4.p1.5.m5.2.3"><subset id="S3.I1.i4.p1.5.m5.2.3.1.cmml" xref="S3.I1.i4.p1.5.m5.2.3.1"></subset><ci id="S3.I1.i4.p1.5.m5.2.3.2.cmml" xref="S3.I1.i4.p1.5.m5.2.3.2">𝒳</ci><apply id="S3.I1.i4.p1.5.m5.2.3.3.cmml" xref="S3.I1.i4.p1.5.m5.2.3.3"><apply id="S3.I1.i4.p1.5.m5.2.3.3.1.cmml" xref="S3.I1.i4.p1.5.m5.2.3.3.1"><csymbol cd="ambiguous" id="S3.I1.i4.p1.5.m5.2.3.3.1.1.cmml" xref="S3.I1.i4.p1.5.m5.2.3.3.1">superscript</csymbol><apply id="S3.I1.i4.p1.5.m5.2.3.3.1.2.cmml" xref="S3.I1.i4.p1.5.m5.2.3.3.1"><csymbol cd="ambiguous" id="S3.I1.i4.p1.5.m5.2.3.3.1.2.1.cmml" xref="S3.I1.i4.p1.5.m5.2.3.3.1">subscript</csymbol><union id="S3.I1.i4.p1.5.m5.2.3.3.1.2.2.cmml" xref="S3.I1.i4.p1.5.m5.2.3.3.1.2.2"></union><apply id="S3.I1.i4.p1.5.m5.2.3.3.1.2.3.cmml" xref="S3.I1.i4.p1.5.m5.2.3.3.1.2.3"><eq id="S3.I1.i4.p1.5.m5.2.3.3.1.2.3.1.cmml" xref="S3.I1.i4.p1.5.m5.2.3.3.1.2.3.1"></eq><ci id="S3.I1.i4.p1.5.m5.2.3.3.1.2.3.2.cmml" xref="S3.I1.i4.p1.5.m5.2.3.3.1.2.3.2">𝑗</ci><cn id="S3.I1.i4.p1.5.m5.2.3.3.1.2.3.3.cmml" type="integer" xref="S3.I1.i4.p1.5.m5.2.3.3.1.2.3.3">1</cn></apply></apply><ci id="S3.I1.i4.p1.5.m5.2.3.3.1.3.cmml" xref="S3.I1.i4.p1.5.m5.2.3.3.1.3">𝐽</ci></apply><apply id="S3.I1.i4.p1.5.m5.2.3.3.2.cmml" xref="S3.I1.i4.p1.5.m5.2.3.3.2"><csymbol cd="ambiguous" id="S3.I1.i4.p1.5.m5.2.3.3.2.1.cmml" xref="S3.I1.i4.p1.5.m5.2.3.3.2">subscript</csymbol><ci id="S3.I1.i4.p1.5.m5.2.3.3.2.2.cmml" xref="S3.I1.i4.p1.5.m5.2.3.3.2.2">𝒳</ci><list id="S3.I1.i4.p1.5.m5.2.2.2.3.cmml" xref="S3.I1.i4.p1.5.m5.2.2.2.4"><ci id="S3.I1.i4.p1.5.m5.1.1.1.1.cmml" xref="S3.I1.i4.p1.5.m5.1.1.1.1">𝑔</ci><ci id="S3.I1.i4.p1.5.m5.2.2.2.2.cmml" xref="S3.I1.i4.p1.5.m5.2.2.2.2">𝑗</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i4.p1.5.m5.2c">\mathcal{X}\subset\bigcup_{j=1}^{J}\mathcal{X}_{g,j}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i4.p1.5.m5.2d">caligraphic_X ⊂ ⋃ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_J end_POSTSUPERSCRIPT caligraphic_X start_POSTSUBSCRIPT italic_g , italic_j end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S3.I1.i5" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(A5)</span> <div class="ltx_para" id="S3.I1.i5.p1"> <p class="ltx_p" id="S3.I1.i5.p1.2">The parameter space <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.I1.i5.p1.1.m1.1"><semantics id="S3.I1.i5.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.I1.i5.p1.1.m1.1.1" xref="S3.I1.i5.p1.1.m1.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i5.p1.1.m1.1b"><ci id="S3.I1.i5.p1.1.m1.1.1.cmml" xref="S3.I1.i5.p1.1.m1.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i5.p1.1.m1.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i5.p1.1.m1.1d">caligraphic_B</annotation></semantics></math> is a non-empty, convex and compact subspace of <math alttext="\mathbb{R}^{d+1}" class="ltx_Math" display="inline" id="S3.I1.i5.p1.2.m2.1"><semantics id="S3.I1.i5.p1.2.m2.1a"><msup id="S3.I1.i5.p1.2.m2.1.1" xref="S3.I1.i5.p1.2.m2.1.1.cmml"><mi id="S3.I1.i5.p1.2.m2.1.1.2" xref="S3.I1.i5.p1.2.m2.1.1.2.cmml">ℝ</mi><mrow id="S3.I1.i5.p1.2.m2.1.1.3" xref="S3.I1.i5.p1.2.m2.1.1.3.cmml"><mi id="S3.I1.i5.p1.2.m2.1.1.3.2" xref="S3.I1.i5.p1.2.m2.1.1.3.2.cmml">d</mi><mo id="S3.I1.i5.p1.2.m2.1.1.3.1" xref="S3.I1.i5.p1.2.m2.1.1.3.1.cmml">+</mo><mn id="S3.I1.i5.p1.2.m2.1.1.3.3" xref="S3.I1.i5.p1.2.m2.1.1.3.3.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.I1.i5.p1.2.m2.1b"><apply id="S3.I1.i5.p1.2.m2.1.1.cmml" xref="S3.I1.i5.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I1.i5.p1.2.m2.1.1.1.cmml" xref="S3.I1.i5.p1.2.m2.1.1">superscript</csymbol><ci id="S3.I1.i5.p1.2.m2.1.1.2.cmml" xref="S3.I1.i5.p1.2.m2.1.1.2">ℝ</ci><apply id="S3.I1.i5.p1.2.m2.1.1.3.cmml" xref="S3.I1.i5.p1.2.m2.1.1.3"><plus id="S3.I1.i5.p1.2.m2.1.1.3.1.cmml" xref="S3.I1.i5.p1.2.m2.1.1.3.1"></plus><ci id="S3.I1.i5.p1.2.m2.1.1.3.2.cmml" xref="S3.I1.i5.p1.2.m2.1.1.3.2">𝑑</ci><cn id="S3.I1.i5.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S3.I1.i5.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i5.p1.2.m2.1c">\mathbb{R}^{d+1}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i5.p1.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_d + 1 end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S3.I1.i6" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(A6)</span> <div class="ltx_para" id="S3.I1.i6.p1"> <p class="ltx_p" id="S3.I1.i6.p1.1">The covariate space <math alttext="\mathcal{X}" class="ltx_Math" display="inline" id="S3.I1.i6.p1.1.m1.1"><semantics id="S3.I1.i6.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.I1.i6.p1.1.m1.1.1" xref="S3.I1.i6.p1.1.m1.1.1.cmml">𝒳</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i6.p1.1.m1.1b"><ci id="S3.I1.i6.p1.1.m1.1.1.cmml" xref="S3.I1.i6.p1.1.m1.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i6.p1.1.m1.1c">\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i6.p1.1.m1.1d">caligraphic_X</annotation></semantics></math> is bounded.</p> </div> </li> <li class="ltx_item" id="S3.I1.i7" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(A7)</span> <div class="ltx_para" id="S3.I1.i7.p1"> <p class="ltx_p" id="S3.I1.i7.p1.3"><math alttext="\exists\eta>0:\forall j\in\{1,\dots,J\},q\in\{1,2\}:\inf_{\beta\in\mathcal{B}}% \sigma_{j,q}(\beta)>\eta" class="ltx_Math" display="inline" id="S3.I1.i7.p1.1.m1.10"><semantics id="S3.I1.i7.p1.1.m1.10a"><mrow id="S3.I1.i7.p1.1.m1.10.10" xref="S3.I1.i7.p1.1.m1.10.10.cmml"><mrow id="S3.I1.i7.p1.1.m1.10.10.4" xref="S3.I1.i7.p1.1.m1.10.10.4.cmml"><mrow id="S3.I1.i7.p1.1.m1.10.10.4.2" xref="S3.I1.i7.p1.1.m1.10.10.4.2.cmml"><mo id="S3.I1.i7.p1.1.m1.10.10.4.2.1" rspace="0.167em" xref="S3.I1.i7.p1.1.m1.10.10.4.2.1.cmml">∃</mo><mi id="S3.I1.i7.p1.1.m1.10.10.4.2.2" xref="S3.I1.i7.p1.1.m1.10.10.4.2.2.cmml">η</mi></mrow><mo id="S3.I1.i7.p1.1.m1.10.10.4.1" xref="S3.I1.i7.p1.1.m1.10.10.4.1.cmml">></mo><mn id="S3.I1.i7.p1.1.m1.10.10.4.3" xref="S3.I1.i7.p1.1.m1.10.10.4.3.cmml">0</mn></mrow><mo id="S3.I1.i7.p1.1.m1.10.10.5" lspace="0.278em" rspace="0.278em" xref="S3.I1.i7.p1.1.m1.10.10.5.cmml">:</mo><mrow id="S3.I1.i7.p1.1.m1.10.10.2.2" xref="S3.I1.i7.p1.1.m1.10.10.2.3.cmml"><mrow id="S3.I1.i7.p1.1.m1.9.9.1.1.1" xref="S3.I1.i7.p1.1.m1.9.9.1.1.1.cmml"><mrow id="S3.I1.i7.p1.1.m1.9.9.1.1.1.2" xref="S3.I1.i7.p1.1.m1.9.9.1.1.1.2.cmml"><mo id="S3.I1.i7.p1.1.m1.9.9.1.1.1.2.1" rspace="0.167em" xref="S3.I1.i7.p1.1.m1.9.9.1.1.1.2.1.cmml">∀</mo><mi id="S3.I1.i7.p1.1.m1.9.9.1.1.1.2.2" xref="S3.I1.i7.p1.1.m1.9.9.1.1.1.2.2.cmml">j</mi></mrow><mo id="S3.I1.i7.p1.1.m1.9.9.1.1.1.1" xref="S3.I1.i7.p1.1.m1.9.9.1.1.1.1.cmml">∈</mo><mrow id="S3.I1.i7.p1.1.m1.9.9.1.1.1.3.2" xref="S3.I1.i7.p1.1.m1.9.9.1.1.1.3.1.cmml"><mo id="S3.I1.i7.p1.1.m1.9.9.1.1.1.3.2.1" stretchy="false" xref="S3.I1.i7.p1.1.m1.9.9.1.1.1.3.1.cmml">{</mo><mn id="S3.I1.i7.p1.1.m1.3.3" xref="S3.I1.i7.p1.1.m1.3.3.cmml">1</mn><mo id="S3.I1.i7.p1.1.m1.9.9.1.1.1.3.2.2" xref="S3.I1.i7.p1.1.m1.9.9.1.1.1.3.1.cmml">,</mo><mi id="S3.I1.i7.p1.1.m1.4.4" mathvariant="normal" xref="S3.I1.i7.p1.1.m1.4.4.cmml">…</mi><mo id="S3.I1.i7.p1.1.m1.9.9.1.1.1.3.2.3" xref="S3.I1.i7.p1.1.m1.9.9.1.1.1.3.1.cmml">,</mo><mi id="S3.I1.i7.p1.1.m1.5.5" xref="S3.I1.i7.p1.1.m1.5.5.cmml">J</mi><mo id="S3.I1.i7.p1.1.m1.9.9.1.1.1.3.2.4" stretchy="false" xref="S3.I1.i7.p1.1.m1.9.9.1.1.1.3.1.cmml">}</mo></mrow></mrow><mo id="S3.I1.i7.p1.1.m1.10.10.2.2.3" xref="S3.I1.i7.p1.1.m1.10.10.2.3a.cmml">,</mo><mrow id="S3.I1.i7.p1.1.m1.10.10.2.2.2" xref="S3.I1.i7.p1.1.m1.10.10.2.2.2.cmml"><mi id="S3.I1.i7.p1.1.m1.10.10.2.2.2.2" xref="S3.I1.i7.p1.1.m1.10.10.2.2.2.2.cmml">q</mi><mo id="S3.I1.i7.p1.1.m1.10.10.2.2.2.1" xref="S3.I1.i7.p1.1.m1.10.10.2.2.2.1.cmml">∈</mo><mrow id="S3.I1.i7.p1.1.m1.10.10.2.2.2.3.2" xref="S3.I1.i7.p1.1.m1.10.10.2.2.2.3.1.cmml"><mo id="S3.I1.i7.p1.1.m1.10.10.2.2.2.3.2.1" stretchy="false" xref="S3.I1.i7.p1.1.m1.10.10.2.2.2.3.1.cmml">{</mo><mn id="S3.I1.i7.p1.1.m1.6.6" xref="S3.I1.i7.p1.1.m1.6.6.cmml">1</mn><mo id="S3.I1.i7.p1.1.m1.10.10.2.2.2.3.2.2" xref="S3.I1.i7.p1.1.m1.10.10.2.2.2.3.1.cmml">,</mo><mn id="S3.I1.i7.p1.1.m1.7.7" xref="S3.I1.i7.p1.1.m1.7.7.cmml">2</mn><mo id="S3.I1.i7.p1.1.m1.10.10.2.2.2.3.2.3" rspace="0.278em" stretchy="false" xref="S3.I1.i7.p1.1.m1.10.10.2.2.2.3.1.cmml">}</mo></mrow></mrow></mrow><mo id="S3.I1.i7.p1.1.m1.10.10.6" xref="S3.I1.i7.p1.1.m1.10.10.6.cmml">:</mo><mrow id="S3.I1.i7.p1.1.m1.10.10.7" xref="S3.I1.i7.p1.1.m1.10.10.7.cmml"><mrow id="S3.I1.i7.p1.1.m1.10.10.7.2" xref="S3.I1.i7.p1.1.m1.10.10.7.2.cmml"><msub id="S3.I1.i7.p1.1.m1.10.10.7.2.1" xref="S3.I1.i7.p1.1.m1.10.10.7.2.1.cmml"><mo id="S3.I1.i7.p1.1.m1.10.10.7.2.1.2" lspace="0.111em" rspace="0.167em" xref="S3.I1.i7.p1.1.m1.10.10.7.2.1.2.cmml">inf</mo><mrow id="S3.I1.i7.p1.1.m1.10.10.7.2.1.3" xref="S3.I1.i7.p1.1.m1.10.10.7.2.1.3.cmml"><mi id="S3.I1.i7.p1.1.m1.10.10.7.2.1.3.2" xref="S3.I1.i7.p1.1.m1.10.10.7.2.1.3.2.cmml">β</mi><mo id="S3.I1.i7.p1.1.m1.10.10.7.2.1.3.1" xref="S3.I1.i7.p1.1.m1.10.10.7.2.1.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.I1.i7.p1.1.m1.10.10.7.2.1.3.3" 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id="S3.I1.i7.p1.3.m3.4c">m_{j,q}(W,\beta)</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i7.p1.3.m3.4d">italic_m start_POSTSUBSCRIPT italic_j , italic_q end_POSTSUBSCRIPT ( italic_W , italic_β )</annotation></semantics></math>.</p> </div> </li> </ol> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.3">The first assumption is standard. Assumptions <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I1.i2" title="item (A2) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A2)</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I1.i3" title="item (A3) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A3)</span></a> hold for all link functions and classes of instrumental functions discussed in this paper. Assumption <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I1.i4" title="item (A4) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A4)</span></a> imposes that the support of each instrumental function has non-empty intersection with the covariate space, and that each point in <math alttext="\mathcal{X}" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.1"><semantics id="S3.SS1.p2.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p2.1.m1.1.1" xref="S3.SS1.p2.1.m1.1.1.cmml">𝒳</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.1b"><ci id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.1c">\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.1d">caligraphic_X</annotation></semantics></math> is covered by at least one instrumental function. For a discussion on how to choose the family of instrumental functions in this way, we refer to Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.SS2" title="3.2 Instrumental functions ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">3.2</span></a>. Assumptions <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I1.i5" title="item (A5) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A5)</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I1.i6" title="item (A6) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A6)</span></a> are regularity conditions on the parameter and covariate space, respectively. In practice, one could select <math alttext="\mathcal{B}=[-M_{\mathcal{B}},M_{\mathcal{B}}]^{d+1}" class="ltx_Math" display="inline" id="S3.SS1.p2.2.m2.2"><semantics id="S3.SS1.p2.2.m2.2a"><mrow id="S3.SS1.p2.2.m2.2.2" xref="S3.SS1.p2.2.m2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p2.2.m2.2.2.4" xref="S3.SS1.p2.2.m2.2.2.4.cmml">ℬ</mi><mo id="S3.SS1.p2.2.m2.2.2.3" xref="S3.SS1.p2.2.m2.2.2.3.cmml">=</mo><msup id="S3.SS1.p2.2.m2.2.2.2" xref="S3.SS1.p2.2.m2.2.2.2.cmml"><mrow id="S3.SS1.p2.2.m2.2.2.2.2.2" xref="S3.SS1.p2.2.m2.2.2.2.2.3.cmml"><mo id="S3.SS1.p2.2.m2.2.2.2.2.2.3" stretchy="false" xref="S3.SS1.p2.2.m2.2.2.2.2.3.cmml">[</mo><mrow id="S3.SS1.p2.2.m2.1.1.1.1.1.1" xref="S3.SS1.p2.2.m2.1.1.1.1.1.1.cmml"><mo id="S3.SS1.p2.2.m2.1.1.1.1.1.1a" xref="S3.SS1.p2.2.m2.1.1.1.1.1.1.cmml">−</mo><msub id="S3.SS1.p2.2.m2.1.1.1.1.1.1.2" xref="S3.SS1.p2.2.m2.1.1.1.1.1.1.2.cmml"><mi id="S3.SS1.p2.2.m2.1.1.1.1.1.1.2.2" xref="S3.SS1.p2.2.m2.1.1.1.1.1.1.2.2.cmml">M</mi><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p2.2.m2.1.1.1.1.1.1.2.3" xref="S3.SS1.p2.2.m2.1.1.1.1.1.1.2.3.cmml">ℬ</mi></msub></mrow><mo id="S3.SS1.p2.2.m2.2.2.2.2.2.4" xref="S3.SS1.p2.2.m2.2.2.2.2.3.cmml">,</mo><msub id="S3.SS1.p2.2.m2.2.2.2.2.2.2" xref="S3.SS1.p2.2.m2.2.2.2.2.2.2.cmml"><mi id="S3.SS1.p2.2.m2.2.2.2.2.2.2.2" xref="S3.SS1.p2.2.m2.2.2.2.2.2.2.2.cmml">M</mi><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p2.2.m2.2.2.2.2.2.2.3" xref="S3.SS1.p2.2.m2.2.2.2.2.2.2.3.cmml">ℬ</mi></msub><mo id="S3.SS1.p2.2.m2.2.2.2.2.2.5" stretchy="false" xref="S3.SS1.p2.2.m2.2.2.2.2.3.cmml">]</mo></mrow><mrow id="S3.SS1.p2.2.m2.2.2.2.4" xref="S3.SS1.p2.2.m2.2.2.2.4.cmml"><mi id="S3.SS1.p2.2.m2.2.2.2.4.2" xref="S3.SS1.p2.2.m2.2.2.2.4.2.cmml">d</mi><mo id="S3.SS1.p2.2.m2.2.2.2.4.1" xref="S3.SS1.p2.2.m2.2.2.2.4.1.cmml">+</mo><mn id="S3.SS1.p2.2.m2.2.2.2.4.3" xref="S3.SS1.p2.2.m2.2.2.2.4.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.2.m2.2b"><apply id="S3.SS1.p2.2.m2.2.2.cmml" xref="S3.SS1.p2.2.m2.2.2"><eq id="S3.SS1.p2.2.m2.2.2.3.cmml" xref="S3.SS1.p2.2.m2.2.2.3"></eq><ci id="S3.SS1.p2.2.m2.2.2.4.cmml" xref="S3.SS1.p2.2.m2.2.2.4">ℬ</ci><apply id="S3.SS1.p2.2.m2.2.2.2.cmml" xref="S3.SS1.p2.2.m2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p2.2.m2.2.2.2.3.cmml" xref="S3.SS1.p2.2.m2.2.2.2">superscript</csymbol><interval closure="closed" id="S3.SS1.p2.2.m2.2.2.2.2.3.cmml" xref="S3.SS1.p2.2.m2.2.2.2.2.2"><apply id="S3.SS1.p2.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1.1.1.1.1"><minus id="S3.SS1.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1.1.1.1.1"></minus><apply id="S3.SS1.p2.2.m2.1.1.1.1.1.1.2.cmml" xref="S3.SS1.p2.2.m2.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p2.2.m2.1.1.1.1.1.1.2.1.cmml" xref="S3.SS1.p2.2.m2.1.1.1.1.1.1.2">subscript</csymbol><ci id="S3.SS1.p2.2.m2.1.1.1.1.1.1.2.2.cmml" xref="S3.SS1.p2.2.m2.1.1.1.1.1.1.2.2">𝑀</ci><ci id="S3.SS1.p2.2.m2.1.1.1.1.1.1.2.3.cmml" xref="S3.SS1.p2.2.m2.1.1.1.1.1.1.2.3">ℬ</ci></apply></apply><apply id="S3.SS1.p2.2.m2.2.2.2.2.2.2.cmml" xref="S3.SS1.p2.2.m2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p2.2.m2.2.2.2.2.2.2.1.cmml" xref="S3.SS1.p2.2.m2.2.2.2.2.2.2">subscript</csymbol><ci id="S3.SS1.p2.2.m2.2.2.2.2.2.2.2.cmml" xref="S3.SS1.p2.2.m2.2.2.2.2.2.2.2">𝑀</ci><ci id="S3.SS1.p2.2.m2.2.2.2.2.2.2.3.cmml" xref="S3.SS1.p2.2.m2.2.2.2.2.2.2.3">ℬ</ci></apply></interval><apply id="S3.SS1.p2.2.m2.2.2.2.4.cmml" xref="S3.SS1.p2.2.m2.2.2.2.4"><plus id="S3.SS1.p2.2.m2.2.2.2.4.1.cmml" xref="S3.SS1.p2.2.m2.2.2.2.4.1"></plus><ci id="S3.SS1.p2.2.m2.2.2.2.4.2.cmml" xref="S3.SS1.p2.2.m2.2.2.2.4.2">𝑑</ci><cn id="S3.SS1.p2.2.m2.2.2.2.4.3.cmml" type="integer" xref="S3.SS1.p2.2.m2.2.2.2.4.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m2.2c">\mathcal{B}=[-M_{\mathcal{B}},M_{\mathcal{B}}]^{d+1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m2.2d">caligraphic_B = [ - italic_M start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT , italic_M start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT italic_d + 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, for a sensible bound <math alttext="M_{\mathcal{B}}>0" class="ltx_Math" display="inline" id="S3.SS1.p2.3.m3.1"><semantics id="S3.SS1.p2.3.m3.1a"><mrow id="S3.SS1.p2.3.m3.1.1" xref="S3.SS1.p2.3.m3.1.1.cmml"><msub id="S3.SS1.p2.3.m3.1.1.2" xref="S3.SS1.p2.3.m3.1.1.2.cmml"><mi id="S3.SS1.p2.3.m3.1.1.2.2" xref="S3.SS1.p2.3.m3.1.1.2.2.cmml">M</mi><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p2.3.m3.1.1.2.3" xref="S3.SS1.p2.3.m3.1.1.2.3.cmml">ℬ</mi></msub><mo id="S3.SS1.p2.3.m3.1.1.1" xref="S3.SS1.p2.3.m3.1.1.1.cmml">></mo><mn id="S3.SS1.p2.3.m3.1.1.3" xref="S3.SS1.p2.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.3.m3.1b"><apply id="S3.SS1.p2.3.m3.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1"><gt id="S3.SS1.p2.3.m3.1.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1.1"></gt><apply id="S3.SS1.p2.3.m3.1.1.2.cmml" xref="S3.SS1.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p2.3.m3.1.1.2.1.cmml" xref="S3.SS1.p2.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS1.p2.3.m3.1.1.2.2.cmml" xref="S3.SS1.p2.3.m3.1.1.2.2">𝑀</ci><ci id="S3.SS1.p2.3.m3.1.1.2.3.cmml" xref="S3.SS1.p2.3.m3.1.1.2.3">ℬ</ci></apply><cn id="S3.SS1.p2.3.m3.1.1.3.cmml" type="integer" xref="S3.SS1.p2.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.3.m3.1c">M_{\mathcal{B}}>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.3.m3.1d">italic_M start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT > 0</annotation></semantics></math>. Assumption <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I1.i7" title="item (A7) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A7)</span></a> imposes that the variances of the moment functions are uniformly bounded away from zero. Example <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:example:_assumption_variance_of_moments</span> in Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Examples</span> illustrates that this is a mild and verifiable assumption.</p> </div> <div class="ltx_para" id="S3.SS1.p3"> <p class="ltx_p" id="S3.SS1.p3.12">Additionally, we require two assumptions of which we present the essence here and defer technical details to Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Technical_assumptions</span>.</p> <ol class="ltx_enumerate" id="S3.I2"> <li class="ltx_item" id="S3.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(A8)</span> <div class="ltx_para" id="S3.I2.i1.p1"> <p class="ltx_p" id="S3.I2.i1.p1.1"><math alttext="\exists c>0:\beta_{\text{true},k}\in\tilde{\mathcal{L}}_{0}(c)" class="ltx_Math" display="inline" id="S3.I2.i1.p1.1.m1.3"><semantics id="S3.I2.i1.p1.1.m1.3a"><mrow 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xref="S3.I2.i1.p1.1.m1.3.4.2.3">0</cn></apply><apply id="S3.I2.i1.p1.1.m1.3.4.3.cmml" xref="S3.I2.i1.p1.1.m1.3.4.3"><in id="S3.I2.i1.p1.1.m1.3.4.3.1.cmml" xref="S3.I2.i1.p1.1.m1.3.4.3.1"></in><apply id="S3.I2.i1.p1.1.m1.3.4.3.2.cmml" xref="S3.I2.i1.p1.1.m1.3.4.3.2"><csymbol cd="ambiguous" id="S3.I2.i1.p1.1.m1.3.4.3.2.1.cmml" xref="S3.I2.i1.p1.1.m1.3.4.3.2">subscript</csymbol><ci id="S3.I2.i1.p1.1.m1.3.4.3.2.2.cmml" xref="S3.I2.i1.p1.1.m1.3.4.3.2.2">𝛽</ci><list id="S3.I2.i1.p1.1.m1.2.2.2.3.cmml" xref="S3.I2.i1.p1.1.m1.2.2.2.4"><ci id="S3.I2.i1.p1.1.m1.1.1.1.1a.cmml" xref="S3.I2.i1.p1.1.m1.1.1.1.1"><mtext id="S3.I2.i1.p1.1.m1.1.1.1.1.cmml" mathsize="70%" xref="S3.I2.i1.p1.1.m1.1.1.1.1">true</mtext></ci><ci id="S3.I2.i1.p1.1.m1.2.2.2.2.cmml" xref="S3.I2.i1.p1.1.m1.2.2.2.2">𝑘</ci></list></apply><apply id="S3.I2.i1.p1.1.m1.3.4.3.3.cmml" xref="S3.I2.i1.p1.1.m1.3.4.3.3"><times id="S3.I2.i1.p1.1.m1.3.4.3.3.1.cmml" xref="S3.I2.i1.p1.1.m1.3.4.3.3.1"></times><apply id="S3.I2.i1.p1.1.m1.3.4.3.3.2.cmml" xref="S3.I2.i1.p1.1.m1.3.4.3.3.2"><csymbol cd="ambiguous" id="S3.I2.i1.p1.1.m1.3.4.3.3.2.1.cmml" xref="S3.I2.i1.p1.1.m1.3.4.3.3.2">subscript</csymbol><apply id="S3.I2.i1.p1.1.m1.3.4.3.3.2.2.cmml" xref="S3.I2.i1.p1.1.m1.3.4.3.3.2.2"><ci id="S3.I2.i1.p1.1.m1.3.4.3.3.2.2.1.cmml" xref="S3.I2.i1.p1.1.m1.3.4.3.3.2.2.1">~</ci><ci id="S3.I2.i1.p1.1.m1.3.4.3.3.2.2.2.cmml" xref="S3.I2.i1.p1.1.m1.3.4.3.3.2.2.2">ℒ</ci></apply><cn id="S3.I2.i1.p1.1.m1.3.4.3.3.2.3.cmml" type="integer" xref="S3.I2.i1.p1.1.m1.3.4.3.3.2.3">0</cn></apply><ci id="S3.I2.i1.p1.1.m1.3.3.cmml" xref="S3.I2.i1.p1.1.m1.3.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.1.m1.3c">\exists c>0:\beta_{\text{true},k}\in\tilde{\mathcal{L}}_{0}(c)</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.1.m1.3d">∃ italic_c > 0 : italic_β start_POSTSUBSCRIPT true , italic_k end_POSTSUBSCRIPT ∈ over~ start_ARG caligraphic_L end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_c )</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S3.SS1.p3.11">The definition of the set <math alttext="\tilde{\mathcal{L}}_{0}(c)\subset\mathcal{B}_{I,k}" class="ltx_Math" display="inline" id="S3.SS1.p3.1.m1.3"><semantics id="S3.SS1.p3.1.m1.3a"><mrow id="S3.SS1.p3.1.m1.3.4" xref="S3.SS1.p3.1.m1.3.4.cmml"><mrow id="S3.SS1.p3.1.m1.3.4.2" xref="S3.SS1.p3.1.m1.3.4.2.cmml"><msub id="S3.SS1.p3.1.m1.3.4.2.2" xref="S3.SS1.p3.1.m1.3.4.2.2.cmml"><mover accent="true" id="S3.SS1.p3.1.m1.3.4.2.2.2" xref="S3.SS1.p3.1.m1.3.4.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p3.1.m1.3.4.2.2.2.2" xref="S3.SS1.p3.1.m1.3.4.2.2.2.2.cmml">ℒ</mi><mo id="S3.SS1.p3.1.m1.3.4.2.2.2.1" xref="S3.SS1.p3.1.m1.3.4.2.2.2.1.cmml">~</mo></mover><mn id="S3.SS1.p3.1.m1.3.4.2.2.3" xref="S3.SS1.p3.1.m1.3.4.2.2.3.cmml">0</mn></msub><mo id="S3.SS1.p3.1.m1.3.4.2.1" xref="S3.SS1.p3.1.m1.3.4.2.1.cmml">⁢</mo><mrow id="S3.SS1.p3.1.m1.3.4.2.3.2" xref="S3.SS1.p3.1.m1.3.4.2.cmml"><mo id="S3.SS1.p3.1.m1.3.4.2.3.2.1" stretchy="false" xref="S3.SS1.p3.1.m1.3.4.2.cmml">(</mo><mi id="S3.SS1.p3.1.m1.3.3" xref="S3.SS1.p3.1.m1.3.3.cmml">c</mi><mo id="S3.SS1.p3.1.m1.3.4.2.3.2.2" stretchy="false" xref="S3.SS1.p3.1.m1.3.4.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.p3.1.m1.3.4.1" xref="S3.SS1.p3.1.m1.3.4.1.cmml">⊂</mo><msub id="S3.SS1.p3.1.m1.3.4.3" xref="S3.SS1.p3.1.m1.3.4.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p3.1.m1.3.4.3.2" xref="S3.SS1.p3.1.m1.3.4.3.2.cmml">ℬ</mi><mrow id="S3.SS1.p3.1.m1.2.2.2.4" xref="S3.SS1.p3.1.m1.2.2.2.3.cmml"><mi id="S3.SS1.p3.1.m1.1.1.1.1" xref="S3.SS1.p3.1.m1.1.1.1.1.cmml">I</mi><mo id="S3.SS1.p3.1.m1.2.2.2.4.1" xref="S3.SS1.p3.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p3.1.m1.2.2.2.2" xref="S3.SS1.p3.1.m1.2.2.2.2.cmml">k</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.1.m1.3b"><apply id="S3.SS1.p3.1.m1.3.4.cmml" xref="S3.SS1.p3.1.m1.3.4"><subset id="S3.SS1.p3.1.m1.3.4.1.cmml" xref="S3.SS1.p3.1.m1.3.4.1"></subset><apply id="S3.SS1.p3.1.m1.3.4.2.cmml" xref="S3.SS1.p3.1.m1.3.4.2"><times id="S3.SS1.p3.1.m1.3.4.2.1.cmml" xref="S3.SS1.p3.1.m1.3.4.2.1"></times><apply id="S3.SS1.p3.1.m1.3.4.2.2.cmml" xref="S3.SS1.p3.1.m1.3.4.2.2"><csymbol cd="ambiguous" id="S3.SS1.p3.1.m1.3.4.2.2.1.cmml" xref="S3.SS1.p3.1.m1.3.4.2.2">subscript</csymbol><apply id="S3.SS1.p3.1.m1.3.4.2.2.2.cmml" xref="S3.SS1.p3.1.m1.3.4.2.2.2"><ci id="S3.SS1.p3.1.m1.3.4.2.2.2.1.cmml" xref="S3.SS1.p3.1.m1.3.4.2.2.2.1">~</ci><ci id="S3.SS1.p3.1.m1.3.4.2.2.2.2.cmml" xref="S3.SS1.p3.1.m1.3.4.2.2.2.2">ℒ</ci></apply><cn id="S3.SS1.p3.1.m1.3.4.2.2.3.cmml" type="integer" xref="S3.SS1.p3.1.m1.3.4.2.2.3">0</cn></apply><ci id="S3.SS1.p3.1.m1.3.3.cmml" xref="S3.SS1.p3.1.m1.3.3">𝑐</ci></apply><apply id="S3.SS1.p3.1.m1.3.4.3.cmml" xref="S3.SS1.p3.1.m1.3.4.3"><csymbol cd="ambiguous" id="S3.SS1.p3.1.m1.3.4.3.1.cmml" xref="S3.SS1.p3.1.m1.3.4.3">subscript</csymbol><ci id="S3.SS1.p3.1.m1.3.4.3.2.cmml" xref="S3.SS1.p3.1.m1.3.4.3.2">ℬ</ci><list id="S3.SS1.p3.1.m1.2.2.2.3.cmml" xref="S3.SS1.p3.1.m1.2.2.2.4"><ci id="S3.SS1.p3.1.m1.1.1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1.1.1">𝐼</ci><ci id="S3.SS1.p3.1.m1.2.2.2.2.cmml" xref="S3.SS1.p3.1.m1.2.2.2.2">𝑘</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.1.m1.3c">\tilde{\mathcal{L}}_{0}(c)\subset\mathcal{B}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.1.m1.3d">over~ start_ARG caligraphic_L end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_c ) ⊂ caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> can be found in Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Technical_assumptions</span>. Assumption <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I2.i1" title="item (A8) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A8)</span></a> deserves special attention, as its purpose is to account for the stringency of Assumption 2 in <cite class="ltx_cite ltx_citemacro_cite">Bei, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib2" title="">2024</a>)</cite>, which has already been the subject of detailed research <cite class="ltx_cite ltx_citemacro_citep">(Kaido et al.,, <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib20" title="">2022</a>)</cite>. In essence, <math alttext="\tilde{\mathcal{L}}_{0}(c)" class="ltx_Math" display="inline" id="S3.SS1.p3.2.m2.1"><semantics id="S3.SS1.p3.2.m2.1a"><mrow id="S3.SS1.p3.2.m2.1.2" xref="S3.SS1.p3.2.m2.1.2.cmml"><msub id="S3.SS1.p3.2.m2.1.2.2" xref="S3.SS1.p3.2.m2.1.2.2.cmml"><mover accent="true" id="S3.SS1.p3.2.m2.1.2.2.2" xref="S3.SS1.p3.2.m2.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p3.2.m2.1.2.2.2.2" xref="S3.SS1.p3.2.m2.1.2.2.2.2.cmml">ℒ</mi><mo id="S3.SS1.p3.2.m2.1.2.2.2.1" xref="S3.SS1.p3.2.m2.1.2.2.2.1.cmml">~</mo></mover><mn id="S3.SS1.p3.2.m2.1.2.2.3" xref="S3.SS1.p3.2.m2.1.2.2.3.cmml">0</mn></msub><mo id="S3.SS1.p3.2.m2.1.2.1" xref="S3.SS1.p3.2.m2.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.p3.2.m2.1.2.3.2" xref="S3.SS1.p3.2.m2.1.2.cmml"><mo id="S3.SS1.p3.2.m2.1.2.3.2.1" stretchy="false" xref="S3.SS1.p3.2.m2.1.2.cmml">(</mo><mi id="S3.SS1.p3.2.m2.1.1" xref="S3.SS1.p3.2.m2.1.1.cmml">c</mi><mo id="S3.SS1.p3.2.m2.1.2.3.2.2" stretchy="false" xref="S3.SS1.p3.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.2.m2.1b"><apply id="S3.SS1.p3.2.m2.1.2.cmml" xref="S3.SS1.p3.2.m2.1.2"><times id="S3.SS1.p3.2.m2.1.2.1.cmml" xref="S3.SS1.p3.2.m2.1.2.1"></times><apply id="S3.SS1.p3.2.m2.1.2.2.cmml" xref="S3.SS1.p3.2.m2.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.p3.2.m2.1.2.2.1.cmml" xref="S3.SS1.p3.2.m2.1.2.2">subscript</csymbol><apply id="S3.SS1.p3.2.m2.1.2.2.2.cmml" xref="S3.SS1.p3.2.m2.1.2.2.2"><ci id="S3.SS1.p3.2.m2.1.2.2.2.1.cmml" xref="S3.SS1.p3.2.m2.1.2.2.2.1">~</ci><ci id="S3.SS1.p3.2.m2.1.2.2.2.2.cmml" xref="S3.SS1.p3.2.m2.1.2.2.2.2">ℒ</ci></apply><cn id="S3.SS1.p3.2.m2.1.2.2.3.cmml" type="integer" xref="S3.SS1.p3.2.m2.1.2.2.3">0</cn></apply><ci id="S3.SS1.p3.2.m2.1.1.cmml" xref="S3.SS1.p3.2.m2.1.1">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.2.m2.1c">\tilde{\mathcal{L}}_{0}(c)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.2.m2.1d">over~ start_ARG caligraphic_L end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_c )</annotation></semantics></math> serves to restrict the values of <math alttext="r\in\mathcal{B}_{I,k}" class="ltx_Math" display="inline" id="S3.SS1.p3.3.m3.2"><semantics id="S3.SS1.p3.3.m3.2a"><mrow id="S3.SS1.p3.3.m3.2.3" xref="S3.SS1.p3.3.m3.2.3.cmml"><mi id="S3.SS1.p3.3.m3.2.3.2" xref="S3.SS1.p3.3.m3.2.3.2.cmml">r</mi><mo id="S3.SS1.p3.3.m3.2.3.1" xref="S3.SS1.p3.3.m3.2.3.1.cmml">∈</mo><msub id="S3.SS1.p3.3.m3.2.3.3" xref="S3.SS1.p3.3.m3.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p3.3.m3.2.3.3.2" xref="S3.SS1.p3.3.m3.2.3.3.2.cmml">ℬ</mi><mrow id="S3.SS1.p3.3.m3.2.2.2.4" xref="S3.SS1.p3.3.m3.2.2.2.3.cmml"><mi id="S3.SS1.p3.3.m3.1.1.1.1" xref="S3.SS1.p3.3.m3.1.1.1.1.cmml">I</mi><mo id="S3.SS1.p3.3.m3.2.2.2.4.1" xref="S3.SS1.p3.3.m3.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p3.3.m3.2.2.2.2" xref="S3.SS1.p3.3.m3.2.2.2.2.cmml">k</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.3.m3.2b"><apply id="S3.SS1.p3.3.m3.2.3.cmml" xref="S3.SS1.p3.3.m3.2.3"><in id="S3.SS1.p3.3.m3.2.3.1.cmml" xref="S3.SS1.p3.3.m3.2.3.1"></in><ci id="S3.SS1.p3.3.m3.2.3.2.cmml" xref="S3.SS1.p3.3.m3.2.3.2">𝑟</ci><apply id="S3.SS1.p3.3.m3.2.3.3.cmml" xref="S3.SS1.p3.3.m3.2.3.3"><csymbol cd="ambiguous" id="S3.SS1.p3.3.m3.2.3.3.1.cmml" xref="S3.SS1.p3.3.m3.2.3.3">subscript</csymbol><ci id="S3.SS1.p3.3.m3.2.3.3.2.cmml" xref="S3.SS1.p3.3.m3.2.3.3.2">ℬ</ci><list id="S3.SS1.p3.3.m3.2.2.2.3.cmml" xref="S3.SS1.p3.3.m3.2.2.2.4"><ci id="S3.SS1.p3.3.m3.1.1.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1.1.1">𝐼</ci><ci id="S3.SS1.p3.3.m3.2.2.2.2.cmml" xref="S3.SS1.p3.3.m3.2.2.2.2">𝑘</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.3.m3.2c">r\in\mathcal{B}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.3.m3.2d">italic_r ∈ caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> for which the hypothesis <math alttext="\mathcal{H}_{0}(r)" class="ltx_Math" display="inline" id="S3.SS1.p3.4.m4.1"><semantics id="S3.SS1.p3.4.m4.1a"><mrow id="S3.SS1.p3.4.m4.1.2" xref="S3.SS1.p3.4.m4.1.2.cmml"><msub id="S3.SS1.p3.4.m4.1.2.2" xref="S3.SS1.p3.4.m4.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p3.4.m4.1.2.2.2" xref="S3.SS1.p3.4.m4.1.2.2.2.cmml">ℋ</mi><mn id="S3.SS1.p3.4.m4.1.2.2.3" xref="S3.SS1.p3.4.m4.1.2.2.3.cmml">0</mn></msub><mo id="S3.SS1.p3.4.m4.1.2.1" xref="S3.SS1.p3.4.m4.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.p3.4.m4.1.2.3.2" xref="S3.SS1.p3.4.m4.1.2.cmml"><mo id="S3.SS1.p3.4.m4.1.2.3.2.1" stretchy="false" xref="S3.SS1.p3.4.m4.1.2.cmml">(</mo><mi id="S3.SS1.p3.4.m4.1.1" xref="S3.SS1.p3.4.m4.1.1.cmml">r</mi><mo id="S3.SS1.p3.4.m4.1.2.3.2.2" stretchy="false" xref="S3.SS1.p3.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.4.m4.1b"><apply id="S3.SS1.p3.4.m4.1.2.cmml" xref="S3.SS1.p3.4.m4.1.2"><times id="S3.SS1.p3.4.m4.1.2.1.cmml" xref="S3.SS1.p3.4.m4.1.2.1"></times><apply id="S3.SS1.p3.4.m4.1.2.2.cmml" xref="S3.SS1.p3.4.m4.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.p3.4.m4.1.2.2.1.cmml" xref="S3.SS1.p3.4.m4.1.2.2">subscript</csymbol><ci id="S3.SS1.p3.4.m4.1.2.2.2.cmml" xref="S3.SS1.p3.4.m4.1.2.2.2">ℋ</ci><cn id="S3.SS1.p3.4.m4.1.2.2.3.cmml" type="integer" xref="S3.SS1.p3.4.m4.1.2.2.3">0</cn></apply><ci id="S3.SS1.p3.4.m4.1.1.cmml" xref="S3.SS1.p3.4.m4.1.1">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.4.m4.1c">\mathcal{H}_{0}(r)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.4.m4.1d">caligraphic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_r )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E9" title="In 2.4 Testing procedure ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">9</span></a>) can be tested with correct type-I error, and Assumption <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I2.i1" title="item (A8) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A8)</span></a> states that <math alttext="\beta_{\text{true},k}" class="ltx_Math" display="inline" id="S3.SS1.p3.5.m5.2"><semantics id="S3.SS1.p3.5.m5.2a"><msub id="S3.SS1.p3.5.m5.2.3" xref="S3.SS1.p3.5.m5.2.3.cmml"><mi id="S3.SS1.p3.5.m5.2.3.2" xref="S3.SS1.p3.5.m5.2.3.2.cmml">β</mi><mrow id="S3.SS1.p3.5.m5.2.2.2.4" xref="S3.SS1.p3.5.m5.2.2.2.3.cmml"><mtext id="S3.SS1.p3.5.m5.1.1.1.1" xref="S3.SS1.p3.5.m5.1.1.1.1a.cmml">true</mtext><mo id="S3.SS1.p3.5.m5.2.2.2.4.1" xref="S3.SS1.p3.5.m5.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p3.5.m5.2.2.2.2" xref="S3.SS1.p3.5.m5.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.5.m5.2b"><apply id="S3.SS1.p3.5.m5.2.3.cmml" xref="S3.SS1.p3.5.m5.2.3"><csymbol cd="ambiguous" id="S3.SS1.p3.5.m5.2.3.1.cmml" xref="S3.SS1.p3.5.m5.2.3">subscript</csymbol><ci id="S3.SS1.p3.5.m5.2.3.2.cmml" xref="S3.SS1.p3.5.m5.2.3.2">𝛽</ci><list id="S3.SS1.p3.5.m5.2.2.2.3.cmml" xref="S3.SS1.p3.5.m5.2.2.2.4"><ci id="S3.SS1.p3.5.m5.1.1.1.1a.cmml" xref="S3.SS1.p3.5.m5.1.1.1.1"><mtext id="S3.SS1.p3.5.m5.1.1.1.1.cmml" mathsize="70%" xref="S3.SS1.p3.5.m5.1.1.1.1">true</mtext></ci><ci id="S3.SS1.p3.5.m5.2.2.2.2.cmml" xref="S3.SS1.p3.5.m5.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.5.m5.2c">\beta_{\text{true},k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.5.m5.2d">italic_β start_POSTSUBSCRIPT true , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is an element of this set. Hence the estimated identified set will contain all values of <math alttext="\tilde{\mathcal{L}}_{0}(c)" class="ltx_Math" display="inline" id="S3.SS1.p3.6.m6.1"><semantics id="S3.SS1.p3.6.m6.1a"><mrow id="S3.SS1.p3.6.m6.1.2" xref="S3.SS1.p3.6.m6.1.2.cmml"><msub id="S3.SS1.p3.6.m6.1.2.2" xref="S3.SS1.p3.6.m6.1.2.2.cmml"><mover accent="true" id="S3.SS1.p3.6.m6.1.2.2.2" xref="S3.SS1.p3.6.m6.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p3.6.m6.1.2.2.2.2" xref="S3.SS1.p3.6.m6.1.2.2.2.2.cmml">ℒ</mi><mo id="S3.SS1.p3.6.m6.1.2.2.2.1" xref="S3.SS1.p3.6.m6.1.2.2.2.1.cmml">~</mo></mover><mn id="S3.SS1.p3.6.m6.1.2.2.3" xref="S3.SS1.p3.6.m6.1.2.2.3.cmml">0</mn></msub><mo id="S3.SS1.p3.6.m6.1.2.1" xref="S3.SS1.p3.6.m6.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.p3.6.m6.1.2.3.2" xref="S3.SS1.p3.6.m6.1.2.cmml"><mo id="S3.SS1.p3.6.m6.1.2.3.2.1" stretchy="false" xref="S3.SS1.p3.6.m6.1.2.cmml">(</mo><mi id="S3.SS1.p3.6.m6.1.1" xref="S3.SS1.p3.6.m6.1.1.cmml">c</mi><mo id="S3.SS1.p3.6.m6.1.2.3.2.2" stretchy="false" xref="S3.SS1.p3.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.6.m6.1b"><apply id="S3.SS1.p3.6.m6.1.2.cmml" xref="S3.SS1.p3.6.m6.1.2"><times id="S3.SS1.p3.6.m6.1.2.1.cmml" xref="S3.SS1.p3.6.m6.1.2.1"></times><apply id="S3.SS1.p3.6.m6.1.2.2.cmml" xref="S3.SS1.p3.6.m6.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.p3.6.m6.1.2.2.1.cmml" xref="S3.SS1.p3.6.m6.1.2.2">subscript</csymbol><apply id="S3.SS1.p3.6.m6.1.2.2.2.cmml" xref="S3.SS1.p3.6.m6.1.2.2.2"><ci id="S3.SS1.p3.6.m6.1.2.2.2.1.cmml" xref="S3.SS1.p3.6.m6.1.2.2.2.1">~</ci><ci id="S3.SS1.p3.6.m6.1.2.2.2.2.cmml" xref="S3.SS1.p3.6.m6.1.2.2.2.2">ℒ</ci></apply><cn id="S3.SS1.p3.6.m6.1.2.2.3.cmml" type="integer" xref="S3.SS1.p3.6.m6.1.2.2.3">0</cn></apply><ci id="S3.SS1.p3.6.m6.1.1.cmml" xref="S3.SS1.p3.6.m6.1.1">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.6.m6.1c">\tilde{\mathcal{L}}_{0}(c)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.6.m6.1d">over~ start_ARG caligraphic_L end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_c )</annotation></semantics></math> with at least the specified confidence level <math alttext="1-\alpha" class="ltx_Math" display="inline" id="S3.SS1.p3.7.m7.1"><semantics id="S3.SS1.p3.7.m7.1a"><mrow id="S3.SS1.p3.7.m7.1.1" xref="S3.SS1.p3.7.m7.1.1.cmml"><mn id="S3.SS1.p3.7.m7.1.1.2" xref="S3.SS1.p3.7.m7.1.1.2.cmml">1</mn><mo id="S3.SS1.p3.7.m7.1.1.1" xref="S3.SS1.p3.7.m7.1.1.1.cmml">−</mo><mi id="S3.SS1.p3.7.m7.1.1.3" xref="S3.SS1.p3.7.m7.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.7.m7.1b"><apply id="S3.SS1.p3.7.m7.1.1.cmml" xref="S3.SS1.p3.7.m7.1.1"><minus id="S3.SS1.p3.7.m7.1.1.1.cmml" xref="S3.SS1.p3.7.m7.1.1.1"></minus><cn id="S3.SS1.p3.7.m7.1.1.2.cmml" type="integer" xref="S3.SS1.p3.7.m7.1.1.2">1</cn><ci id="S3.SS1.p3.7.m7.1.1.3.cmml" xref="S3.SS1.p3.7.m7.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.7.m7.1c">1-\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.7.m7.1d">1 - italic_α</annotation></semantics></math>, in particular including <math alttext="\beta_{\text{true},k}" class="ltx_Math" display="inline" id="S3.SS1.p3.8.m8.2"><semantics id="S3.SS1.p3.8.m8.2a"><msub id="S3.SS1.p3.8.m8.2.3" xref="S3.SS1.p3.8.m8.2.3.cmml"><mi id="S3.SS1.p3.8.m8.2.3.2" xref="S3.SS1.p3.8.m8.2.3.2.cmml">β</mi><mrow id="S3.SS1.p3.8.m8.2.2.2.4" xref="S3.SS1.p3.8.m8.2.2.2.3.cmml"><mtext id="S3.SS1.p3.8.m8.1.1.1.1" xref="S3.SS1.p3.8.m8.1.1.1.1a.cmml">true</mtext><mo id="S3.SS1.p3.8.m8.2.2.2.4.1" xref="S3.SS1.p3.8.m8.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p3.8.m8.2.2.2.2" xref="S3.SS1.p3.8.m8.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.8.m8.2b"><apply id="S3.SS1.p3.8.m8.2.3.cmml" xref="S3.SS1.p3.8.m8.2.3"><csymbol cd="ambiguous" id="S3.SS1.p3.8.m8.2.3.1.cmml" xref="S3.SS1.p3.8.m8.2.3">subscript</csymbol><ci id="S3.SS1.p3.8.m8.2.3.2.cmml" xref="S3.SS1.p3.8.m8.2.3.2">𝛽</ci><list id="S3.SS1.p3.8.m8.2.2.2.3.cmml" xref="S3.SS1.p3.8.m8.2.2.2.4"><ci id="S3.SS1.p3.8.m8.1.1.1.1a.cmml" xref="S3.SS1.p3.8.m8.1.1.1.1"><mtext id="S3.SS1.p3.8.m8.1.1.1.1.cmml" mathsize="70%" xref="S3.SS1.p3.8.m8.1.1.1.1">true</mtext></ci><ci id="S3.SS1.p3.8.m8.2.2.2.2.cmml" xref="S3.SS1.p3.8.m8.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.8.m8.2c">\beta_{\text{true},k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.8.m8.2d">italic_β start_POSTSUBSCRIPT true , italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="\tilde{\mathcal{L}}_{0}(c)" class="ltx_Math" display="inline" id="S3.SS1.p3.9.m9.1"><semantics id="S3.SS1.p3.9.m9.1a"><mrow id="S3.SS1.p3.9.m9.1.2" xref="S3.SS1.p3.9.m9.1.2.cmml"><msub id="S3.SS1.p3.9.m9.1.2.2" xref="S3.SS1.p3.9.m9.1.2.2.cmml"><mover accent="true" id="S3.SS1.p3.9.m9.1.2.2.2" xref="S3.SS1.p3.9.m9.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p3.9.m9.1.2.2.2.2" xref="S3.SS1.p3.9.m9.1.2.2.2.2.cmml">ℒ</mi><mo id="S3.SS1.p3.9.m9.1.2.2.2.1" xref="S3.SS1.p3.9.m9.1.2.2.2.1.cmml">~</mo></mover><mn id="S3.SS1.p3.9.m9.1.2.2.3" xref="S3.SS1.p3.9.m9.1.2.2.3.cmml">0</mn></msub><mo id="S3.SS1.p3.9.m9.1.2.1" xref="S3.SS1.p3.9.m9.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.p3.9.m9.1.2.3.2" xref="S3.SS1.p3.9.m9.1.2.cmml"><mo id="S3.SS1.p3.9.m9.1.2.3.2.1" stretchy="false" xref="S3.SS1.p3.9.m9.1.2.cmml">(</mo><mi id="S3.SS1.p3.9.m9.1.1" xref="S3.SS1.p3.9.m9.1.1.cmml">c</mi><mo id="S3.SS1.p3.9.m9.1.2.3.2.2" stretchy="false" xref="S3.SS1.p3.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.9.m9.1b"><apply id="S3.SS1.p3.9.m9.1.2.cmml" xref="S3.SS1.p3.9.m9.1.2"><times id="S3.SS1.p3.9.m9.1.2.1.cmml" xref="S3.SS1.p3.9.m9.1.2.1"></times><apply id="S3.SS1.p3.9.m9.1.2.2.cmml" xref="S3.SS1.p3.9.m9.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.p3.9.m9.1.2.2.1.cmml" xref="S3.SS1.p3.9.m9.1.2.2">subscript</csymbol><apply id="S3.SS1.p3.9.m9.1.2.2.2.cmml" xref="S3.SS1.p3.9.m9.1.2.2.2"><ci id="S3.SS1.p3.9.m9.1.2.2.2.1.cmml" xref="S3.SS1.p3.9.m9.1.2.2.2.1">~</ci><ci id="S3.SS1.p3.9.m9.1.2.2.2.2.cmml" xref="S3.SS1.p3.9.m9.1.2.2.2.2">ℒ</ci></apply><cn id="S3.SS1.p3.9.m9.1.2.2.3.cmml" type="integer" xref="S3.SS1.p3.9.m9.1.2.2.3">0</cn></apply><ci id="S3.SS1.p3.9.m9.1.1.cmml" xref="S3.SS1.p3.9.m9.1.1">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.9.m9.1c">\tilde{\mathcal{L}}_{0}(c)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.9.m9.1d">over~ start_ARG caligraphic_L end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_c )</annotation></semantics></math> may seem abstract at first glance, we provide an example (Example <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:example:_assumption_boundary_BI</span>) in Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Examples</span> that illustrates which values are excluded. By nature of the unidentifiablity of <math alttext="\beta_{\text{true},k}" class="ltx_Math" display="inline" id="S3.SS1.p3.10.m10.2"><semantics id="S3.SS1.p3.10.m10.2a"><msub id="S3.SS1.p3.10.m10.2.3" xref="S3.SS1.p3.10.m10.2.3.cmml"><mi id="S3.SS1.p3.10.m10.2.3.2" xref="S3.SS1.p3.10.m10.2.3.2.cmml">β</mi><mrow id="S3.SS1.p3.10.m10.2.2.2.4" xref="S3.SS1.p3.10.m10.2.2.2.3.cmml"><mtext id="S3.SS1.p3.10.m10.1.1.1.1" xref="S3.SS1.p3.10.m10.1.1.1.1a.cmml">true</mtext><mo id="S3.SS1.p3.10.m10.2.2.2.4.1" xref="S3.SS1.p3.10.m10.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p3.10.m10.2.2.2.2" xref="S3.SS1.p3.10.m10.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.10.m10.2b"><apply id="S3.SS1.p3.10.m10.2.3.cmml" xref="S3.SS1.p3.10.m10.2.3"><csymbol cd="ambiguous" id="S3.SS1.p3.10.m10.2.3.1.cmml" xref="S3.SS1.p3.10.m10.2.3">subscript</csymbol><ci id="S3.SS1.p3.10.m10.2.3.2.cmml" xref="S3.SS1.p3.10.m10.2.3.2">𝛽</ci><list id="S3.SS1.p3.10.m10.2.2.2.3.cmml" xref="S3.SS1.p3.10.m10.2.2.2.4"><ci id="S3.SS1.p3.10.m10.1.1.1.1a.cmml" xref="S3.SS1.p3.10.m10.1.1.1.1"><mtext id="S3.SS1.p3.10.m10.1.1.1.1.cmml" mathsize="70%" xref="S3.SS1.p3.10.m10.1.1.1.1">true</mtext></ci><ci id="S3.SS1.p3.10.m10.2.2.2.2.cmml" xref="S3.SS1.p3.10.m10.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.10.m10.2c">\beta_{\text{true},k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.10.m10.2d">italic_β start_POSTSUBSCRIPT true , italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, Assumption <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I2.i1" title="item (A8) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A8)</span></a> is untestable. However, the aforementioned example illustrates that for well-behaved <math alttext="\mathcal{B}_{I}" class="ltx_Math" display="inline" id="S3.SS1.p3.11.m11.1"><semantics id="S3.SS1.p3.11.m11.1a"><msub id="S3.SS1.p3.11.m11.1.1" xref="S3.SS1.p3.11.m11.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p3.11.m11.1.1.2" xref="S3.SS1.p3.11.m11.1.1.2.cmml">ℬ</mi><mi id="S3.SS1.p3.11.m11.1.1.3" xref="S3.SS1.p3.11.m11.1.1.3.cmml">I</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.11.m11.1b"><apply id="S3.SS1.p3.11.m11.1.1.cmml" xref="S3.SS1.p3.11.m11.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.11.m11.1.1.1.cmml" xref="S3.SS1.p3.11.m11.1.1">subscript</csymbol><ci id="S3.SS1.p3.11.m11.1.1.2.cmml" xref="S3.SS1.p3.11.m11.1.1.2">ℬ</ci><ci id="S3.SS1.p3.11.m11.1.1.3.cmml" xref="S3.SS1.p3.11.m11.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.11.m11.1c">\mathcal{B}_{I}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.11.m11.1d">caligraphic_B start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT</annotation></semantics></math> the set of excluded values is very small.</p> <ol class="ltx_enumerate" id="S3.I3"> <li class="ltx_item" id="S3.I3.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(A9)</span> <div class="ltx_para" id="S3.I3.i1.p1"> <p class="ltx_p" id="S3.I3.i1.p1.2"><math alttext="\exists\eta_{\mathcal{L}}>0:\forall r\in\tilde{\mathcal{L}}_{0}(c):\exists% \beta\in\mathcal{B}(r):\mathbb{E}[m(W,\beta)]>\eta_{\mathcal{L}}" class="ltx_Math" display="inline" id="S3.I3.i1.p1.1.m1.5"><semantics id="S3.I3.i1.p1.1.m1.5a"><mrow id="S3.I3.i1.p1.1.m1.5.5" xref="S3.I3.i1.p1.1.m1.5.5.cmml"><mrow id="S3.I3.i1.p1.1.m1.5.5.3" xref="S3.I3.i1.p1.1.m1.5.5.3.cmml"><mrow id="S3.I3.i1.p1.1.m1.5.5.3.2" xref="S3.I3.i1.p1.1.m1.5.5.3.2.cmml"><mo id="S3.I3.i1.p1.1.m1.5.5.3.2.1" rspace="0.167em" xref="S3.I3.i1.p1.1.m1.5.5.3.2.1.cmml">∃</mo><msub id="S3.I3.i1.p1.1.m1.5.5.3.2.2" xref="S3.I3.i1.p1.1.m1.5.5.3.2.2.cmml"><mi id="S3.I3.i1.p1.1.m1.5.5.3.2.2.2" xref="S3.I3.i1.p1.1.m1.5.5.3.2.2.2.cmml">η</mi><mi class="ltx_font_mathcaligraphic" id="S3.I3.i1.p1.1.m1.5.5.3.2.2.3" xref="S3.I3.i1.p1.1.m1.5.5.3.2.2.3.cmml">ℒ</mi></msub></mrow><mo id="S3.I3.i1.p1.1.m1.5.5.3.1" xref="S3.I3.i1.p1.1.m1.5.5.3.1.cmml">></mo><mn id="S3.I3.i1.p1.1.m1.5.5.3.3" xref="S3.I3.i1.p1.1.m1.5.5.3.3.cmml">0</mn></mrow><mo id="S3.I3.i1.p1.1.m1.5.5.4" lspace="0.278em" rspace="0.278em" xref="S3.I3.i1.p1.1.m1.5.5.4.cmml">:</mo><mrow id="S3.I3.i1.p1.1.m1.5.5.5" xref="S3.I3.i1.p1.1.m1.5.5.5.cmml"><mrow id="S3.I3.i1.p1.1.m1.5.5.5.2" xref="S3.I3.i1.p1.1.m1.5.5.5.2.cmml"><mo id="S3.I3.i1.p1.1.m1.5.5.5.2.1" rspace="0.167em" xref="S3.I3.i1.p1.1.m1.5.5.5.2.1.cmml">∀</mo><mi id="S3.I3.i1.p1.1.m1.5.5.5.2.2" xref="S3.I3.i1.p1.1.m1.5.5.5.2.2.cmml">r</mi></mrow><mo id="S3.I3.i1.p1.1.m1.5.5.5.1" xref="S3.I3.i1.p1.1.m1.5.5.5.1.cmml">∈</mo><mrow id="S3.I3.i1.p1.1.m1.5.5.5.3" xref="S3.I3.i1.p1.1.m1.5.5.5.3.cmml"><msub id="S3.I3.i1.p1.1.m1.5.5.5.3.2" xref="S3.I3.i1.p1.1.m1.5.5.5.3.2.cmml"><mover accent="true" id="S3.I3.i1.p1.1.m1.5.5.5.3.2.2" xref="S3.I3.i1.p1.1.m1.5.5.5.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I3.i1.p1.1.m1.5.5.5.3.2.2.2" xref="S3.I3.i1.p1.1.m1.5.5.5.3.2.2.2.cmml">ℒ</mi><mo id="S3.I3.i1.p1.1.m1.5.5.5.3.2.2.1" xref="S3.I3.i1.p1.1.m1.5.5.5.3.2.2.1.cmml">~</mo></mover><mn id="S3.I3.i1.p1.1.m1.5.5.5.3.2.3" xref="S3.I3.i1.p1.1.m1.5.5.5.3.2.3.cmml">0</mn></msub><mo id="S3.I3.i1.p1.1.m1.5.5.5.3.1" xref="S3.I3.i1.p1.1.m1.5.5.5.3.1.cmml">⁢</mo><mrow id="S3.I3.i1.p1.1.m1.5.5.5.3.3.2" xref="S3.I3.i1.p1.1.m1.5.5.5.3.cmml"><mo id="S3.I3.i1.p1.1.m1.5.5.5.3.3.2.1" stretchy="false" xref="S3.I3.i1.p1.1.m1.5.5.5.3.cmml">(</mo><mi id="S3.I3.i1.p1.1.m1.1.1" xref="S3.I3.i1.p1.1.m1.1.1.cmml">c</mi><mo id="S3.I3.i1.p1.1.m1.5.5.5.3.3.2.2" rspace="0.278em" stretchy="false" xref="S3.I3.i1.p1.1.m1.5.5.5.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.I3.i1.p1.1.m1.5.5.6" rspace="0.278em" xref="S3.I3.i1.p1.1.m1.5.5.6.cmml">:</mo><mrow id="S3.I3.i1.p1.1.m1.5.5.7" xref="S3.I3.i1.p1.1.m1.5.5.7.cmml"><mrow id="S3.I3.i1.p1.1.m1.5.5.7.2" xref="S3.I3.i1.p1.1.m1.5.5.7.2.cmml"><mo id="S3.I3.i1.p1.1.m1.5.5.7.2.1" rspace="0.167em" xref="S3.I3.i1.p1.1.m1.5.5.7.2.1.cmml">∃</mo><mi id="S3.I3.i1.p1.1.m1.5.5.7.2.2" xref="S3.I3.i1.p1.1.m1.5.5.7.2.2.cmml">β</mi></mrow><mo id="S3.I3.i1.p1.1.m1.5.5.7.1" xref="S3.I3.i1.p1.1.m1.5.5.7.1.cmml">∈</mo><mrow id="S3.I3.i1.p1.1.m1.5.5.7.3" xref="S3.I3.i1.p1.1.m1.5.5.7.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I3.i1.p1.1.m1.5.5.7.3.2" xref="S3.I3.i1.p1.1.m1.5.5.7.3.2.cmml">ℬ</mi><mo id="S3.I3.i1.p1.1.m1.5.5.7.3.1" xref="S3.I3.i1.p1.1.m1.5.5.7.3.1.cmml">⁢</mo><mrow id="S3.I3.i1.p1.1.m1.5.5.7.3.3.2" xref="S3.I3.i1.p1.1.m1.5.5.7.3.cmml"><mo id="S3.I3.i1.p1.1.m1.5.5.7.3.3.2.1" stretchy="false" xref="S3.I3.i1.p1.1.m1.5.5.7.3.cmml">(</mo><mi id="S3.I3.i1.p1.1.m1.2.2" xref="S3.I3.i1.p1.1.m1.2.2.cmml">r</mi><mo id="S3.I3.i1.p1.1.m1.5.5.7.3.3.2.2" rspace="0.278em" stretchy="false" xref="S3.I3.i1.p1.1.m1.5.5.7.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.I3.i1.p1.1.m1.5.5.8" rspace="0.278em" xref="S3.I3.i1.p1.1.m1.5.5.8.cmml">:</mo><mrow id="S3.I3.i1.p1.1.m1.5.5.1" xref="S3.I3.i1.p1.1.m1.5.5.1.cmml"><mrow id="S3.I3.i1.p1.1.m1.5.5.1.1" xref="S3.I3.i1.p1.1.m1.5.5.1.1.cmml"><mi id="S3.I3.i1.p1.1.m1.5.5.1.1.3" xref="S3.I3.i1.p1.1.m1.5.5.1.1.3.cmml">𝔼</mi><mo id="S3.I3.i1.p1.1.m1.5.5.1.1.2" xref="S3.I3.i1.p1.1.m1.5.5.1.1.2.cmml">⁢</mo><mrow id="S3.I3.i1.p1.1.m1.5.5.1.1.1.1" xref="S3.I3.i1.p1.1.m1.5.5.1.1.1.2.cmml"><mo id="S3.I3.i1.p1.1.m1.5.5.1.1.1.1.2" stretchy="false" xref="S3.I3.i1.p1.1.m1.5.5.1.1.1.2.1.cmml">[</mo><mrow id="S3.I3.i1.p1.1.m1.5.5.1.1.1.1.1" xref="S3.I3.i1.p1.1.m1.5.5.1.1.1.1.1.cmml"><mi id="S3.I3.i1.p1.1.m1.5.5.1.1.1.1.1.2" xref="S3.I3.i1.p1.1.m1.5.5.1.1.1.1.1.2.cmml">m</mi><mo id="S3.I3.i1.p1.1.m1.5.5.1.1.1.1.1.1" 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xref="S3.I3.i1.p1.1.m1.5.5.1.2"></gt><apply id="S3.I3.i1.p1.1.m1.5.5.1.1.cmml" xref="S3.I3.i1.p1.1.m1.5.5.1.1"><times id="S3.I3.i1.p1.1.m1.5.5.1.1.2.cmml" xref="S3.I3.i1.p1.1.m1.5.5.1.1.2"></times><ci id="S3.I3.i1.p1.1.m1.5.5.1.1.3.cmml" xref="S3.I3.i1.p1.1.m1.5.5.1.1.3">𝔼</ci><apply id="S3.I3.i1.p1.1.m1.5.5.1.1.1.2.cmml" xref="S3.I3.i1.p1.1.m1.5.5.1.1.1.1"><csymbol cd="latexml" id="S3.I3.i1.p1.1.m1.5.5.1.1.1.2.1.cmml" xref="S3.I3.i1.p1.1.m1.5.5.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.I3.i1.p1.1.m1.5.5.1.1.1.1.1.cmml" xref="S3.I3.i1.p1.1.m1.5.5.1.1.1.1.1"><times id="S3.I3.i1.p1.1.m1.5.5.1.1.1.1.1.1.cmml" xref="S3.I3.i1.p1.1.m1.5.5.1.1.1.1.1.1"></times><ci id="S3.I3.i1.p1.1.m1.5.5.1.1.1.1.1.2.cmml" xref="S3.I3.i1.p1.1.m1.5.5.1.1.1.1.1.2">𝑚</ci><interval closure="open" id="S3.I3.i1.p1.1.m1.5.5.1.1.1.1.1.3.1.cmml" xref="S3.I3.i1.p1.1.m1.5.5.1.1.1.1.1.3.2"><ci id="S3.I3.i1.p1.1.m1.3.3.cmml" xref="S3.I3.i1.p1.1.m1.3.3">𝑊</ci><ci id="S3.I3.i1.p1.1.m1.4.4.cmml" xref="S3.I3.i1.p1.1.m1.4.4">𝛽</ci></interval></apply></apply></apply><apply id="S3.I3.i1.p1.1.m1.5.5.1.3.cmml" xref="S3.I3.i1.p1.1.m1.5.5.1.3"><csymbol cd="ambiguous" id="S3.I3.i1.p1.1.m1.5.5.1.3.1.cmml" xref="S3.I3.i1.p1.1.m1.5.5.1.3">subscript</csymbol><ci id="S3.I3.i1.p1.1.m1.5.5.1.3.2.cmml" xref="S3.I3.i1.p1.1.m1.5.5.1.3.2">𝜂</ci><ci id="S3.I3.i1.p1.1.m1.5.5.1.3.3.cmml" xref="S3.I3.i1.p1.1.m1.5.5.1.3.3">ℒ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.1.m1.5c">\exists\eta_{\mathcal{L}}>0:\forall r\in\tilde{\mathcal{L}}_{0}(c):\exists% \beta\in\mathcal{B}(r):\mathbb{E}[m(W,\beta)]>\eta_{\mathcal{L}}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.1.m1.5d">∃ italic_η start_POSTSUBSCRIPT caligraphic_L end_POSTSUBSCRIPT > 0 : ∀ italic_r ∈ over~ start_ARG caligraphic_L end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_c ) : ∃ italic_β ∈ caligraphic_B ( italic_r ) : blackboard_E [ italic_m ( italic_W , italic_β ) ] > italic_η start_POSTSUBSCRIPT caligraphic_L end_POSTSUBSCRIPT</annotation></semantics></math>, for <math alttext="c" class="ltx_Math" display="inline" id="S3.I3.i1.p1.2.m2.1"><semantics id="S3.I3.i1.p1.2.m2.1a"><mi id="S3.I3.i1.p1.2.m2.1.1" xref="S3.I3.i1.p1.2.m2.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.I3.i1.p1.2.m2.1b"><ci id="S3.I3.i1.p1.2.m2.1.1.cmml" xref="S3.I3.i1.p1.2.m2.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.2.m2.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.2.m2.1d">italic_c</annotation></semantics></math> resulting from Assumption <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I2.i1" title="item (A8) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A8)</span></a> and where the inequality holds elementwise.</p> </div> </li> </ol> <p class="ltx_p" id="S3.SS1.p3.13">Lastly, Assumption <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I3.i1" title="item (A9) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A9)</span></a> facilitates certain technical difficulties in the theory of our method. It will in many cases be satisfied when Assumption <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I2.i1" title="item (A8) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A8)</span></a> holds, though it is not implied by it. We can now state the main theorem.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmtheorem1.1.1.1">Theorem 1</span></span></h6> <div class="ltx_para" id="Thmtheorem1.p1"> <p class="ltx_p" id="Thmtheorem1.p1.3"><span class="ltx_text ltx_font_italic" id="Thmtheorem1.p1.3.3">Under Assumptions <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I1.i1" title="item (A1) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A1)</span></a>–<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I3.i1" title="item (A9) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A9)</span></a> it holds that <math alttext="\beta_{\text{true},k}" class="ltx_Math" display="inline" id="Thmtheorem1.p1.1.1.m1.2"><semantics id="Thmtheorem1.p1.1.1.m1.2a"><msub id="Thmtheorem1.p1.1.1.m1.2.3" xref="Thmtheorem1.p1.1.1.m1.2.3.cmml"><mi id="Thmtheorem1.p1.1.1.m1.2.3.2" xref="Thmtheorem1.p1.1.1.m1.2.3.2.cmml">β</mi><mrow id="Thmtheorem1.p1.1.1.m1.2.2.2.4" xref="Thmtheorem1.p1.1.1.m1.2.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="Thmtheorem1.p1.1.1.m1.1.1.1.1" xref="Thmtheorem1.p1.1.1.m1.1.1.1.1a.cmml">true</mtext><mo id="Thmtheorem1.p1.1.1.m1.2.2.2.4.1" xref="Thmtheorem1.p1.1.1.m1.2.2.2.3.cmml">,</mo><mi id="Thmtheorem1.p1.1.1.m1.2.2.2.2" xref="Thmtheorem1.p1.1.1.m1.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.1.1.m1.2b"><apply id="Thmtheorem1.p1.1.1.m1.2.3.cmml" xref="Thmtheorem1.p1.1.1.m1.2.3"><csymbol cd="ambiguous" id="Thmtheorem1.p1.1.1.m1.2.3.1.cmml" xref="Thmtheorem1.p1.1.1.m1.2.3">subscript</csymbol><ci id="Thmtheorem1.p1.1.1.m1.2.3.2.cmml" xref="Thmtheorem1.p1.1.1.m1.2.3.2">𝛽</ci><list id="Thmtheorem1.p1.1.1.m1.2.2.2.3.cmml" xref="Thmtheorem1.p1.1.1.m1.2.2.2.4"><ci id="Thmtheorem1.p1.1.1.m1.1.1.1.1a.cmml" xref="Thmtheorem1.p1.1.1.m1.1.1.1.1"><mtext class="ltx_mathvariant_italic" id="Thmtheorem1.p1.1.1.m1.1.1.1.1.cmml" mathsize="70%" xref="Thmtheorem1.p1.1.1.m1.1.1.1.1">true</mtext></ci><ci id="Thmtheorem1.p1.1.1.m1.2.2.2.2.cmml" xref="Thmtheorem1.p1.1.1.m1.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.1.1.m1.2c">\beta_{\text{true},k}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.1.1.m1.2d">italic_β start_POSTSUBSCRIPT true , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> will be contained in <math alttext="\hat{\mathcal{B}}_{I,k}" class="ltx_Math" display="inline" id="Thmtheorem1.p1.2.2.m2.2"><semantics id="Thmtheorem1.p1.2.2.m2.2a"><msub id="Thmtheorem1.p1.2.2.m2.2.3" xref="Thmtheorem1.p1.2.2.m2.2.3.cmml"><mover accent="true" id="Thmtheorem1.p1.2.2.m2.2.3.2" xref="Thmtheorem1.p1.2.2.m2.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmtheorem1.p1.2.2.m2.2.3.2.2" xref="Thmtheorem1.p1.2.2.m2.2.3.2.2.cmml">ℬ</mi><mo id="Thmtheorem1.p1.2.2.m2.2.3.2.1" xref="Thmtheorem1.p1.2.2.m2.2.3.2.1.cmml">^</mo></mover><mrow id="Thmtheorem1.p1.2.2.m2.2.2.2.4" xref="Thmtheorem1.p1.2.2.m2.2.2.2.3.cmml"><mi id="Thmtheorem1.p1.2.2.m2.1.1.1.1" xref="Thmtheorem1.p1.2.2.m2.1.1.1.1.cmml">I</mi><mo id="Thmtheorem1.p1.2.2.m2.2.2.2.4.1" xref="Thmtheorem1.p1.2.2.m2.2.2.2.3.cmml">,</mo><mi id="Thmtheorem1.p1.2.2.m2.2.2.2.2" xref="Thmtheorem1.p1.2.2.m2.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.2.2.m2.2b"><apply id="Thmtheorem1.p1.2.2.m2.2.3.cmml" xref="Thmtheorem1.p1.2.2.m2.2.3"><csymbol cd="ambiguous" id="Thmtheorem1.p1.2.2.m2.2.3.1.cmml" xref="Thmtheorem1.p1.2.2.m2.2.3">subscript</csymbol><apply id="Thmtheorem1.p1.2.2.m2.2.3.2.cmml" xref="Thmtheorem1.p1.2.2.m2.2.3.2"><ci id="Thmtheorem1.p1.2.2.m2.2.3.2.1.cmml" xref="Thmtheorem1.p1.2.2.m2.2.3.2.1">^</ci><ci id="Thmtheorem1.p1.2.2.m2.2.3.2.2.cmml" xref="Thmtheorem1.p1.2.2.m2.2.3.2.2">ℬ</ci></apply><list id="Thmtheorem1.p1.2.2.m2.2.2.2.3.cmml" xref="Thmtheorem1.p1.2.2.m2.2.2.2.4"><ci id="Thmtheorem1.p1.2.2.m2.1.1.1.1.cmml" xref="Thmtheorem1.p1.2.2.m2.1.1.1.1">𝐼</ci><ci id="Thmtheorem1.p1.2.2.m2.2.2.2.2.cmml" xref="Thmtheorem1.p1.2.2.m2.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.2.2.m2.2c">\hat{\mathcal{B}}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.2.2.m2.2d">over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> with at least the specified confidence level <math alttext="1-\alpha" class="ltx_Math" display="inline" id="Thmtheorem1.p1.3.3.m3.1"><semantics id="Thmtheorem1.p1.3.3.m3.1a"><mrow id="Thmtheorem1.p1.3.3.m3.1.1" xref="Thmtheorem1.p1.3.3.m3.1.1.cmml"><mn id="Thmtheorem1.p1.3.3.m3.1.1.2" xref="Thmtheorem1.p1.3.3.m3.1.1.2.cmml">1</mn><mo id="Thmtheorem1.p1.3.3.m3.1.1.1" xref="Thmtheorem1.p1.3.3.m3.1.1.1.cmml">−</mo><mi id="Thmtheorem1.p1.3.3.m3.1.1.3" xref="Thmtheorem1.p1.3.3.m3.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmtheorem1.p1.3.3.m3.1b"><apply id="Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="Thmtheorem1.p1.3.3.m3.1.1"><minus id="Thmtheorem1.p1.3.3.m3.1.1.1.cmml" xref="Thmtheorem1.p1.3.3.m3.1.1.1"></minus><cn id="Thmtheorem1.p1.3.3.m3.1.1.2.cmml" type="integer" xref="Thmtheorem1.p1.3.3.m3.1.1.2">1</cn><ci id="Thmtheorem1.p1.3.3.m3.1.1.3.cmml" xref="Thmtheorem1.p1.3.3.m3.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmtheorem1.p1.3.3.m3.1c">1-\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmtheorem1.p1.3.3.m3.1d">1 - italic_α</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS1.p4"> <p class="ltx_p" id="S3.SS1.p4.1">The proof of this result, presented in Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Lemmas_and_theorems</span>, consists in verifying that our context meets each of the underpinning assumptions of the test of <cite class="ltx_cite ltx_citemacro_cite">Bei, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib2" title="">2024</a>)</cite>, and can be of independent interest to researchers who want to apply this test in different contexts.</p> </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.2 </span>Instrumental functions</h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.2">We elaborate on the family <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.1"><semantics id="S3.SS2.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.1b"><ci id="S3.SS2.p1.1.m1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.1d">caligraphic_G</annotation></semantics></math> of instrumental functions. First, we will consider a class of instrumental functions for a discrete covariate. Next, we discuss a class for a continuous covariate. An additional class of instrumental functions is discussed in Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Instrumental_functions</span>. By Assumption <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I1.i6" title="item (A6) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A6)</span></a>, we consider covariates that take values in <math alttext="[0,1]" class="ltx_Math" display="inline" id="S3.SS2.p1.2.m2.2"><semantics id="S3.SS2.p1.2.m2.2a"><mrow id="S3.SS2.p1.2.m2.2.3.2" xref="S3.SS2.p1.2.m2.2.3.1.cmml"><mo id="S3.SS2.p1.2.m2.2.3.2.1" stretchy="false" xref="S3.SS2.p1.2.m2.2.3.1.cmml">[</mo><mn id="S3.SS2.p1.2.m2.1.1" xref="S3.SS2.p1.2.m2.1.1.cmml">0</mn><mo id="S3.SS2.p1.2.m2.2.3.2.2" xref="S3.SS2.p1.2.m2.2.3.1.cmml">,</mo><mn id="S3.SS2.p1.2.m2.2.2" xref="S3.SS2.p1.2.m2.2.2.cmml">1</mn><mo id="S3.SS2.p1.2.m2.2.3.2.3" stretchy="false" xref="S3.SS2.p1.2.m2.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.2.m2.2b"><interval closure="closed" id="S3.SS2.p1.2.m2.2.3.1.cmml" xref="S3.SS2.p1.2.m2.2.3.2"><cn id="S3.SS2.p1.2.m2.1.1.cmml" type="integer" xref="S3.SS2.p1.2.m2.1.1">0</cn><cn id="S3.SS2.p1.2.m2.2.2.cmml" type="integer" xref="S3.SS2.p1.2.m2.2.2">1</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.2.m2.2c">[0,1]</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.2.m2.2d">[ 0 , 1 ]</annotation></semantics></math> after scaling. Instrumental functions for vectors of covariates can be obtained by multiplying the instrumental functions for each of its components:</p> <table class="ltx_equation ltx_eqn_table" id="S3.E10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="g_{j_{0},j_{1},\dots,j_{d}}:[0,1]^{d+1}\to\mathbb{R}_{+}:(x_{0},x_{1},\dots,x_% {d})\mapsto g_{j_{0},j_{1},\dots,j_{d}}(x_{0},x_{1},\dots,x_{d})=\prod_{k^{% \prime}=0}^{d}g_{j_{k^{\prime}}}(x_{k^{\prime}})." class="ltx_Math" display="block" id="S3.E10.m1.13"><semantics id="S3.E10.m1.13a"><mrow id="S3.E10.m1.13.13.1" xref="S3.E10.m1.13.13.1.1.cmml"><mrow id="S3.E10.m1.13.13.1.1" xref="S3.E10.m1.13.13.1.1.cmml"><msub id="S3.E10.m1.13.13.1.1.9" xref="S3.E10.m1.13.13.1.1.9.cmml"><mi id="S3.E10.m1.13.13.1.1.9.2" xref="S3.E10.m1.13.13.1.1.9.2.cmml">g</mi><mrow id="S3.E10.m1.4.4.4.4" xref="S3.E10.m1.4.4.4.5.cmml"><msub 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id="S3.E10.m1.13.13.1.1.7.7.1.3.3.1.cmml" xref="S3.E10.m1.13.13.1.1.7.7.1.3.3">subscript</csymbol><ci id="S3.E10.m1.13.13.1.1.7.7.1.3.3.2.cmml" xref="S3.E10.m1.13.13.1.1.7.7.1.3.3.2">𝑗</ci><apply id="S3.E10.m1.13.13.1.1.7.7.1.3.3.3.cmml" xref="S3.E10.m1.13.13.1.1.7.7.1.3.3.3"><csymbol cd="ambiguous" id="S3.E10.m1.13.13.1.1.7.7.1.3.3.3.1.cmml" xref="S3.E10.m1.13.13.1.1.7.7.1.3.3.3">superscript</csymbol><ci id="S3.E10.m1.13.13.1.1.7.7.1.3.3.3.2.cmml" xref="S3.E10.m1.13.13.1.1.7.7.1.3.3.3.2">𝑘</ci><ci id="S3.E10.m1.13.13.1.1.7.7.1.3.3.3.3.cmml" xref="S3.E10.m1.13.13.1.1.7.7.1.3.3.3.3">′</ci></apply></apply></apply><apply id="S3.E10.m1.13.13.1.1.7.7.1.1.1.1.cmml" xref="S3.E10.m1.13.13.1.1.7.7.1.1.1"><csymbol cd="ambiguous" id="S3.E10.m1.13.13.1.1.7.7.1.1.1.1.1.cmml" xref="S3.E10.m1.13.13.1.1.7.7.1.1.1">subscript</csymbol><ci id="S3.E10.m1.13.13.1.1.7.7.1.1.1.1.2.cmml" xref="S3.E10.m1.13.13.1.1.7.7.1.1.1.1.2">𝑥</ci><apply id="S3.E10.m1.13.13.1.1.7.7.1.1.1.1.3.cmml" xref="S3.E10.m1.13.13.1.1.7.7.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.E10.m1.13.13.1.1.7.7.1.1.1.1.3.1.cmml" xref="S3.E10.m1.13.13.1.1.7.7.1.1.1.1.3">superscript</csymbol><ci id="S3.E10.m1.13.13.1.1.7.7.1.1.1.1.3.2.cmml" xref="S3.E10.m1.13.13.1.1.7.7.1.1.1.1.3.2">𝑘</ci><ci id="S3.E10.m1.13.13.1.1.7.7.1.1.1.1.3.3.cmml" xref="S3.E10.m1.13.13.1.1.7.7.1.1.1.1.3.3">′</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E10.m1.13c">g_{j_{0},j_{1},\dots,j_{d}}:[0,1]^{d+1}\to\mathbb{R}_{+}:(x_{0},x_{1},\dots,x_% {d})\mapsto g_{j_{0},j_{1},\dots,j_{d}}(x_{0},x_{1},\dots,x_{d})=\prod_{k^{% \prime}=0}^{d}g_{j_{k^{\prime}}}(x_{k^{\prime}}).</annotation><annotation encoding="application/x-llamapun" id="S3.E10.m1.13d">italic_g start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_j start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT : [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d + 1 end_POSTSUPERSCRIPT → blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT : ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) ↦ italic_g start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_j start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) = ∏ start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.p1.4">Remark that <math alttext="x_{0}\equiv 1" class="ltx_Math" display="inline" id="S3.SS2.p1.3.m1.1"><semantics id="S3.SS2.p1.3.m1.1a"><mrow id="S3.SS2.p1.3.m1.1.1" xref="S3.SS2.p1.3.m1.1.1.cmml"><msub id="S3.SS2.p1.3.m1.1.1.2" xref="S3.SS2.p1.3.m1.1.1.2.cmml"><mi id="S3.SS2.p1.3.m1.1.1.2.2" xref="S3.SS2.p1.3.m1.1.1.2.2.cmml">x</mi><mn id="S3.SS2.p1.3.m1.1.1.2.3" xref="S3.SS2.p1.3.m1.1.1.2.3.cmml">0</mn></msub><mo id="S3.SS2.p1.3.m1.1.1.1" xref="S3.SS2.p1.3.m1.1.1.1.cmml">≡</mo><mn id="S3.SS2.p1.3.m1.1.1.3" xref="S3.SS2.p1.3.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.3.m1.1b"><apply id="S3.SS2.p1.3.m1.1.1.cmml" xref="S3.SS2.p1.3.m1.1.1"><equivalent id="S3.SS2.p1.3.m1.1.1.1.cmml" xref="S3.SS2.p1.3.m1.1.1.1"></equivalent><apply id="S3.SS2.p1.3.m1.1.1.2.cmml" xref="S3.SS2.p1.3.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p1.3.m1.1.1.2.1.cmml" xref="S3.SS2.p1.3.m1.1.1.2">subscript</csymbol><ci id="S3.SS2.p1.3.m1.1.1.2.2.cmml" xref="S3.SS2.p1.3.m1.1.1.2.2">𝑥</ci><cn id="S3.SS2.p1.3.m1.1.1.2.3.cmml" type="integer" xref="S3.SS2.p1.3.m1.1.1.2.3">0</cn></apply><cn id="S3.SS2.p1.3.m1.1.1.3.cmml" type="integer" xref="S3.SS2.p1.3.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.3.m1.1c">x_{0}\equiv 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.3.m1.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≡ 1</annotation></semantics></math> and hence we can simply set <math alttext="g_{j_{0}}(\cdot)=1" class="ltx_Math" display="inline" id="S3.SS2.p1.4.m2.1"><semantics id="S3.SS2.p1.4.m2.1a"><mrow id="S3.SS2.p1.4.m2.1.2" xref="S3.SS2.p1.4.m2.1.2.cmml"><mrow id="S3.SS2.p1.4.m2.1.2.2" xref="S3.SS2.p1.4.m2.1.2.2.cmml"><msub id="S3.SS2.p1.4.m2.1.2.2.2" xref="S3.SS2.p1.4.m2.1.2.2.2.cmml"><mi id="S3.SS2.p1.4.m2.1.2.2.2.2" xref="S3.SS2.p1.4.m2.1.2.2.2.2.cmml">g</mi><msub id="S3.SS2.p1.4.m2.1.2.2.2.3" xref="S3.SS2.p1.4.m2.1.2.2.2.3.cmml"><mi id="S3.SS2.p1.4.m2.1.2.2.2.3.2" xref="S3.SS2.p1.4.m2.1.2.2.2.3.2.cmml">j</mi><mn id="S3.SS2.p1.4.m2.1.2.2.2.3.3" xref="S3.SS2.p1.4.m2.1.2.2.2.3.3.cmml">0</mn></msub></msub><mo id="S3.SS2.p1.4.m2.1.2.2.1" xref="S3.SS2.p1.4.m2.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS2.p1.4.m2.1.2.2.3.2" xref="S3.SS2.p1.4.m2.1.2.2.cmml"><mo id="S3.SS2.p1.4.m2.1.2.2.3.2.1" stretchy="false" xref="S3.SS2.p1.4.m2.1.2.2.cmml">(</mo><mo id="S3.SS2.p1.4.m2.1.1" lspace="0em" rspace="0em" xref="S3.SS2.p1.4.m2.1.1.cmml">⋅</mo><mo id="S3.SS2.p1.4.m2.1.2.2.3.2.2" stretchy="false" xref="S3.SS2.p1.4.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p1.4.m2.1.2.1" xref="S3.SS2.p1.4.m2.1.2.1.cmml">=</mo><mn id="S3.SS2.p1.4.m2.1.2.3" xref="S3.SS2.p1.4.m2.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.4.m2.1b"><apply id="S3.SS2.p1.4.m2.1.2.cmml" xref="S3.SS2.p1.4.m2.1.2"><eq id="S3.SS2.p1.4.m2.1.2.1.cmml" xref="S3.SS2.p1.4.m2.1.2.1"></eq><apply id="S3.SS2.p1.4.m2.1.2.2.cmml" xref="S3.SS2.p1.4.m2.1.2.2"><times id="S3.SS2.p1.4.m2.1.2.2.1.cmml" xref="S3.SS2.p1.4.m2.1.2.2.1"></times><apply id="S3.SS2.p1.4.m2.1.2.2.2.cmml" xref="S3.SS2.p1.4.m2.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.p1.4.m2.1.2.2.2.1.cmml" xref="S3.SS2.p1.4.m2.1.2.2.2">subscript</csymbol><ci id="S3.SS2.p1.4.m2.1.2.2.2.2.cmml" xref="S3.SS2.p1.4.m2.1.2.2.2.2">𝑔</ci><apply id="S3.SS2.p1.4.m2.1.2.2.2.3.cmml" xref="S3.SS2.p1.4.m2.1.2.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.p1.4.m2.1.2.2.2.3.1.cmml" xref="S3.SS2.p1.4.m2.1.2.2.2.3">subscript</csymbol><ci id="S3.SS2.p1.4.m2.1.2.2.2.3.2.cmml" xref="S3.SS2.p1.4.m2.1.2.2.2.3.2">𝑗</ci><cn id="S3.SS2.p1.4.m2.1.2.2.2.3.3.cmml" type="integer" xref="S3.SS2.p1.4.m2.1.2.2.2.3.3">0</cn></apply></apply><ci id="S3.SS2.p1.4.m2.1.1.cmml" xref="S3.SS2.p1.4.m2.1.1">⋅</ci></apply><cn id="S3.SS2.p1.4.m2.1.2.3.cmml" type="integer" xref="S3.SS2.p1.4.m2.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.4.m2.1c">g_{j_{0}}(\cdot)=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.4.m2.1d">italic_g start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( ⋅ ) = 1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.3"><span class="ltx_text ltx_font_bold" id="S3.SS2.p2.3.1">Indicator functions.</span> The most natural class of instrumental functions for a discrete covariate <math alttext="X_{k^{\prime}}" class="ltx_Math" display="inline" id="S3.SS2.p2.1.m1.1"><semantics id="S3.SS2.p2.1.m1.1a"><msub id="S3.SS2.p2.1.m1.1.1" xref="S3.SS2.p2.1.m1.1.1.cmml"><mi id="S3.SS2.p2.1.m1.1.1.2" xref="S3.SS2.p2.1.m1.1.1.2.cmml">X</mi><msup id="S3.SS2.p2.1.m1.1.1.3" xref="S3.SS2.p2.1.m1.1.1.3.cmml"><mi id="S3.SS2.p2.1.m1.1.1.3.2" xref="S3.SS2.p2.1.m1.1.1.3.2.cmml">k</mi><mo id="S3.SS2.p2.1.m1.1.1.3.3" xref="S3.SS2.p2.1.m1.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.1.m1.1b"><apply id="S3.SS2.p2.1.m1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.1.m1.1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p2.1.m1.1.1.2.cmml" xref="S3.SS2.p2.1.m1.1.1.2">𝑋</ci><apply id="S3.SS2.p2.1.m1.1.1.3.cmml" xref="S3.SS2.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p2.1.m1.1.1.3.1.cmml" xref="S3.SS2.p2.1.m1.1.1.3">superscript</csymbol><ci id="S3.SS2.p2.1.m1.1.1.3.2.cmml" xref="S3.SS2.p2.1.m1.1.1.3.2">𝑘</ci><ci id="S3.SS2.p2.1.m1.1.1.3.3.cmml" xref="S3.SS2.p2.1.m1.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.1.m1.1c">X_{k^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.1.m1.1d">italic_X start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="l" class="ltx_Math" display="inline" id="S3.SS2.p2.2.m2.1"><semantics id="S3.SS2.p2.2.m2.1a"><mi id="S3.SS2.p2.2.m2.1.1" xref="S3.SS2.p2.2.m2.1.1.cmml">l</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.2.m2.1b"><ci id="S3.SS2.p2.2.m2.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1">𝑙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.2.m2.1c">l</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.2.m2.1d">italic_l</annotation></semantics></math> levels – say, <math alttext="a_{1},\dots,a_{l}" class="ltx_Math" display="inline" id="S3.SS2.p2.3.m3.3"><semantics id="S3.SS2.p2.3.m3.3a"><mrow id="S3.SS2.p2.3.m3.3.3.2" xref="S3.SS2.p2.3.m3.3.3.3.cmml"><msub id="S3.SS2.p2.3.m3.2.2.1.1" xref="S3.SS2.p2.3.m3.2.2.1.1.cmml"><mi id="S3.SS2.p2.3.m3.2.2.1.1.2" xref="S3.SS2.p2.3.m3.2.2.1.1.2.cmml">a</mi><mn id="S3.SS2.p2.3.m3.2.2.1.1.3" xref="S3.SS2.p2.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.SS2.p2.3.m3.3.3.2.3" xref="S3.SS2.p2.3.m3.3.3.3.cmml">,</mo><mi id="S3.SS2.p2.3.m3.1.1" mathvariant="normal" xref="S3.SS2.p2.3.m3.1.1.cmml">…</mi><mo id="S3.SS2.p2.3.m3.3.3.2.4" xref="S3.SS2.p2.3.m3.3.3.3.cmml">,</mo><msub id="S3.SS2.p2.3.m3.3.3.2.2" xref="S3.SS2.p2.3.m3.3.3.2.2.cmml"><mi id="S3.SS2.p2.3.m3.3.3.2.2.2" xref="S3.SS2.p2.3.m3.3.3.2.2.2.cmml">a</mi><mi id="S3.SS2.p2.3.m3.3.3.2.2.3" xref="S3.SS2.p2.3.m3.3.3.2.2.3.cmml">l</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.3.m3.3b"><list id="S3.SS2.p2.3.m3.3.3.3.cmml" xref="S3.SS2.p2.3.m3.3.3.2"><apply id="S3.SS2.p2.3.m3.2.2.1.1.cmml" xref="S3.SS2.p2.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.3.m3.2.2.1.1.1.cmml" xref="S3.SS2.p2.3.m3.2.2.1.1">subscript</csymbol><ci id="S3.SS2.p2.3.m3.2.2.1.1.2.cmml" xref="S3.SS2.p2.3.m3.2.2.1.1.2">𝑎</ci><cn id="S3.SS2.p2.3.m3.2.2.1.1.3.cmml" type="integer" xref="S3.SS2.p2.3.m3.2.2.1.1.3">1</cn></apply><ci id="S3.SS2.p2.3.m3.1.1.cmml" xref="S3.SS2.p2.3.m3.1.1">…</ci><apply id="S3.SS2.p2.3.m3.3.3.2.2.cmml" xref="S3.SS2.p2.3.m3.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS2.p2.3.m3.3.3.2.2.1.cmml" xref="S3.SS2.p2.3.m3.3.3.2.2">subscript</csymbol><ci id="S3.SS2.p2.3.m3.3.3.2.2.2.cmml" xref="S3.SS2.p2.3.m3.3.3.2.2.2">𝑎</ci><ci id="S3.SS2.p2.3.m3.3.3.2.2.3.cmml" xref="S3.SS2.p2.3.m3.3.3.2.2.3">𝑙</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.3.m3.3c">a_{1},\dots,a_{l}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.3.m3.3d">italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> – are the indicator functions,</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{G}_{\text{disc}}=\{g_{j}:x_{k^{\prime}}\mapsto g_{j}(x_{k^{\prime}})=% \mathbbm{1}(x_{k^{\prime}}=a_{j}),j=1,\dots,l\}." class="ltx_Math" display="block" id="S3.Ex3.m1.4"><semantics id="S3.Ex3.m1.4a"><mrow id="S3.Ex3.m1.4.4.1" xref="S3.Ex3.m1.4.4.1.1.cmml"><mrow id="S3.Ex3.m1.4.4.1.1" xref="S3.Ex3.m1.4.4.1.1.cmml"><msub id="S3.Ex3.m1.4.4.1.1.4" xref="S3.Ex3.m1.4.4.1.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex3.m1.4.4.1.1.4.2" xref="S3.Ex3.m1.4.4.1.1.4.2.cmml">𝒢</mi><mtext id="S3.Ex3.m1.4.4.1.1.4.3" xref="S3.Ex3.m1.4.4.1.1.4.3a.cmml">disc</mtext></msub><mo id="S3.Ex3.m1.4.4.1.1.3" xref="S3.Ex3.m1.4.4.1.1.3.cmml">=</mo><mrow id="S3.Ex3.m1.4.4.1.1.2.2" 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.</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.p2.5">When the discrete covariate is not ordinal, one usually encodes it with <math alttext="l-1" class="ltx_Math" display="inline" id="S3.SS2.p2.4.m1.1"><semantics id="S3.SS2.p2.4.m1.1a"><mrow id="S3.SS2.p2.4.m1.1.1" xref="S3.SS2.p2.4.m1.1.1.cmml"><mi id="S3.SS2.p2.4.m1.1.1.2" xref="S3.SS2.p2.4.m1.1.1.2.cmml">l</mi><mo id="S3.SS2.p2.4.m1.1.1.1" xref="S3.SS2.p2.4.m1.1.1.1.cmml">−</mo><mn id="S3.SS2.p2.4.m1.1.1.3" xref="S3.SS2.p2.4.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.4.m1.1b"><apply id="S3.SS2.p2.4.m1.1.1.cmml" xref="S3.SS2.p2.4.m1.1.1"><minus id="S3.SS2.p2.4.m1.1.1.1.cmml" xref="S3.SS2.p2.4.m1.1.1.1"></minus><ci id="S3.SS2.p2.4.m1.1.1.2.cmml" xref="S3.SS2.p2.4.m1.1.1.2">𝑙</ci><cn id="S3.SS2.p2.4.m1.1.1.3.cmml" type="integer" xref="S3.SS2.p2.4.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.4.m1.1c">l-1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.4.m1.1d">italic_l - 1</annotation></semantics></math> dummy variables when including it in the analysis. Denote these as <math alttext="X_{k^{\prime},1},\dots,X_{k^{\prime},l-1}" class="ltx_Math" display="inline" id="S3.SS2.p2.5.m2.7"><semantics id="S3.SS2.p2.5.m2.7a"><mrow id="S3.SS2.p2.5.m2.7.7.2" xref="S3.SS2.p2.5.m2.7.7.3.cmml"><msub id="S3.SS2.p2.5.m2.6.6.1.1" xref="S3.SS2.p2.5.m2.6.6.1.1.cmml"><mi id="S3.SS2.p2.5.m2.6.6.1.1.2" xref="S3.SS2.p2.5.m2.6.6.1.1.2.cmml">X</mi><mrow id="S3.SS2.p2.5.m2.2.2.2.2" xref="S3.SS2.p2.5.m2.2.2.2.3.cmml"><msup id="S3.SS2.p2.5.m2.2.2.2.2.1" xref="S3.SS2.p2.5.m2.2.2.2.2.1.cmml"><mi id="S3.SS2.p2.5.m2.2.2.2.2.1.2" xref="S3.SS2.p2.5.m2.2.2.2.2.1.2.cmml">k</mi><mo id="S3.SS2.p2.5.m2.2.2.2.2.1.3" xref="S3.SS2.p2.5.m2.2.2.2.2.1.3.cmml">′</mo></msup><mo id="S3.SS2.p2.5.m2.2.2.2.2.2" xref="S3.SS2.p2.5.m2.2.2.2.3.cmml">,</mo><mn id="S3.SS2.p2.5.m2.1.1.1.1" xref="S3.SS2.p2.5.m2.1.1.1.1.cmml">1</mn></mrow></msub><mo id="S3.SS2.p2.5.m2.7.7.2.3" xref="S3.SS2.p2.5.m2.7.7.3.cmml">,</mo><mi id="S3.SS2.p2.5.m2.5.5" mathvariant="normal" xref="S3.SS2.p2.5.m2.5.5.cmml">…</mi><mo id="S3.SS2.p2.5.m2.7.7.2.4" xref="S3.SS2.p2.5.m2.7.7.3.cmml">,</mo><msub id="S3.SS2.p2.5.m2.7.7.2.2" xref="S3.SS2.p2.5.m2.7.7.2.2.cmml"><mi id="S3.SS2.p2.5.m2.7.7.2.2.2" xref="S3.SS2.p2.5.m2.7.7.2.2.2.cmml">X</mi><mrow id="S3.SS2.p2.5.m2.4.4.2.2" xref="S3.SS2.p2.5.m2.4.4.2.3.cmml"><msup id="S3.SS2.p2.5.m2.3.3.1.1.1" xref="S3.SS2.p2.5.m2.3.3.1.1.1.cmml"><mi id="S3.SS2.p2.5.m2.3.3.1.1.1.2" xref="S3.SS2.p2.5.m2.3.3.1.1.1.2.cmml">k</mi><mo id="S3.SS2.p2.5.m2.3.3.1.1.1.3" xref="S3.SS2.p2.5.m2.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S3.SS2.p2.5.m2.4.4.2.2.3" xref="S3.SS2.p2.5.m2.4.4.2.3.cmml">,</mo><mrow id="S3.SS2.p2.5.m2.4.4.2.2.2" xref="S3.SS2.p2.5.m2.4.4.2.2.2.cmml"><mi id="S3.SS2.p2.5.m2.4.4.2.2.2.2" xref="S3.SS2.p2.5.m2.4.4.2.2.2.2.cmml">l</mi><mo id="S3.SS2.p2.5.m2.4.4.2.2.2.1" xref="S3.SS2.p2.5.m2.4.4.2.2.2.1.cmml">−</mo><mn id="S3.SS2.p2.5.m2.4.4.2.2.2.3" xref="S3.SS2.p2.5.m2.4.4.2.2.2.3.cmml">1</mn></mrow></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.5.m2.7b"><list id="S3.SS2.p2.5.m2.7.7.3.cmml" xref="S3.SS2.p2.5.m2.7.7.2"><apply id="S3.SS2.p2.5.m2.6.6.1.1.cmml" xref="S3.SS2.p2.5.m2.6.6.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.5.m2.6.6.1.1.1.cmml" xref="S3.SS2.p2.5.m2.6.6.1.1">subscript</csymbol><ci id="S3.SS2.p2.5.m2.6.6.1.1.2.cmml" xref="S3.SS2.p2.5.m2.6.6.1.1.2">𝑋</ci><list id="S3.SS2.p2.5.m2.2.2.2.3.cmml" xref="S3.SS2.p2.5.m2.2.2.2.2"><apply id="S3.SS2.p2.5.m2.2.2.2.2.1.cmml" xref="S3.SS2.p2.5.m2.2.2.2.2.1"><csymbol cd="ambiguous" id="S3.SS2.p2.5.m2.2.2.2.2.1.1.cmml" xref="S3.SS2.p2.5.m2.2.2.2.2.1">superscript</csymbol><ci id="S3.SS2.p2.5.m2.2.2.2.2.1.2.cmml" xref="S3.SS2.p2.5.m2.2.2.2.2.1.2">𝑘</ci><ci id="S3.SS2.p2.5.m2.2.2.2.2.1.3.cmml" xref="S3.SS2.p2.5.m2.2.2.2.2.1.3">′</ci></apply><cn id="S3.SS2.p2.5.m2.1.1.1.1.cmml" type="integer" xref="S3.SS2.p2.5.m2.1.1.1.1">1</cn></list></apply><ci id="S3.SS2.p2.5.m2.5.5.cmml" xref="S3.SS2.p2.5.m2.5.5">…</ci><apply id="S3.SS2.p2.5.m2.7.7.2.2.cmml" xref="S3.SS2.p2.5.m2.7.7.2.2"><csymbol cd="ambiguous" id="S3.SS2.p2.5.m2.7.7.2.2.1.cmml" xref="S3.SS2.p2.5.m2.7.7.2.2">subscript</csymbol><ci id="S3.SS2.p2.5.m2.7.7.2.2.2.cmml" xref="S3.SS2.p2.5.m2.7.7.2.2.2">𝑋</ci><list id="S3.SS2.p2.5.m2.4.4.2.3.cmml" xref="S3.SS2.p2.5.m2.4.4.2.2"><apply id="S3.SS2.p2.5.m2.3.3.1.1.1.cmml" xref="S3.SS2.p2.5.m2.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.5.m2.3.3.1.1.1.1.cmml" xref="S3.SS2.p2.5.m2.3.3.1.1.1">superscript</csymbol><ci id="S3.SS2.p2.5.m2.3.3.1.1.1.2.cmml" xref="S3.SS2.p2.5.m2.3.3.1.1.1.2">𝑘</ci><ci id="S3.SS2.p2.5.m2.3.3.1.1.1.3.cmml" xref="S3.SS2.p2.5.m2.3.3.1.1.1.3">′</ci></apply><apply id="S3.SS2.p2.5.m2.4.4.2.2.2.cmml" xref="S3.SS2.p2.5.m2.4.4.2.2.2"><minus id="S3.SS2.p2.5.m2.4.4.2.2.2.1.cmml" xref="S3.SS2.p2.5.m2.4.4.2.2.2.1"></minus><ci id="S3.SS2.p2.5.m2.4.4.2.2.2.2.cmml" xref="S3.SS2.p2.5.m2.4.4.2.2.2.2">𝑙</ci><cn id="S3.SS2.p2.5.m2.4.4.2.2.2.3.cmml" type="integer" xref="S3.SS2.p2.5.m2.4.4.2.2.2.3">1</cn></apply></list></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.5.m2.7c">X_{k^{\prime},1},\dots,X_{k^{\prime},l-1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.5.m2.7d">italic_X start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , 1 end_POSTSUBSCRIPT , … , italic_X start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_l - 1 end_POSTSUBSCRIPT</annotation></semantics></math>. In this case, one could use the class</p> <table class="ltx_equation ltx_eqn_table" id="S3.SS2.p2.6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{G}_{\text{cat}}=\bigg{\{}g_{j}:(x_{k^{\prime},1},\dots,x_{k^{\prime},% l-1})\mapsto g_{j}(x_{k^{\prime},1},\dots,x_{k^{\prime},l-1})=\\ \mathbbm{1}(x_{k^{\prime},j}=1)\prod_{1\leq j^{\prime}<l,j^{\prime}\neq j}% \mathbbm{1}(x_{k^{\prime},j^{\prime}}=0),j=1,\dots,l\bigg{\}}," class="ltx_Math" display="block" id="S3.SS2.p2.6.m1.56"><semantics id="S3.SS2.p2.6.m1.56a"><mtable displaystyle="true" id="S3.SS2.p2.6.m1.55.55" rowspacing="0pt"><mtr id="S3.SS2.p2.6.m1.55.55a"><mtd class="ltx_align_left" columnalign="left" id="S3.SS2.p2.6.m1.55.55b"><mrow id="S3.SS2.p2.6.m1.29.29.29.29.29"><msub id="S3.SS2.p2.6.m1.29.29.29.29.29.30"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p2.6.m1.1.1.1.1.1.1" 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\mathbbm{1}(x_{k^{\prime},j^{\prime}}=0),j=1,\dots,l\bigg{\}},</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.6.m1.56d">start_ROW start_CELL caligraphic_G start_POSTSUBSCRIPT cat end_POSTSUBSCRIPT = { italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT : ( italic_x start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_l - 1 end_POSTSUBSCRIPT ) ↦ italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_l - 1 end_POSTSUBSCRIPT ) = end_CELL end_ROW start_ROW start_CELL blackboard_1 ( italic_x start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_j end_POSTSUBSCRIPT = 1 ) ∏ start_POSTSUBSCRIPT 1 ≤ italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_l , italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≠ italic_j end_POSTSUBSCRIPT blackboard_1 ( italic_x start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 0 ) , italic_j = 1 , … , italic_l } , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.p2.15">with <math alttext="x_{k^{\prime},l}\equiv 1" class="ltx_Math" display="inline" id="S3.SS2.p2.7.m1.2"><semantics id="S3.SS2.p2.7.m1.2a"><mrow id="S3.SS2.p2.7.m1.2.3" xref="S3.SS2.p2.7.m1.2.3.cmml"><msub id="S3.SS2.p2.7.m1.2.3.2" xref="S3.SS2.p2.7.m1.2.3.2.cmml"><mi id="S3.SS2.p2.7.m1.2.3.2.2" xref="S3.SS2.p2.7.m1.2.3.2.2.cmml">x</mi><mrow id="S3.SS2.p2.7.m1.2.2.2.2" xref="S3.SS2.p2.7.m1.2.2.2.3.cmml"><msup id="S3.SS2.p2.7.m1.2.2.2.2.1" xref="S3.SS2.p2.7.m1.2.2.2.2.1.cmml"><mi id="S3.SS2.p2.7.m1.2.2.2.2.1.2" xref="S3.SS2.p2.7.m1.2.2.2.2.1.2.cmml">k</mi><mo id="S3.SS2.p2.7.m1.2.2.2.2.1.3" xref="S3.SS2.p2.7.m1.2.2.2.2.1.3.cmml">′</mo></msup><mo id="S3.SS2.p2.7.m1.2.2.2.2.2" xref="S3.SS2.p2.7.m1.2.2.2.3.cmml">,</mo><mi id="S3.SS2.p2.7.m1.1.1.1.1" xref="S3.SS2.p2.7.m1.1.1.1.1.cmml">l</mi></mrow></msub><mo id="S3.SS2.p2.7.m1.2.3.1" xref="S3.SS2.p2.7.m1.2.3.1.cmml">≡</mo><mn id="S3.SS2.p2.7.m1.2.3.3" xref="S3.SS2.p2.7.m1.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.7.m1.2b"><apply id="S3.SS2.p2.7.m1.2.3.cmml" xref="S3.SS2.p2.7.m1.2.3"><equivalent id="S3.SS2.p2.7.m1.2.3.1.cmml" xref="S3.SS2.p2.7.m1.2.3.1"></equivalent><apply id="S3.SS2.p2.7.m1.2.3.2.cmml" xref="S3.SS2.p2.7.m1.2.3.2"><csymbol cd="ambiguous" id="S3.SS2.p2.7.m1.2.3.2.1.cmml" xref="S3.SS2.p2.7.m1.2.3.2">subscript</csymbol><ci id="S3.SS2.p2.7.m1.2.3.2.2.cmml" xref="S3.SS2.p2.7.m1.2.3.2.2">𝑥</ci><list id="S3.SS2.p2.7.m1.2.2.2.3.cmml" xref="S3.SS2.p2.7.m1.2.2.2.2"><apply id="S3.SS2.p2.7.m1.2.2.2.2.1.cmml" xref="S3.SS2.p2.7.m1.2.2.2.2.1"><csymbol cd="ambiguous" id="S3.SS2.p2.7.m1.2.2.2.2.1.1.cmml" xref="S3.SS2.p2.7.m1.2.2.2.2.1">superscript</csymbol><ci id="S3.SS2.p2.7.m1.2.2.2.2.1.2.cmml" xref="S3.SS2.p2.7.m1.2.2.2.2.1.2">𝑘</ci><ci id="S3.SS2.p2.7.m1.2.2.2.2.1.3.cmml" xref="S3.SS2.p2.7.m1.2.2.2.2.1.3">′</ci></apply><ci id="S3.SS2.p2.7.m1.1.1.1.1.cmml" xref="S3.SS2.p2.7.m1.1.1.1.1">𝑙</ci></list></apply><cn id="S3.SS2.p2.7.m1.2.3.3.cmml" type="integer" xref="S3.SS2.p2.7.m1.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.7.m1.2c">x_{k^{\prime},l}\equiv 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.7.m1.2d">italic_x start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_l end_POSTSUBSCRIPT ≡ 1</annotation></semantics></math>. Simply viewing each dummy variable as a separate binary covariate and considering <math alttext="\mathcal{G}_{\text{disc}}" class="ltx_Math" display="inline" id="S3.SS2.p2.8.m2.1"><semantics id="S3.SS2.p2.8.m2.1a"><msub id="S3.SS2.p2.8.m2.1.1" xref="S3.SS2.p2.8.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p2.8.m2.1.1.2" xref="S3.SS2.p2.8.m2.1.1.2.cmml">𝒢</mi><mtext id="S3.SS2.p2.8.m2.1.1.3" xref="S3.SS2.p2.8.m2.1.1.3a.cmml">disc</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.8.m2.1b"><apply id="S3.SS2.p2.8.m2.1.1.cmml" xref="S3.SS2.p2.8.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.8.m2.1.1.1.cmml" xref="S3.SS2.p2.8.m2.1.1">subscript</csymbol><ci id="S3.SS2.p2.8.m2.1.1.2.cmml" xref="S3.SS2.p2.8.m2.1.1.2">𝒢</ci><ci id="S3.SS2.p2.8.m2.1.1.3a.cmml" xref="S3.SS2.p2.8.m2.1.1.3"><mtext id="S3.SS2.p2.8.m2.1.1.3.cmml" mathsize="70%" xref="S3.SS2.p2.8.m2.1.1.3">disc</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.8.m2.1c">\mathcal{G}_{\text{disc}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.8.m2.1d">caligraphic_G start_POSTSUBSCRIPT disc end_POSTSUBSCRIPT</annotation></semantics></math> for each will not work, since it can be seen that this will lead to instrumental functions of the form (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.E10" title="In 3.2 Instrumental functions ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">10</span></a>) that will always be zero (which violates Assumption <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I1.i7" title="item (A7) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A7)</span></a>). Indeed, consider for example the case in which there is one discrete covariate <math alttext="X_{1}" class="ltx_Math" display="inline" id="S3.SS2.p2.9.m3.1"><semantics id="S3.SS2.p2.9.m3.1a"><msub id="S3.SS2.p2.9.m3.1.1" xref="S3.SS2.p2.9.m3.1.1.cmml"><mi id="S3.SS2.p2.9.m3.1.1.2" xref="S3.SS2.p2.9.m3.1.1.2.cmml">X</mi><mn id="S3.SS2.p2.9.m3.1.1.3" xref="S3.SS2.p2.9.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.9.m3.1b"><apply id="S3.SS2.p2.9.m3.1.1.cmml" xref="S3.SS2.p2.9.m3.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.9.m3.1.1.1.cmml" xref="S3.SS2.p2.9.m3.1.1">subscript</csymbol><ci id="S3.SS2.p2.9.m3.1.1.2.cmml" xref="S3.SS2.p2.9.m3.1.1.2">𝑋</ci><cn id="S3.SS2.p2.9.m3.1.1.3.cmml" type="integer" xref="S3.SS2.p2.9.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.9.m3.1c">X_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.9.m3.1d">italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> that has four levels, which we encode using three binary variables <math alttext="(X_{1,1},X_{1,2},X_{1,3})" class="ltx_Math" display="inline" id="S3.SS2.p2.10.m4.9"><semantics id="S3.SS2.p2.10.m4.9a"><mrow id="S3.SS2.p2.10.m4.9.9.3" xref="S3.SS2.p2.10.m4.9.9.4.cmml"><mo id="S3.SS2.p2.10.m4.9.9.3.4" stretchy="false" xref="S3.SS2.p2.10.m4.9.9.4.cmml">(</mo><msub id="S3.SS2.p2.10.m4.7.7.1.1" xref="S3.SS2.p2.10.m4.7.7.1.1.cmml"><mi id="S3.SS2.p2.10.m4.7.7.1.1.2" xref="S3.SS2.p2.10.m4.7.7.1.1.2.cmml">X</mi><mrow id="S3.SS2.p2.10.m4.2.2.2.4" xref="S3.SS2.p2.10.m4.2.2.2.3.cmml"><mn id="S3.SS2.p2.10.m4.1.1.1.1" xref="S3.SS2.p2.10.m4.1.1.1.1.cmml">1</mn><mo id="S3.SS2.p2.10.m4.2.2.2.4.1" xref="S3.SS2.p2.10.m4.2.2.2.3.cmml">,</mo><mn id="S3.SS2.p2.10.m4.2.2.2.2" xref="S3.SS2.p2.10.m4.2.2.2.2.cmml">1</mn></mrow></msub><mo id="S3.SS2.p2.10.m4.9.9.3.5" xref="S3.SS2.p2.10.m4.9.9.4.cmml">,</mo><msub id="S3.SS2.p2.10.m4.8.8.2.2" xref="S3.SS2.p2.10.m4.8.8.2.2.cmml"><mi id="S3.SS2.p2.10.m4.8.8.2.2.2" xref="S3.SS2.p2.10.m4.8.8.2.2.2.cmml">X</mi><mrow id="S3.SS2.p2.10.m4.4.4.2.4" xref="S3.SS2.p2.10.m4.4.4.2.3.cmml"><mn id="S3.SS2.p2.10.m4.3.3.1.1" xref="S3.SS2.p2.10.m4.3.3.1.1.cmml">1</mn><mo id="S3.SS2.p2.10.m4.4.4.2.4.1" xref="S3.SS2.p2.10.m4.4.4.2.3.cmml">,</mo><mn id="S3.SS2.p2.10.m4.4.4.2.2" xref="S3.SS2.p2.10.m4.4.4.2.2.cmml">2</mn></mrow></msub><mo id="S3.SS2.p2.10.m4.9.9.3.6" xref="S3.SS2.p2.10.m4.9.9.4.cmml">,</mo><msub id="S3.SS2.p2.10.m4.9.9.3.3" xref="S3.SS2.p2.10.m4.9.9.3.3.cmml"><mi id="S3.SS2.p2.10.m4.9.9.3.3.2" xref="S3.SS2.p2.10.m4.9.9.3.3.2.cmml">X</mi><mrow id="S3.SS2.p2.10.m4.6.6.2.4" xref="S3.SS2.p2.10.m4.6.6.2.3.cmml"><mn id="S3.SS2.p2.10.m4.5.5.1.1" xref="S3.SS2.p2.10.m4.5.5.1.1.cmml">1</mn><mo id="S3.SS2.p2.10.m4.6.6.2.4.1" xref="S3.SS2.p2.10.m4.6.6.2.3.cmml">,</mo><mn id="S3.SS2.p2.10.m4.6.6.2.2" xref="S3.SS2.p2.10.m4.6.6.2.2.cmml">3</mn></mrow></msub><mo id="S3.SS2.p2.10.m4.9.9.3.7" stretchy="false" xref="S3.SS2.p2.10.m4.9.9.4.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.10.m4.9b"><vector id="S3.SS2.p2.10.m4.9.9.4.cmml" xref="S3.SS2.p2.10.m4.9.9.3"><apply id="S3.SS2.p2.10.m4.7.7.1.1.cmml" xref="S3.SS2.p2.10.m4.7.7.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.10.m4.7.7.1.1.1.cmml" xref="S3.SS2.p2.10.m4.7.7.1.1">subscript</csymbol><ci id="S3.SS2.p2.10.m4.7.7.1.1.2.cmml" xref="S3.SS2.p2.10.m4.7.7.1.1.2">𝑋</ci><list id="S3.SS2.p2.10.m4.2.2.2.3.cmml" xref="S3.SS2.p2.10.m4.2.2.2.4"><cn id="S3.SS2.p2.10.m4.1.1.1.1.cmml" type="integer" xref="S3.SS2.p2.10.m4.1.1.1.1">1</cn><cn id="S3.SS2.p2.10.m4.2.2.2.2.cmml" type="integer" xref="S3.SS2.p2.10.m4.2.2.2.2">1</cn></list></apply><apply id="S3.SS2.p2.10.m4.8.8.2.2.cmml" xref="S3.SS2.p2.10.m4.8.8.2.2"><csymbol cd="ambiguous" id="S3.SS2.p2.10.m4.8.8.2.2.1.cmml" xref="S3.SS2.p2.10.m4.8.8.2.2">subscript</csymbol><ci id="S3.SS2.p2.10.m4.8.8.2.2.2.cmml" xref="S3.SS2.p2.10.m4.8.8.2.2.2">𝑋</ci><list id="S3.SS2.p2.10.m4.4.4.2.3.cmml" xref="S3.SS2.p2.10.m4.4.4.2.4"><cn id="S3.SS2.p2.10.m4.3.3.1.1.cmml" type="integer" xref="S3.SS2.p2.10.m4.3.3.1.1">1</cn><cn id="S3.SS2.p2.10.m4.4.4.2.2.cmml" type="integer" xref="S3.SS2.p2.10.m4.4.4.2.2">2</cn></list></apply><apply id="S3.SS2.p2.10.m4.9.9.3.3.cmml" xref="S3.SS2.p2.10.m4.9.9.3.3"><csymbol cd="ambiguous" id="S3.SS2.p2.10.m4.9.9.3.3.1.cmml" xref="S3.SS2.p2.10.m4.9.9.3.3">subscript</csymbol><ci id="S3.SS2.p2.10.m4.9.9.3.3.2.cmml" xref="S3.SS2.p2.10.m4.9.9.3.3.2">𝑋</ci><list id="S3.SS2.p2.10.m4.6.6.2.3.cmml" xref="S3.SS2.p2.10.m4.6.6.2.4"><cn id="S3.SS2.p2.10.m4.5.5.1.1.cmml" type="integer" xref="S3.SS2.p2.10.m4.5.5.1.1">1</cn><cn id="S3.SS2.p2.10.m4.6.6.2.2.cmml" type="integer" xref="S3.SS2.p2.10.m4.6.6.2.2">3</cn></list></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.10.m4.9c">(X_{1,1},X_{1,2},X_{1,3})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.10.m4.9d">( italic_X start_POSTSUBSCRIPT 1 , 1 end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT 1 , 3 end_POSTSUBSCRIPT )</annotation></semantics></math>. Considering <math alttext="\mathcal{G}_{\text{disc}}" class="ltx_Math" display="inline" id="S3.SS2.p2.11.m5.1"><semantics id="S3.SS2.p2.11.m5.1a"><msub id="S3.SS2.p2.11.m5.1.1" xref="S3.SS2.p2.11.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p2.11.m5.1.1.2" xref="S3.SS2.p2.11.m5.1.1.2.cmml">𝒢</mi><mtext id="S3.SS2.p2.11.m5.1.1.3" xref="S3.SS2.p2.11.m5.1.1.3a.cmml">disc</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.11.m5.1b"><apply id="S3.SS2.p2.11.m5.1.1.cmml" xref="S3.SS2.p2.11.m5.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.11.m5.1.1.1.cmml" xref="S3.SS2.p2.11.m5.1.1">subscript</csymbol><ci id="S3.SS2.p2.11.m5.1.1.2.cmml" xref="S3.SS2.p2.11.m5.1.1.2">𝒢</ci><ci id="S3.SS2.p2.11.m5.1.1.3a.cmml" xref="S3.SS2.p2.11.m5.1.1.3"><mtext id="S3.SS2.p2.11.m5.1.1.3.cmml" mathsize="70%" xref="S3.SS2.p2.11.m5.1.1.3">disc</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.11.m5.1c">\mathcal{G}_{\text{disc}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.11.m5.1d">caligraphic_G start_POSTSUBSCRIPT disc end_POSTSUBSCRIPT</annotation></semantics></math> for each and combining them via Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.E10" title="In 3.2 Instrumental functions ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">10</span></a>), we would obtain a class <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S3.SS2.p2.12.m6.1"><semantics id="S3.SS2.p2.12.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p2.12.m6.1.1" xref="S3.SS2.p2.12.m6.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.12.m6.1b"><ci id="S3.SS2.p2.12.m6.1.1.cmml" xref="S3.SS2.p2.12.m6.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.12.m6.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.12.m6.1d">caligraphic_G</annotation></semantics></math> that contains the instrumental function <math alttext="g(x_{1})=\mathbbm{1}(x_{1,1}=1)\mathbbm{1}(x_{1,2}=0)\mathbbm{1}(x_{1,3}=1)" class="ltx_Math" display="inline" id="S3.SS2.p2.13.m7.10"><semantics id="S3.SS2.p2.13.m7.10a"><mrow id="S3.SS2.p2.13.m7.10.10" xref="S3.SS2.p2.13.m7.10.10.cmml"><mrow id="S3.SS2.p2.13.m7.7.7.1" xref="S3.SS2.p2.13.m7.7.7.1.cmml"><mi id="S3.SS2.p2.13.m7.7.7.1.3" xref="S3.SS2.p2.13.m7.7.7.1.3.cmml">g</mi><mo id="S3.SS2.p2.13.m7.7.7.1.2" xref="S3.SS2.p2.13.m7.7.7.1.2.cmml">⁢</mo><mrow id="S3.SS2.p2.13.m7.7.7.1.1.1" xref="S3.SS2.p2.13.m7.7.7.1.1.1.1.cmml"><mo id="S3.SS2.p2.13.m7.7.7.1.1.1.2" stretchy="false" xref="S3.SS2.p2.13.m7.7.7.1.1.1.1.cmml">(</mo><msub id="S3.SS2.p2.13.m7.7.7.1.1.1.1" xref="S3.SS2.p2.13.m7.7.7.1.1.1.1.cmml"><mi id="S3.SS2.p2.13.m7.7.7.1.1.1.1.2" xref="S3.SS2.p2.13.m7.7.7.1.1.1.1.2.cmml">x</mi><mn id="S3.SS2.p2.13.m7.7.7.1.1.1.1.3" xref="S3.SS2.p2.13.m7.7.7.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS2.p2.13.m7.7.7.1.1.1.3" stretchy="false" xref="S3.SS2.p2.13.m7.7.7.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p2.13.m7.10.10.5" xref="S3.SS2.p2.13.m7.10.10.5.cmml">=</mo><mrow id="S3.SS2.p2.13.m7.10.10.4" xref="S3.SS2.p2.13.m7.10.10.4.cmml"><mn id="S3.SS2.p2.13.m7.10.10.4.5" xref="S3.SS2.p2.13.m7.10.10.4.5.cmml">𝟙</mn><mo id="S3.SS2.p2.13.m7.10.10.4.4" xref="S3.SS2.p2.13.m7.10.10.4.4.cmml">⁢</mo><mrow id="S3.SS2.p2.13.m7.8.8.2.1.1" xref="S3.SS2.p2.13.m7.8.8.2.1.1.1.cmml"><mo id="S3.SS2.p2.13.m7.8.8.2.1.1.2" stretchy="false" xref="S3.SS2.p2.13.m7.8.8.2.1.1.1.cmml">(</mo><mrow id="S3.SS2.p2.13.m7.8.8.2.1.1.1" xref="S3.SS2.p2.13.m7.8.8.2.1.1.1.cmml"><msub id="S3.SS2.p2.13.m7.8.8.2.1.1.1.2" xref="S3.SS2.p2.13.m7.8.8.2.1.1.1.2.cmml"><mi id="S3.SS2.p2.13.m7.8.8.2.1.1.1.2.2" xref="S3.SS2.p2.13.m7.8.8.2.1.1.1.2.2.cmml">x</mi><mrow 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xref="S3.SS2.p2.13.m7.4.4.2.2">2</cn></list></apply><cn id="S3.SS2.p2.13.m7.9.9.3.2.1.1.3.cmml" type="integer" xref="S3.SS2.p2.13.m7.9.9.3.2.1.1.3">0</cn></apply><cn id="S3.SS2.p2.13.m7.10.10.4.7.cmml" type="integer" xref="S3.SS2.p2.13.m7.10.10.4.7">1</cn><apply id="S3.SS2.p2.13.m7.10.10.4.3.1.1.cmml" xref="S3.SS2.p2.13.m7.10.10.4.3.1"><eq id="S3.SS2.p2.13.m7.10.10.4.3.1.1.1.cmml" xref="S3.SS2.p2.13.m7.10.10.4.3.1.1.1"></eq><apply id="S3.SS2.p2.13.m7.10.10.4.3.1.1.2.cmml" xref="S3.SS2.p2.13.m7.10.10.4.3.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p2.13.m7.10.10.4.3.1.1.2.1.cmml" xref="S3.SS2.p2.13.m7.10.10.4.3.1.1.2">subscript</csymbol><ci id="S3.SS2.p2.13.m7.10.10.4.3.1.1.2.2.cmml" xref="S3.SS2.p2.13.m7.10.10.4.3.1.1.2.2">𝑥</ci><list id="S3.SS2.p2.13.m7.6.6.2.3.cmml" xref="S3.SS2.p2.13.m7.6.6.2.4"><cn id="S3.SS2.p2.13.m7.5.5.1.1.cmml" type="integer" xref="S3.SS2.p2.13.m7.5.5.1.1">1</cn><cn id="S3.SS2.p2.13.m7.6.6.2.2.cmml" type="integer" xref="S3.SS2.p2.13.m7.6.6.2.2">3</cn></list></apply><cn id="S3.SS2.p2.13.m7.10.10.4.3.1.1.3.cmml" type="integer" xref="S3.SS2.p2.13.m7.10.10.4.3.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.13.m7.10c">g(x_{1})=\mathbbm{1}(x_{1,1}=1)\mathbbm{1}(x_{1,2}=0)\mathbbm{1}(x_{1,3}=1)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.13.m7.10d">italic_g ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = blackboard_1 ( italic_x start_POSTSUBSCRIPT 1 , 1 end_POSTSUBSCRIPT = 1 ) blackboard_1 ( italic_x start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT = 0 ) blackboard_1 ( italic_x start_POSTSUBSCRIPT 1 , 3 end_POSTSUBSCRIPT = 1 )</annotation></semantics></math>. 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id="S3.SS2.p2.15.m9.1c">g</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.15.m9.1d">italic_g</annotation></semantics></math> in this case is always zero.</p> </div> <div class="ltx_para" id="S3.SS2.p3"> <p class="ltx_p" id="S3.SS2.p3.5"><span class="ltx_text ltx_font_bold" id="S3.SS2.p3.5.1">Spline functions.</span> A commonly used class of instrumental functions for continuous covariates is the class of cubic B-splines. Let <math alttext="\{p_{j}\}_{j=-2,\dots,l+2}" class="ltx_Math" display="inline" id="S3.SS2.p3.1.m1.4"><semantics id="S3.SS2.p3.1.m1.4a"><msub id="S3.SS2.p3.1.m1.4.4" xref="S3.SS2.p3.1.m1.4.4.cmml"><mrow id="S3.SS2.p3.1.m1.4.4.1.1" xref="S3.SS2.p3.1.m1.4.4.1.2.cmml"><mo id="S3.SS2.p3.1.m1.4.4.1.1.2" stretchy="false" xref="S3.SS2.p3.1.m1.4.4.1.2.cmml">{</mo><msub id="S3.SS2.p3.1.m1.4.4.1.1.1" xref="S3.SS2.p3.1.m1.4.4.1.1.1.cmml"><mi id="S3.SS2.p3.1.m1.4.4.1.1.1.2" xref="S3.SS2.p3.1.m1.4.4.1.1.1.2.cmml">p</mi><mi id="S3.SS2.p3.1.m1.4.4.1.1.1.3" xref="S3.SS2.p3.1.m1.4.4.1.1.1.3.cmml">j</mi></msub><mo id="S3.SS2.p3.1.m1.4.4.1.1.3" stretchy="false" xref="S3.SS2.p3.1.m1.4.4.1.2.cmml">}</mo></mrow><mrow id="S3.SS2.p3.1.m1.3.3.3" xref="S3.SS2.p3.1.m1.3.3.3.cmml"><mi id="S3.SS2.p3.1.m1.3.3.3.5" xref="S3.SS2.p3.1.m1.3.3.3.5.cmml">j</mi><mo id="S3.SS2.p3.1.m1.3.3.3.4" xref="S3.SS2.p3.1.m1.3.3.3.4.cmml">=</mo><mrow id="S3.SS2.p3.1.m1.3.3.3.3.2" xref="S3.SS2.p3.1.m1.3.3.3.3.3.cmml"><mrow 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cd="ambiguous" id="S3.SS2.p3.1.m1.4.4.2.cmml" xref="S3.SS2.p3.1.m1.4.4">subscript</csymbol><set id="S3.SS2.p3.1.m1.4.4.1.2.cmml" xref="S3.SS2.p3.1.m1.4.4.1.1"><apply id="S3.SS2.p3.1.m1.4.4.1.1.1.cmml" xref="S3.SS2.p3.1.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.1.m1.4.4.1.1.1.1.cmml" xref="S3.SS2.p3.1.m1.4.4.1.1.1">subscript</csymbol><ci id="S3.SS2.p3.1.m1.4.4.1.1.1.2.cmml" xref="S3.SS2.p3.1.m1.4.4.1.1.1.2">𝑝</ci><ci id="S3.SS2.p3.1.m1.4.4.1.1.1.3.cmml" xref="S3.SS2.p3.1.m1.4.4.1.1.1.3">𝑗</ci></apply></set><apply id="S3.SS2.p3.1.m1.3.3.3.cmml" xref="S3.SS2.p3.1.m1.3.3.3"><eq id="S3.SS2.p3.1.m1.3.3.3.4.cmml" xref="S3.SS2.p3.1.m1.3.3.3.4"></eq><ci id="S3.SS2.p3.1.m1.3.3.3.5.cmml" xref="S3.SS2.p3.1.m1.3.3.3.5">𝑗</ci><list id="S3.SS2.p3.1.m1.3.3.3.3.3.cmml" xref="S3.SS2.p3.1.m1.3.3.3.3.2"><apply id="S3.SS2.p3.1.m1.2.2.2.2.1.1.cmml" xref="S3.SS2.p3.1.m1.2.2.2.2.1.1"><minus id="S3.SS2.p3.1.m1.2.2.2.2.1.1.1.cmml" xref="S3.SS2.p3.1.m1.2.2.2.2.1.1"></minus><cn id="S3.SS2.p3.1.m1.2.2.2.2.1.1.2.cmml" type="integer" xref="S3.SS2.p3.1.m1.2.2.2.2.1.1.2">2</cn></apply><ci id="S3.SS2.p3.1.m1.1.1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1.1.1">…</ci><apply id="S3.SS2.p3.1.m1.3.3.3.3.2.2.cmml" xref="S3.SS2.p3.1.m1.3.3.3.3.2.2"><plus id="S3.SS2.p3.1.m1.3.3.3.3.2.2.1.cmml" xref="S3.SS2.p3.1.m1.3.3.3.3.2.2.1"></plus><ci id="S3.SS2.p3.1.m1.3.3.3.3.2.2.2.cmml" xref="S3.SS2.p3.1.m1.3.3.3.3.2.2.2">𝑙</ci><cn id="S3.SS2.p3.1.m1.3.3.3.3.2.2.3.cmml" type="integer" xref="S3.SS2.p3.1.m1.3.3.3.3.2.2.3">2</cn></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.1.m1.4c">\{p_{j}\}_{j=-2,\dots,l+2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.1.m1.4d">{ italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_j = - 2 , … , italic_l + 2 end_POSTSUBSCRIPT</annotation></semantics></math> denote an increasing sequence of knot points, where <math alttext="p_{0}=0" class="ltx_Math" display="inline" id="S3.SS2.p3.2.m2.1"><semantics id="S3.SS2.p3.2.m2.1a"><mrow id="S3.SS2.p3.2.m2.1.1" xref="S3.SS2.p3.2.m2.1.1.cmml"><msub id="S3.SS2.p3.2.m2.1.1.2" xref="S3.SS2.p3.2.m2.1.1.2.cmml"><mi id="S3.SS2.p3.2.m2.1.1.2.2" xref="S3.SS2.p3.2.m2.1.1.2.2.cmml">p</mi><mn id="S3.SS2.p3.2.m2.1.1.2.3" xref="S3.SS2.p3.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="S3.SS2.p3.2.m2.1.1.1" xref="S3.SS2.p3.2.m2.1.1.1.cmml">=</mo><mn id="S3.SS2.p3.2.m2.1.1.3" xref="S3.SS2.p3.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.2.m2.1b"><apply id="S3.SS2.p3.2.m2.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1"><eq id="S3.SS2.p3.2.m2.1.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1.1"></eq><apply id="S3.SS2.p3.2.m2.1.1.2.cmml" xref="S3.SS2.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p3.2.m2.1.1.2.1.cmml" xref="S3.SS2.p3.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS2.p3.2.m2.1.1.2.2.cmml" xref="S3.SS2.p3.2.m2.1.1.2.2">𝑝</ci><cn id="S3.SS2.p3.2.m2.1.1.2.3.cmml" type="integer" xref="S3.SS2.p3.2.m2.1.1.2.3">0</cn></apply><cn id="S3.SS2.p3.2.m2.1.1.3.cmml" type="integer" xref="S3.SS2.p3.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.2.m2.1c">p_{0}=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.2.m2.1d">italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 0</annotation></semantics></math> and <math alttext="p_{l}=1" class="ltx_Math" display="inline" id="S3.SS2.p3.3.m3.1"><semantics id="S3.SS2.p3.3.m3.1a"><mrow id="S3.SS2.p3.3.m3.1.1" xref="S3.SS2.p3.3.m3.1.1.cmml"><msub id="S3.SS2.p3.3.m3.1.1.2" xref="S3.SS2.p3.3.m3.1.1.2.cmml"><mi id="S3.SS2.p3.3.m3.1.1.2.2" xref="S3.SS2.p3.3.m3.1.1.2.2.cmml">p</mi><mi id="S3.SS2.p3.3.m3.1.1.2.3" xref="S3.SS2.p3.3.m3.1.1.2.3.cmml">l</mi></msub><mo id="S3.SS2.p3.3.m3.1.1.1" xref="S3.SS2.p3.3.m3.1.1.1.cmml">=</mo><mn id="S3.SS2.p3.3.m3.1.1.3" xref="S3.SS2.p3.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.3.m3.1b"><apply id="S3.SS2.p3.3.m3.1.1.cmml" xref="S3.SS2.p3.3.m3.1.1"><eq id="S3.SS2.p3.3.m3.1.1.1.cmml" xref="S3.SS2.p3.3.m3.1.1.1"></eq><apply id="S3.SS2.p3.3.m3.1.1.2.cmml" xref="S3.SS2.p3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p3.3.m3.1.1.2.1.cmml" xref="S3.SS2.p3.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS2.p3.3.m3.1.1.2.2.cmml" xref="S3.SS2.p3.3.m3.1.1.2.2">𝑝</ci><ci id="S3.SS2.p3.3.m3.1.1.2.3.cmml" xref="S3.SS2.p3.3.m3.1.1.2.3">𝑙</ci></apply><cn id="S3.SS2.p3.3.m3.1.1.3.cmml" type="integer" xref="S3.SS2.p3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.3.m3.1c">p_{l}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.3.m3.1d">italic_p start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT = 1</annotation></semantics></math>, and let <math alttext="B_{3}(\cdot;p_{j})" class="ltx_Math" display="inline" id="S3.SS2.p3.4.m4.2"><semantics id="S3.SS2.p3.4.m4.2a"><mrow id="S3.SS2.p3.4.m4.2.2" xref="S3.SS2.p3.4.m4.2.2.cmml"><msub id="S3.SS2.p3.4.m4.2.2.3" xref="S3.SS2.p3.4.m4.2.2.3.cmml"><mi id="S3.SS2.p3.4.m4.2.2.3.2" xref="S3.SS2.p3.4.m4.2.2.3.2.cmml">B</mi><mn id="S3.SS2.p3.4.m4.2.2.3.3" xref="S3.SS2.p3.4.m4.2.2.3.3.cmml">3</mn></msub><mo id="S3.SS2.p3.4.m4.2.2.2" xref="S3.SS2.p3.4.m4.2.2.2.cmml">⁢</mo><mrow id="S3.SS2.p3.4.m4.2.2.1.1" xref="S3.SS2.p3.4.m4.2.2.1.2.cmml"><mo id="S3.SS2.p3.4.m4.2.2.1.1.2" stretchy="false" xref="S3.SS2.p3.4.m4.2.2.1.2.cmml">(</mo><mo id="S3.SS2.p3.4.m4.1.1" lspace="0em" rspace="0em" xref="S3.SS2.p3.4.m4.1.1.cmml">⋅</mo><mo id="S3.SS2.p3.4.m4.2.2.1.1.3" xref="S3.SS2.p3.4.m4.2.2.1.2.cmml">;</mo><msub id="S3.SS2.p3.4.m4.2.2.1.1.1" xref="S3.SS2.p3.4.m4.2.2.1.1.1.cmml"><mi id="S3.SS2.p3.4.m4.2.2.1.1.1.2" xref="S3.SS2.p3.4.m4.2.2.1.1.1.2.cmml">p</mi><mi id="S3.SS2.p3.4.m4.2.2.1.1.1.3" xref="S3.SS2.p3.4.m4.2.2.1.1.1.3.cmml">j</mi></msub><mo id="S3.SS2.p3.4.m4.2.2.1.1.4" stretchy="false" xref="S3.SS2.p3.4.m4.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.4.m4.2b"><apply id="S3.SS2.p3.4.m4.2.2.cmml" xref="S3.SS2.p3.4.m4.2.2"><times id="S3.SS2.p3.4.m4.2.2.2.cmml" xref="S3.SS2.p3.4.m4.2.2.2"></times><apply id="S3.SS2.p3.4.m4.2.2.3.cmml" xref="S3.SS2.p3.4.m4.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.p3.4.m4.2.2.3.1.cmml" xref="S3.SS2.p3.4.m4.2.2.3">subscript</csymbol><ci id="S3.SS2.p3.4.m4.2.2.3.2.cmml" xref="S3.SS2.p3.4.m4.2.2.3.2">𝐵</ci><cn id="S3.SS2.p3.4.m4.2.2.3.3.cmml" type="integer" xref="S3.SS2.p3.4.m4.2.2.3.3">3</cn></apply><list id="S3.SS2.p3.4.m4.2.2.1.2.cmml" xref="S3.SS2.p3.4.m4.2.2.1.1"><ci id="S3.SS2.p3.4.m4.1.1.cmml" xref="S3.SS2.p3.4.m4.1.1">⋅</ci><apply id="S3.SS2.p3.4.m4.2.2.1.1.1.cmml" xref="S3.SS2.p3.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.4.m4.2.2.1.1.1.1.cmml" xref="S3.SS2.p3.4.m4.2.2.1.1.1">subscript</csymbol><ci id="S3.SS2.p3.4.m4.2.2.1.1.1.2.cmml" xref="S3.SS2.p3.4.m4.2.2.1.1.1.2">𝑝</ci><ci id="S3.SS2.p3.4.m4.2.2.1.1.1.3.cmml" xref="S3.SS2.p3.4.m4.2.2.1.1.1.3">𝑗</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.4.m4.2c">B_{3}(\cdot;p_{j})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.4.m4.2d">italic_B start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( ⋅ ; italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math> denote the cubic B-spline defined by the knot points <math alttext="p_{j-2},p_{j-1},p_{j},p_{j+1},p_{j+2}" class="ltx_Math" display="inline" id="S3.SS2.p3.5.m5.5"><semantics id="S3.SS2.p3.5.m5.5a"><mrow id="S3.SS2.p3.5.m5.5.5.5" xref="S3.SS2.p3.5.m5.5.5.6.cmml"><msub id="S3.SS2.p3.5.m5.1.1.1.1" xref="S3.SS2.p3.5.m5.1.1.1.1.cmml"><mi id="S3.SS2.p3.5.m5.1.1.1.1.2" xref="S3.SS2.p3.5.m5.1.1.1.1.2.cmml">p</mi><mrow id="S3.SS2.p3.5.m5.1.1.1.1.3" xref="S3.SS2.p3.5.m5.1.1.1.1.3.cmml"><mi id="S3.SS2.p3.5.m5.1.1.1.1.3.2" xref="S3.SS2.p3.5.m5.1.1.1.1.3.2.cmml">j</mi><mo id="S3.SS2.p3.5.m5.1.1.1.1.3.1" xref="S3.SS2.p3.5.m5.1.1.1.1.3.1.cmml">−</mo><mn id="S3.SS2.p3.5.m5.1.1.1.1.3.3" xref="S3.SS2.p3.5.m5.1.1.1.1.3.3.cmml">2</mn></mrow></msub><mo id="S3.SS2.p3.5.m5.5.5.5.6" xref="S3.SS2.p3.5.m5.5.5.6.cmml">,</mo><msub id="S3.SS2.p3.5.m5.2.2.2.2" xref="S3.SS2.p3.5.m5.2.2.2.2.cmml"><mi id="S3.SS2.p3.5.m5.2.2.2.2.2" xref="S3.SS2.p3.5.m5.2.2.2.2.2.cmml">p</mi><mrow id="S3.SS2.p3.5.m5.2.2.2.2.3" xref="S3.SS2.p3.5.m5.2.2.2.2.3.cmml"><mi id="S3.SS2.p3.5.m5.2.2.2.2.3.2" xref="S3.SS2.p3.5.m5.2.2.2.2.3.2.cmml">j</mi><mo id="S3.SS2.p3.5.m5.2.2.2.2.3.1" xref="S3.SS2.p3.5.m5.2.2.2.2.3.1.cmml">−</mo><mn id="S3.SS2.p3.5.m5.2.2.2.2.3.3" xref="S3.SS2.p3.5.m5.2.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.SS2.p3.5.m5.5.5.5.7" xref="S3.SS2.p3.5.m5.5.5.6.cmml">,</mo><msub id="S3.SS2.p3.5.m5.3.3.3.3" xref="S3.SS2.p3.5.m5.3.3.3.3.cmml"><mi id="S3.SS2.p3.5.m5.3.3.3.3.2" xref="S3.SS2.p3.5.m5.3.3.3.3.2.cmml">p</mi><mi id="S3.SS2.p3.5.m5.3.3.3.3.3" xref="S3.SS2.p3.5.m5.3.3.3.3.3.cmml">j</mi></msub><mo id="S3.SS2.p3.5.m5.5.5.5.8" xref="S3.SS2.p3.5.m5.5.5.6.cmml">,</mo><msub id="S3.SS2.p3.5.m5.4.4.4.4" xref="S3.SS2.p3.5.m5.4.4.4.4.cmml"><mi id="S3.SS2.p3.5.m5.4.4.4.4.2" xref="S3.SS2.p3.5.m5.4.4.4.4.2.cmml">p</mi><mrow id="S3.SS2.p3.5.m5.4.4.4.4.3" xref="S3.SS2.p3.5.m5.4.4.4.4.3.cmml"><mi id="S3.SS2.p3.5.m5.4.4.4.4.3.2" xref="S3.SS2.p3.5.m5.4.4.4.4.3.2.cmml">j</mi><mo id="S3.SS2.p3.5.m5.4.4.4.4.3.1" xref="S3.SS2.p3.5.m5.4.4.4.4.3.1.cmml">+</mo><mn id="S3.SS2.p3.5.m5.4.4.4.4.3.3" xref="S3.SS2.p3.5.m5.4.4.4.4.3.3.cmml">1</mn></mrow></msub><mo id="S3.SS2.p3.5.m5.5.5.5.9" xref="S3.SS2.p3.5.m5.5.5.6.cmml">,</mo><msub id="S3.SS2.p3.5.m5.5.5.5.5" xref="S3.SS2.p3.5.m5.5.5.5.5.cmml"><mi id="S3.SS2.p3.5.m5.5.5.5.5.2" xref="S3.SS2.p3.5.m5.5.5.5.5.2.cmml">p</mi><mrow id="S3.SS2.p3.5.m5.5.5.5.5.3" xref="S3.SS2.p3.5.m5.5.5.5.5.3.cmml"><mi id="S3.SS2.p3.5.m5.5.5.5.5.3.2" xref="S3.SS2.p3.5.m5.5.5.5.5.3.2.cmml">j</mi><mo id="S3.SS2.p3.5.m5.5.5.5.5.3.1" xref="S3.SS2.p3.5.m5.5.5.5.5.3.1.cmml">+</mo><mn id="S3.SS2.p3.5.m5.5.5.5.5.3.3" xref="S3.SS2.p3.5.m5.5.5.5.5.3.3.cmml">2</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.5.m5.5b"><list id="S3.SS2.p3.5.m5.5.5.6.cmml" xref="S3.SS2.p3.5.m5.5.5.5"><apply id="S3.SS2.p3.5.m5.1.1.1.1.cmml" xref="S3.SS2.p3.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.5.m5.1.1.1.1.1.cmml" xref="S3.SS2.p3.5.m5.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p3.5.m5.1.1.1.1.2.cmml" xref="S3.SS2.p3.5.m5.1.1.1.1.2">𝑝</ci><apply id="S3.SS2.p3.5.m5.1.1.1.1.3.cmml" xref="S3.SS2.p3.5.m5.1.1.1.1.3"><minus id="S3.SS2.p3.5.m5.1.1.1.1.3.1.cmml" xref="S3.SS2.p3.5.m5.1.1.1.1.3.1"></minus><ci id="S3.SS2.p3.5.m5.1.1.1.1.3.2.cmml" xref="S3.SS2.p3.5.m5.1.1.1.1.3.2">𝑗</ci><cn id="S3.SS2.p3.5.m5.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS2.p3.5.m5.1.1.1.1.3.3">2</cn></apply></apply><apply id="S3.SS2.p3.5.m5.2.2.2.2.cmml" xref="S3.SS2.p3.5.m5.2.2.2.2"><csymbol 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Then we can construct the class of cubic B-splines</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{G}_{\text{spline}}=\{g_{j}:x\mapsto g_{j}(x)=B_{3}(x;p_{j})\}." class="ltx_Math" display="block" id="S3.Ex4.m1.3"><semantics id="S3.Ex4.m1.3a"><mrow id="S3.Ex4.m1.3.3.1" xref="S3.Ex4.m1.3.3.1.1.cmml"><mrow id="S3.Ex4.m1.3.3.1.1" xref="S3.Ex4.m1.3.3.1.1.cmml"><msub id="S3.Ex4.m1.3.3.1.1.4" xref="S3.Ex4.m1.3.3.1.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex4.m1.3.3.1.1.4.2" xref="S3.Ex4.m1.3.3.1.1.4.2.cmml">𝒢</mi><mtext id="S3.Ex4.m1.3.3.1.1.4.3" xref="S3.Ex4.m1.3.3.1.1.4.3a.cmml">spline</mtext></msub><mo id="S3.Ex4.m1.3.3.1.1.3" xref="S3.Ex4.m1.3.3.1.1.3.cmml">=</mo><mrow id="S3.Ex4.m1.3.3.1.1.2.2" xref="S3.Ex4.m1.3.3.1.1.2.3.cmml"><mo id="S3.Ex4.m1.3.3.1.1.2.2.3" stretchy="false" 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end_POSTSUBSCRIPT ( italic_x ) = italic_B start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_x ; italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS2.p4"> <p class="ltx_p" id="S3.SS2.p4.6"><span class="ltx_text ltx_font_bold" id="S3.SS2.p4.1.1">Instrumental functions on <math alttext="\mathcal{X}" class="ltx_Math" display="inline" id="S3.SS2.p4.1.1.m1.1"><semantics id="S3.SS2.p4.1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p4.1.1.m1.1.1" xref="S3.SS2.p4.1.1.m1.1.1.cmml">𝒳</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.1.1.m1.1b"><ci id="S3.SS2.p4.1.1.m1.1.1.cmml" xref="S3.SS2.p4.1.1.m1.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.1.1.m1.1c">\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.1.1.m1.1d">caligraphic_X</annotation></semantics></math>.</span> So far, we have defined all instrumental functions on <math alttext="[0,1]" class="ltx_Math" display="inline" id="S3.SS2.p4.2.m1.2"><semantics id="S3.SS2.p4.2.m1.2a"><mrow id="S3.SS2.p4.2.m1.2.3.2" xref="S3.SS2.p4.2.m1.2.3.1.cmml"><mo id="S3.SS2.p4.2.m1.2.3.2.1" stretchy="false" xref="S3.SS2.p4.2.m1.2.3.1.cmml">[</mo><mn id="S3.SS2.p4.2.m1.1.1" xref="S3.SS2.p4.2.m1.1.1.cmml">0</mn><mo id="S3.SS2.p4.2.m1.2.3.2.2" xref="S3.SS2.p4.2.m1.2.3.1.cmml">,</mo><mn id="S3.SS2.p4.2.m1.2.2" xref="S3.SS2.p4.2.m1.2.2.cmml">1</mn><mo id="S3.SS2.p4.2.m1.2.3.2.3" stretchy="false" xref="S3.SS2.p4.2.m1.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.2.m1.2b"><interval closure="closed" id="S3.SS2.p4.2.m1.2.3.1.cmml" xref="S3.SS2.p4.2.m1.2.3.2"><cn id="S3.SS2.p4.2.m1.1.1.cmml" type="integer" xref="S3.SS2.p4.2.m1.1.1">0</cn><cn id="S3.SS2.p4.2.m1.2.2.cmml" type="integer" xref="S3.SS2.p4.2.m1.2.2">1</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.2.m1.2c">[0,1]</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.2.m1.2d">[ 0 , 1 ]</annotation></semantics></math>, as by Assumption <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I1.i6" title="item (A6) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A6)</span></a> <math alttext="\mathcal{X}" class="ltx_Math" display="inline" id="S3.SS2.p4.3.m2.1"><semantics id="S3.SS2.p4.3.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p4.3.m2.1.1" xref="S3.SS2.p4.3.m2.1.1.cmml">𝒳</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.3.m2.1b"><ci id="S3.SS2.p4.3.m2.1.1.cmml" xref="S3.SS2.p4.3.m2.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.3.m2.1c">\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.3.m2.1d">caligraphic_X</annotation></semantics></math> is bounded, and hence each component of <math alttext="X" class="ltx_Math" display="inline" id="S3.SS2.p4.4.m3.1"><semantics id="S3.SS2.p4.4.m3.1a"><mi id="S3.SS2.p4.4.m3.1.1" xref="S3.SS2.p4.4.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.4.m3.1b"><ci id="S3.SS2.p4.4.m3.1.1.cmml" xref="S3.SS2.p4.4.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.4.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.4.m3.1d">italic_X</annotation></semantics></math> can easily be rescaled to take values in the unit interval. We will denote this rescaling transformation as <math alttext="N:\mathcal{X}\to[0,1]^{d+1}" class="ltx_Math" display="inline" id="S3.SS2.p4.5.m4.2"><semantics id="S3.SS2.p4.5.m4.2a"><mrow id="S3.SS2.p4.5.m4.2.3" xref="S3.SS2.p4.5.m4.2.3.cmml"><mi id="S3.SS2.p4.5.m4.2.3.2" xref="S3.SS2.p4.5.m4.2.3.2.cmml">N</mi><mo id="S3.SS2.p4.5.m4.2.3.1" lspace="0.278em" rspace="0.278em" xref="S3.SS2.p4.5.m4.2.3.1.cmml">:</mo><mrow id="S3.SS2.p4.5.m4.2.3.3" xref="S3.SS2.p4.5.m4.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p4.5.m4.2.3.3.2" xref="S3.SS2.p4.5.m4.2.3.3.2.cmml">𝒳</mi><mo id="S3.SS2.p4.5.m4.2.3.3.1" stretchy="false" xref="S3.SS2.p4.5.m4.2.3.3.1.cmml">→</mo><msup id="S3.SS2.p4.5.m4.2.3.3.3" xref="S3.SS2.p4.5.m4.2.3.3.3.cmml"><mrow id="S3.SS2.p4.5.m4.2.3.3.3.2.2" xref="S3.SS2.p4.5.m4.2.3.3.3.2.1.cmml"><mo id="S3.SS2.p4.5.m4.2.3.3.3.2.2.1" stretchy="false" xref="S3.SS2.p4.5.m4.2.3.3.3.2.1.cmml">[</mo><mn id="S3.SS2.p4.5.m4.1.1" xref="S3.SS2.p4.5.m4.1.1.cmml">0</mn><mo id="S3.SS2.p4.5.m4.2.3.3.3.2.2.2" xref="S3.SS2.p4.5.m4.2.3.3.3.2.1.cmml">,</mo><mn id="S3.SS2.p4.5.m4.2.2" xref="S3.SS2.p4.5.m4.2.2.cmml">1</mn><mo id="S3.SS2.p4.5.m4.2.3.3.3.2.2.3" stretchy="false" xref="S3.SS2.p4.5.m4.2.3.3.3.2.1.cmml">]</mo></mrow><mrow id="S3.SS2.p4.5.m4.2.3.3.3.3" xref="S3.SS2.p4.5.m4.2.3.3.3.3.cmml"><mi id="S3.SS2.p4.5.m4.2.3.3.3.3.2" xref="S3.SS2.p4.5.m4.2.3.3.3.3.2.cmml">d</mi><mo id="S3.SS2.p4.5.m4.2.3.3.3.3.1" xref="S3.SS2.p4.5.m4.2.3.3.3.3.1.cmml">+</mo><mn id="S3.SS2.p4.5.m4.2.3.3.3.3.3" xref="S3.SS2.p4.5.m4.2.3.3.3.3.3.cmml">1</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.5.m4.2b"><apply id="S3.SS2.p4.5.m4.2.3.cmml" xref="S3.SS2.p4.5.m4.2.3"><ci id="S3.SS2.p4.5.m4.2.3.1.cmml" xref="S3.SS2.p4.5.m4.2.3.1">:</ci><ci id="S3.SS2.p4.5.m4.2.3.2.cmml" xref="S3.SS2.p4.5.m4.2.3.2">𝑁</ci><apply id="S3.SS2.p4.5.m4.2.3.3.cmml" xref="S3.SS2.p4.5.m4.2.3.3"><ci id="S3.SS2.p4.5.m4.2.3.3.1.cmml" xref="S3.SS2.p4.5.m4.2.3.3.1">→</ci><ci id="S3.SS2.p4.5.m4.2.3.3.2.cmml" xref="S3.SS2.p4.5.m4.2.3.3.2">𝒳</ci><apply id="S3.SS2.p4.5.m4.2.3.3.3.cmml" xref="S3.SS2.p4.5.m4.2.3.3.3"><csymbol cd="ambiguous" id="S3.SS2.p4.5.m4.2.3.3.3.1.cmml" xref="S3.SS2.p4.5.m4.2.3.3.3">superscript</csymbol><interval closure="closed" id="S3.SS2.p4.5.m4.2.3.3.3.2.1.cmml" xref="S3.SS2.p4.5.m4.2.3.3.3.2.2"><cn id="S3.SS2.p4.5.m4.1.1.cmml" type="integer" xref="S3.SS2.p4.5.m4.1.1">0</cn><cn id="S3.SS2.p4.5.m4.2.2.cmml" type="integer" xref="S3.SS2.p4.5.m4.2.2">1</cn></interval><apply id="S3.SS2.p4.5.m4.2.3.3.3.3.cmml" xref="S3.SS2.p4.5.m4.2.3.3.3.3"><plus id="S3.SS2.p4.5.m4.2.3.3.3.3.1.cmml" xref="S3.SS2.p4.5.m4.2.3.3.3.3.1"></plus><ci id="S3.SS2.p4.5.m4.2.3.3.3.3.2.cmml" xref="S3.SS2.p4.5.m4.2.3.3.3.3.2">𝑑</ci><cn id="S3.SS2.p4.5.m4.2.3.3.3.3.3.cmml" type="integer" xref="S3.SS2.p4.5.m4.2.3.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.5.m4.2c">N:\mathcal{X}\to[0,1]^{d+1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.5.m4.2d">italic_N : caligraphic_X → [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d + 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. In the simplest case, we can consider <math alttext="N" class="ltx_Math" display="inline" id="S3.SS2.p4.6.m5.1"><semantics id="S3.SS2.p4.6.m5.1a"><mi id="S3.SS2.p4.6.m5.1.1" xref="S3.SS2.p4.6.m5.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.6.m5.1b"><ci id="S3.SS2.p4.6.m5.1.1.cmml" xref="S3.SS2.p4.6.m5.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.6.m5.1c">N</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.6.m5.1d">italic_N</annotation></semantics></math> to be the component-wise min-max scaler (e.g. as proposed by <cite class="ltx_cite ltx_citemacro_cite">Andrews and Shi, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib1" title="">2013</a>)</cite>), that is,</p> <table class="ltx_equation ltx_eqn_table" id="S3.SS2.p4.7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell 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xref="S3.SS2.p4.7.m1.5.5.5.5.5.5"><csymbol cd="latexml" id="S3.SS2.p4.7.m1.56.56.56.11.11.11.cmml" xref="S3.SS2.p4.7.m1.56.56.56.11.11.11">for-all</csymbol><ci id="S3.SS2.p4.7.m1.57.57.57.12.12.12.cmml" xref="S3.SS2.p4.7.m1.57.57.57.12.12.12">𝑗</ci></apply><set id="S3.SS2.p4.7.m1.67.67.1.1.1.2.2.2.2.2.3.cmml" xref="S3.SS2.p4.7.m1.5.5.5.5.5.5"><cn id="S3.SS2.p4.7.m1.60.60.60.15.15.15.cmml" type="integer" xref="S3.SS2.p4.7.m1.60.60.60.15.15.15">1</cn><ci id="S3.SS2.p4.7.m1.62.62.62.17.17.17.cmml" xref="S3.SS2.p4.7.m1.62.62.62.17.17.17">…</ci><ci id="S3.SS2.p4.7.m1.64.64.64.19.19.19.cmml" xref="S3.SS2.p4.7.m1.64.64.64.19.19.19">𝑑</ci></set></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.7.m1.69c">N:\mathcal{X}\to[0,1]^{d+1}:(x_{0},x_{1},\dots,x_{d})\mapsto(x_{0},N_{1}(x_{1}% ),\dots,N_{d}(x_{d})),\\ \text{with}\quad N_{j}(x_{j})=\frac{x_{j}+M_{j}}{2M_{j}},\forall j\in\{1,\dots% ,d\},</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.7.m1.69d">start_ROW start_CELL italic_N : caligraphic_X → [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d + 1 end_POSTSUPERSCRIPT : ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) ↦ ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , … , italic_N start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) ) , end_CELL end_ROW start_ROW start_CELL with italic_N start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = divide start_ARG italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + italic_M start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG start_ARG 2 italic_M start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG , ∀ italic_j ∈ { 1 , … , italic_d } , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.p4.11">where <math alttext="M_{j}\in\mathbb{R}" class="ltx_Math" display="inline" id="S3.SS2.p4.8.m1.1"><semantics id="S3.SS2.p4.8.m1.1a"><mrow id="S3.SS2.p4.8.m1.1.1" xref="S3.SS2.p4.8.m1.1.1.cmml"><msub id="S3.SS2.p4.8.m1.1.1.2" xref="S3.SS2.p4.8.m1.1.1.2.cmml"><mi id="S3.SS2.p4.8.m1.1.1.2.2" xref="S3.SS2.p4.8.m1.1.1.2.2.cmml">M</mi><mi id="S3.SS2.p4.8.m1.1.1.2.3" xref="S3.SS2.p4.8.m1.1.1.2.3.cmml">j</mi></msub><mo id="S3.SS2.p4.8.m1.1.1.1" xref="S3.SS2.p4.8.m1.1.1.1.cmml">∈</mo><mi id="S3.SS2.p4.8.m1.1.1.3" xref="S3.SS2.p4.8.m1.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.8.m1.1b"><apply id="S3.SS2.p4.8.m1.1.1.cmml" xref="S3.SS2.p4.8.m1.1.1"><in id="S3.SS2.p4.8.m1.1.1.1.cmml" xref="S3.SS2.p4.8.m1.1.1.1"></in><apply id="S3.SS2.p4.8.m1.1.1.2.cmml" xref="S3.SS2.p4.8.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p4.8.m1.1.1.2.1.cmml" xref="S3.SS2.p4.8.m1.1.1.2">subscript</csymbol><ci id="S3.SS2.p4.8.m1.1.1.2.2.cmml" xref="S3.SS2.p4.8.m1.1.1.2.2">𝑀</ci><ci id="S3.SS2.p4.8.m1.1.1.2.3.cmml" xref="S3.SS2.p4.8.m1.1.1.2.3">𝑗</ci></apply><ci id="S3.SS2.p4.8.m1.1.1.3.cmml" xref="S3.SS2.p4.8.m1.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.8.m1.1c">M_{j}\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.8.m1.1d">italic_M start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ blackboard_R</annotation></semantics></math> are such that <math alttext="\mathcal{X}\subset\prod_{j=1}^{d+1}[-M_{j},M_{j}]" class="ltx_Math" display="inline" id="S3.SS2.p4.9.m2.2"><semantics id="S3.SS2.p4.9.m2.2a"><mrow id="S3.SS2.p4.9.m2.2.2" xref="S3.SS2.p4.9.m2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p4.9.m2.2.2.4" xref="S3.SS2.p4.9.m2.2.2.4.cmml">𝒳</mi><mo id="S3.SS2.p4.9.m2.2.2.3" rspace="0.111em" xref="S3.SS2.p4.9.m2.2.2.3.cmml">⊂</mo><mrow id="S3.SS2.p4.9.m2.2.2.2" xref="S3.SS2.p4.9.m2.2.2.2.cmml"><msubsup id="S3.SS2.p4.9.m2.2.2.2.3" xref="S3.SS2.p4.9.m2.2.2.2.3.cmml"><mo id="S3.SS2.p4.9.m2.2.2.2.3.2.2" rspace="0em" xref="S3.SS2.p4.9.m2.2.2.2.3.2.2.cmml">∏</mo><mrow id="S3.SS2.p4.9.m2.2.2.2.3.2.3" xref="S3.SS2.p4.9.m2.2.2.2.3.2.3.cmml"><mi id="S3.SS2.p4.9.m2.2.2.2.3.2.3.2" xref="S3.SS2.p4.9.m2.2.2.2.3.2.3.2.cmml">j</mi><mo id="S3.SS2.p4.9.m2.2.2.2.3.2.3.1" xref="S3.SS2.p4.9.m2.2.2.2.3.2.3.1.cmml">=</mo><mn id="S3.SS2.p4.9.m2.2.2.2.3.2.3.3" xref="S3.SS2.p4.9.m2.2.2.2.3.2.3.3.cmml">1</mn></mrow><mrow id="S3.SS2.p4.9.m2.2.2.2.3.3" xref="S3.SS2.p4.9.m2.2.2.2.3.3.cmml"><mi id="S3.SS2.p4.9.m2.2.2.2.3.3.2" xref="S3.SS2.p4.9.m2.2.2.2.3.3.2.cmml">d</mi><mo id="S3.SS2.p4.9.m2.2.2.2.3.3.1" xref="S3.SS2.p4.9.m2.2.2.2.3.3.1.cmml">+</mo><mn id="S3.SS2.p4.9.m2.2.2.2.3.3.3" xref="S3.SS2.p4.9.m2.2.2.2.3.3.3.cmml">1</mn></mrow></msubsup><mrow 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xref="S3.SS2.p4.9.m2.2.2.2.2.3.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.9.m2.2b"><apply id="S3.SS2.p4.9.m2.2.2.cmml" xref="S3.SS2.p4.9.m2.2.2"><subset id="S3.SS2.p4.9.m2.2.2.3.cmml" xref="S3.SS2.p4.9.m2.2.2.3"></subset><ci id="S3.SS2.p4.9.m2.2.2.4.cmml" xref="S3.SS2.p4.9.m2.2.2.4">𝒳</ci><apply id="S3.SS2.p4.9.m2.2.2.2.cmml" xref="S3.SS2.p4.9.m2.2.2.2"><apply id="S3.SS2.p4.9.m2.2.2.2.3.cmml" xref="S3.SS2.p4.9.m2.2.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.p4.9.m2.2.2.2.3.1.cmml" xref="S3.SS2.p4.9.m2.2.2.2.3">superscript</csymbol><apply id="S3.SS2.p4.9.m2.2.2.2.3.2.cmml" xref="S3.SS2.p4.9.m2.2.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.p4.9.m2.2.2.2.3.2.1.cmml" xref="S3.SS2.p4.9.m2.2.2.2.3">subscript</csymbol><csymbol cd="latexml" id="S3.SS2.p4.9.m2.2.2.2.3.2.2.cmml" xref="S3.SS2.p4.9.m2.2.2.2.3.2.2">product</csymbol><apply id="S3.SS2.p4.9.m2.2.2.2.3.2.3.cmml" xref="S3.SS2.p4.9.m2.2.2.2.3.2.3"><eq id="S3.SS2.p4.9.m2.2.2.2.3.2.3.1.cmml" 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xref="S3.SS2.p4.9.m2.1.1.1.1.1.1.2">subscript</csymbol><ci id="S3.SS2.p4.9.m2.1.1.1.1.1.1.2.2.cmml" xref="S3.SS2.p4.9.m2.1.1.1.1.1.1.2.2">𝑀</ci><ci id="S3.SS2.p4.9.m2.1.1.1.1.1.1.2.3.cmml" xref="S3.SS2.p4.9.m2.1.1.1.1.1.1.2.3">𝑗</ci></apply></apply><apply id="S3.SS2.p4.9.m2.2.2.2.2.2.2.cmml" xref="S3.SS2.p4.9.m2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.p4.9.m2.2.2.2.2.2.2.1.cmml" xref="S3.SS2.p4.9.m2.2.2.2.2.2.2">subscript</csymbol><ci id="S3.SS2.p4.9.m2.2.2.2.2.2.2.2.cmml" xref="S3.SS2.p4.9.m2.2.2.2.2.2.2.2">𝑀</ci><ci id="S3.SS2.p4.9.m2.2.2.2.2.2.2.3.cmml" xref="S3.SS2.p4.9.m2.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.9.m2.2c">\mathcal{X}\subset\prod_{j=1}^{d+1}[-M_{j},M_{j}]</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.9.m2.2d">caligraphic_X ⊂ ∏ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d + 1 end_POSTSUPERSCRIPT [ - italic_M start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_M start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ]</annotation></semantics></math>. One problem, however, that could present itself is that <math alttext="N" class="ltx_Math" display="inline" id="S3.SS2.p4.10.m3.1"><semantics id="S3.SS2.p4.10.m3.1a"><mi id="S3.SS2.p4.10.m3.1.1" xref="S3.SS2.p4.10.m3.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.10.m3.1b"><ci id="S3.SS2.p4.10.m3.1.1.cmml" xref="S3.SS2.p4.10.m3.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.10.m3.1c">N</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.10.m3.1d">italic_N</annotation></semantics></math> is not <em class="ltx_emph ltx_font_italic" id="S3.SS2.p4.11.1">sufficiently surjective</em>, in the sense that there exists an instrumental function of the form (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.E10" title="In 3.2 Instrumental functions ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">10</span></a>) with support that falls entirely outside of <math alttext="N(\mathcal{X})" class="ltx_Math" display="inline" id="S3.SS2.p4.11.m4.1"><semantics id="S3.SS2.p4.11.m4.1a"><mrow id="S3.SS2.p4.11.m4.1.2" xref="S3.SS2.p4.11.m4.1.2.cmml"><mi id="S3.SS2.p4.11.m4.1.2.2" xref="S3.SS2.p4.11.m4.1.2.2.cmml">N</mi><mo id="S3.SS2.p4.11.m4.1.2.1" xref="S3.SS2.p4.11.m4.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.p4.11.m4.1.2.3.2" xref="S3.SS2.p4.11.m4.1.2.cmml"><mo id="S3.SS2.p4.11.m4.1.2.3.2.1" stretchy="false" xref="S3.SS2.p4.11.m4.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p4.11.m4.1.1" xref="S3.SS2.p4.11.m4.1.1.cmml">𝒳</mi><mo id="S3.SS2.p4.11.m4.1.2.3.2.2" stretchy="false" xref="S3.SS2.p4.11.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.11.m4.1b"><apply id="S3.SS2.p4.11.m4.1.2.cmml" xref="S3.SS2.p4.11.m4.1.2"><times id="S3.SS2.p4.11.m4.1.2.1.cmml" xref="S3.SS2.p4.11.m4.1.2.1"></times><ci id="S3.SS2.p4.11.m4.1.2.2.cmml" xref="S3.SS2.p4.11.m4.1.2.2">𝑁</ci><ci id="S3.SS2.p4.11.m4.1.1.cmml" xref="S3.SS2.p4.11.m4.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.11.m4.1c">N(\mathcal{X})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.11.m4.1d">italic_N ( caligraphic_X )</annotation></semantics></math>. An example of this is given in the middle panel of Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.F1" title="Figure 1 ‣ 3.2 Instrumental functions ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">1</span></a>. In what follows, we present a different rescaling transformation that is designed to overcome this issue.</p> </div> <div class="ltx_para" id="S3.SS2.p5"> <p class="ltx_p" id="S3.SS2.p5.18">Let <math alttext="\mathcal{X}_{\text{disc}}" class="ltx_Math" display="inline" id="S3.SS2.p5.1.m1.1"><semantics id="S3.SS2.p5.1.m1.1a"><msub id="S3.SS2.p5.1.m1.1.1" xref="S3.SS2.p5.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p5.1.m1.1.1.2" xref="S3.SS2.p5.1.m1.1.1.2.cmml">𝒳</mi><mtext id="S3.SS2.p5.1.m1.1.1.3" xref="S3.SS2.p5.1.m1.1.1.3a.cmml">disc</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.1.m1.1b"><apply id="S3.SS2.p5.1.m1.1.1.cmml" xref="S3.SS2.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.1.m1.1.1.1.cmml" xref="S3.SS2.p5.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p5.1.m1.1.1.2.cmml" xref="S3.SS2.p5.1.m1.1.1.2">𝒳</ci><ci id="S3.SS2.p5.1.m1.1.1.3a.cmml" xref="S3.SS2.p5.1.m1.1.1.3"><mtext id="S3.SS2.p5.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.SS2.p5.1.m1.1.1.3">disc</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.1.m1.1c">\mathcal{X}_{\text{disc}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.1.m1.1d">caligraphic_X start_POSTSUBSCRIPT disc end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{X}_{\text{cont}}" class="ltx_Math" display="inline" id="S3.SS2.p5.2.m2.1"><semantics id="S3.SS2.p5.2.m2.1a"><msub id="S3.SS2.p5.2.m2.1.1" xref="S3.SS2.p5.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p5.2.m2.1.1.2" xref="S3.SS2.p5.2.m2.1.1.2.cmml">𝒳</mi><mtext id="S3.SS2.p5.2.m2.1.1.3" xref="S3.SS2.p5.2.m2.1.1.3a.cmml">cont</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.2.m2.1b"><apply id="S3.SS2.p5.2.m2.1.1.cmml" xref="S3.SS2.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.2.m2.1.1.1.cmml" xref="S3.SS2.p5.2.m2.1.1">subscript</csymbol><ci id="S3.SS2.p5.2.m2.1.1.2.cmml" xref="S3.SS2.p5.2.m2.1.1.2">𝒳</ci><ci id="S3.SS2.p5.2.m2.1.1.3a.cmml" xref="S3.SS2.p5.2.m2.1.1.3"><mtext id="S3.SS2.p5.2.m2.1.1.3.cmml" mathsize="70%" xref="S3.SS2.p5.2.m2.1.1.3">cont</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.2.m2.1c">\mathcal{X}_{\text{cont}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.2.m2.1d">caligraphic_X start_POSTSUBSCRIPT cont end_POSTSUBSCRIPT</annotation></semantics></math> denote the subspaces of <math alttext="\mathcal{X}" class="ltx_Math" display="inline" id="S3.SS2.p5.3.m3.1"><semantics id="S3.SS2.p5.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p5.3.m3.1.1" xref="S3.SS2.p5.3.m3.1.1.cmml">𝒳</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.3.m3.1b"><ci id="S3.SS2.p5.3.m3.1.1.cmml" xref="S3.SS2.p5.3.m3.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.3.m3.1c">\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.3.m3.1d">caligraphic_X</annotation></semantics></math> of discrete and continuous covariates respectively and define their dimensions as <math alttext="d_{\text{disc}}" class="ltx_Math" display="inline" id="S3.SS2.p5.4.m4.1"><semantics id="S3.SS2.p5.4.m4.1a"><msub id="S3.SS2.p5.4.m4.1.1" xref="S3.SS2.p5.4.m4.1.1.cmml"><mi id="S3.SS2.p5.4.m4.1.1.2" xref="S3.SS2.p5.4.m4.1.1.2.cmml">d</mi><mtext id="S3.SS2.p5.4.m4.1.1.3" xref="S3.SS2.p5.4.m4.1.1.3a.cmml">disc</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.4.m4.1b"><apply id="S3.SS2.p5.4.m4.1.1.cmml" xref="S3.SS2.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.4.m4.1.1.1.cmml" xref="S3.SS2.p5.4.m4.1.1">subscript</csymbol><ci id="S3.SS2.p5.4.m4.1.1.2.cmml" xref="S3.SS2.p5.4.m4.1.1.2">𝑑</ci><ci id="S3.SS2.p5.4.m4.1.1.3a.cmml" xref="S3.SS2.p5.4.m4.1.1.3"><mtext id="S3.SS2.p5.4.m4.1.1.3.cmml" mathsize="70%" xref="S3.SS2.p5.4.m4.1.1.3">disc</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.4.m4.1c">d_{\text{disc}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.4.m4.1d">italic_d start_POSTSUBSCRIPT disc end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="d_{\text{cont}}" class="ltx_Math" display="inline" id="S3.SS2.p5.5.m5.1"><semantics id="S3.SS2.p5.5.m5.1a"><msub id="S3.SS2.p5.5.m5.1.1" xref="S3.SS2.p5.5.m5.1.1.cmml"><mi id="S3.SS2.p5.5.m5.1.1.2" xref="S3.SS2.p5.5.m5.1.1.2.cmml">d</mi><mtext id="S3.SS2.p5.5.m5.1.1.3" xref="S3.SS2.p5.5.m5.1.1.3a.cmml">cont</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.5.m5.1b"><apply id="S3.SS2.p5.5.m5.1.1.cmml" xref="S3.SS2.p5.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.5.m5.1.1.1.cmml" xref="S3.SS2.p5.5.m5.1.1">subscript</csymbol><ci id="S3.SS2.p5.5.m5.1.1.2.cmml" xref="S3.SS2.p5.5.m5.1.1.2">𝑑</ci><ci id="S3.SS2.p5.5.m5.1.1.3a.cmml" xref="S3.SS2.p5.5.m5.1.1.3"><mtext id="S3.SS2.p5.5.m5.1.1.3.cmml" mathsize="70%" xref="S3.SS2.p5.5.m5.1.1.3">cont</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.5.m5.1c">d_{\text{cont}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.5.m5.1d">italic_d start_POSTSUBSCRIPT cont end_POSTSUBSCRIPT</annotation></semantics></math>. We will split the construction of <math alttext="N" class="ltx_Math" display="inline" id="S3.SS2.p5.6.m6.1"><semantics id="S3.SS2.p5.6.m6.1a"><mi id="S3.SS2.p5.6.m6.1.1" xref="S3.SS2.p5.6.m6.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.6.m6.1b"><ci id="S3.SS2.p5.6.m6.1.1.cmml" xref="S3.SS2.p5.6.m6.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.6.m6.1c">N</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.6.m6.1d">italic_N</annotation></semantics></math> into its effect on the continuous covariates on the one hand, and discrete covariates on the other hand. Denote the transformation <math alttext="N" class="ltx_Math" display="inline" id="S3.SS2.p5.7.m7.1"><semantics id="S3.SS2.p5.7.m7.1a"><mi id="S3.SS2.p5.7.m7.1.1" xref="S3.SS2.p5.7.m7.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.7.m7.1b"><ci id="S3.SS2.p5.7.m7.1.1.cmml" xref="S3.SS2.p5.7.m7.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.7.m7.1c">N</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.7.m7.1d">italic_N</annotation></semantics></math> with its domain restricted to <math alttext="\mathcal{X}_{\text{cont}}" class="ltx_Math" display="inline" id="S3.SS2.p5.8.m8.1"><semantics id="S3.SS2.p5.8.m8.1a"><msub id="S3.SS2.p5.8.m8.1.1" xref="S3.SS2.p5.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p5.8.m8.1.1.2" xref="S3.SS2.p5.8.m8.1.1.2.cmml">𝒳</mi><mtext id="S3.SS2.p5.8.m8.1.1.3" xref="S3.SS2.p5.8.m8.1.1.3a.cmml">cont</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.8.m8.1b"><apply id="S3.SS2.p5.8.m8.1.1.cmml" xref="S3.SS2.p5.8.m8.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.8.m8.1.1.1.cmml" xref="S3.SS2.p5.8.m8.1.1">subscript</csymbol><ci id="S3.SS2.p5.8.m8.1.1.2.cmml" xref="S3.SS2.p5.8.m8.1.1.2">𝒳</ci><ci id="S3.SS2.p5.8.m8.1.1.3a.cmml" xref="S3.SS2.p5.8.m8.1.1.3"><mtext id="S3.SS2.p5.8.m8.1.1.3.cmml" mathsize="70%" xref="S3.SS2.p5.8.m8.1.1.3">cont</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.8.m8.1c">\mathcal{X}_{\text{cont}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.8.m8.1d">caligraphic_X start_POSTSUBSCRIPT cont end_POSTSUBSCRIPT</annotation></semantics></math> (<math alttext="\mathcal{X}_{\text{disc}}" class="ltx_Math" display="inline" id="S3.SS2.p5.9.m9.1"><semantics id="S3.SS2.p5.9.m9.1a"><msub id="S3.SS2.p5.9.m9.1.1" xref="S3.SS2.p5.9.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p5.9.m9.1.1.2" xref="S3.SS2.p5.9.m9.1.1.2.cmml">𝒳</mi><mtext id="S3.SS2.p5.9.m9.1.1.3" xref="S3.SS2.p5.9.m9.1.1.3a.cmml">disc</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.9.m9.1b"><apply id="S3.SS2.p5.9.m9.1.1.cmml" xref="S3.SS2.p5.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.9.m9.1.1.1.cmml" xref="S3.SS2.p5.9.m9.1.1">subscript</csymbol><ci id="S3.SS2.p5.9.m9.1.1.2.cmml" xref="S3.SS2.p5.9.m9.1.1.2">𝒳</ci><ci id="S3.SS2.p5.9.m9.1.1.3a.cmml" xref="S3.SS2.p5.9.m9.1.1.3"><mtext id="S3.SS2.p5.9.m9.1.1.3.cmml" mathsize="70%" xref="S3.SS2.p5.9.m9.1.1.3">disc</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.9.m9.1c">\mathcal{X}_{\text{disc}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.9.m9.1d">caligraphic_X start_POSTSUBSCRIPT disc end_POSTSUBSCRIPT</annotation></semantics></math>) as <math alttext="N_{\text{cont}}" class="ltx_Math" display="inline" id="S3.SS2.p5.10.m10.1"><semantics id="S3.SS2.p5.10.m10.1a"><msub id="S3.SS2.p5.10.m10.1.1" xref="S3.SS2.p5.10.m10.1.1.cmml"><mi id="S3.SS2.p5.10.m10.1.1.2" xref="S3.SS2.p5.10.m10.1.1.2.cmml">N</mi><mtext id="S3.SS2.p5.10.m10.1.1.3" xref="S3.SS2.p5.10.m10.1.1.3a.cmml">cont</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.10.m10.1b"><apply id="S3.SS2.p5.10.m10.1.1.cmml" xref="S3.SS2.p5.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.10.m10.1.1.1.cmml" xref="S3.SS2.p5.10.m10.1.1">subscript</csymbol><ci id="S3.SS2.p5.10.m10.1.1.2.cmml" xref="S3.SS2.p5.10.m10.1.1.2">𝑁</ci><ci id="S3.SS2.p5.10.m10.1.1.3a.cmml" xref="S3.SS2.p5.10.m10.1.1.3"><mtext id="S3.SS2.p5.10.m10.1.1.3.cmml" mathsize="70%" xref="S3.SS2.p5.10.m10.1.1.3">cont</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.10.m10.1c">N_{\text{cont}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.10.m10.1d">italic_N start_POSTSUBSCRIPT cont end_POSTSUBSCRIPT</annotation></semantics></math> (<math alttext="N_{\text{disc}}" class="ltx_Math" display="inline" id="S3.SS2.p5.11.m11.1"><semantics id="S3.SS2.p5.11.m11.1a"><msub id="S3.SS2.p5.11.m11.1.1" xref="S3.SS2.p5.11.m11.1.1.cmml"><mi id="S3.SS2.p5.11.m11.1.1.2" xref="S3.SS2.p5.11.m11.1.1.2.cmml">N</mi><mtext id="S3.SS2.p5.11.m11.1.1.3" xref="S3.SS2.p5.11.m11.1.1.3a.cmml">disc</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.11.m11.1b"><apply id="S3.SS2.p5.11.m11.1.1.cmml" xref="S3.SS2.p5.11.m11.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.11.m11.1.1.1.cmml" xref="S3.SS2.p5.11.m11.1.1">subscript</csymbol><ci id="S3.SS2.p5.11.m11.1.1.2.cmml" xref="S3.SS2.p5.11.m11.1.1.2">𝑁</ci><ci id="S3.SS2.p5.11.m11.1.1.3a.cmml" xref="S3.SS2.p5.11.m11.1.1.3"><mtext id="S3.SS2.p5.11.m11.1.1.3.cmml" mathsize="70%" xref="S3.SS2.p5.11.m11.1.1.3">disc</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.11.m11.1c">N_{\text{disc}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.11.m11.1d">italic_N start_POSTSUBSCRIPT disc end_POSTSUBSCRIPT</annotation></semantics></math>). For <math alttext="N_{\text{disc}}" class="ltx_Math" display="inline" id="S3.SS2.p5.12.m12.1"><semantics id="S3.SS2.p5.12.m12.1a"><msub id="S3.SS2.p5.12.m12.1.1" xref="S3.SS2.p5.12.m12.1.1.cmml"><mi id="S3.SS2.p5.12.m12.1.1.2" xref="S3.SS2.p5.12.m12.1.1.2.cmml">N</mi><mtext id="S3.SS2.p5.12.m12.1.1.3" xref="S3.SS2.p5.12.m12.1.1.3a.cmml">disc</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.12.m12.1b"><apply id="S3.SS2.p5.12.m12.1.1.cmml" xref="S3.SS2.p5.12.m12.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.12.m12.1.1.1.cmml" xref="S3.SS2.p5.12.m12.1.1">subscript</csymbol><ci id="S3.SS2.p5.12.m12.1.1.2.cmml" xref="S3.SS2.p5.12.m12.1.1.2">𝑁</ci><ci id="S3.SS2.p5.12.m12.1.1.3a.cmml" xref="S3.SS2.p5.12.m12.1.1.3"><mtext id="S3.SS2.p5.12.m12.1.1.3.cmml" mathsize="70%" xref="S3.SS2.p5.12.m12.1.1.3">disc</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.12.m12.1c">N_{\text{disc}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.12.m12.1d">italic_N start_POSTSUBSCRIPT disc end_POSTSUBSCRIPT</annotation></semantics></math>, any injective transformation from <math alttext="\mathcal{X}_{\text{disc}}\to[0,1]^{d_{\text{disc}}}" class="ltx_Math" display="inline" id="S3.SS2.p5.13.m13.2"><semantics id="S3.SS2.p5.13.m13.2a"><mrow id="S3.SS2.p5.13.m13.2.3" xref="S3.SS2.p5.13.m13.2.3.cmml"><msub id="S3.SS2.p5.13.m13.2.3.2" xref="S3.SS2.p5.13.m13.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p5.13.m13.2.3.2.2" xref="S3.SS2.p5.13.m13.2.3.2.2.cmml">𝒳</mi><mtext id="S3.SS2.p5.13.m13.2.3.2.3" xref="S3.SS2.p5.13.m13.2.3.2.3a.cmml">disc</mtext></msub><mo id="S3.SS2.p5.13.m13.2.3.1" stretchy="false" xref="S3.SS2.p5.13.m13.2.3.1.cmml">→</mo><msup id="S3.SS2.p5.13.m13.2.3.3" xref="S3.SS2.p5.13.m13.2.3.3.cmml"><mrow id="S3.SS2.p5.13.m13.2.3.3.2.2" xref="S3.SS2.p5.13.m13.2.3.3.2.1.cmml"><mo id="S3.SS2.p5.13.m13.2.3.3.2.2.1" stretchy="false" xref="S3.SS2.p5.13.m13.2.3.3.2.1.cmml">[</mo><mn id="S3.SS2.p5.13.m13.1.1" xref="S3.SS2.p5.13.m13.1.1.cmml">0</mn><mo id="S3.SS2.p5.13.m13.2.3.3.2.2.2" xref="S3.SS2.p5.13.m13.2.3.3.2.1.cmml">,</mo><mn id="S3.SS2.p5.13.m13.2.2" xref="S3.SS2.p5.13.m13.2.2.cmml">1</mn><mo id="S3.SS2.p5.13.m13.2.3.3.2.2.3" stretchy="false" xref="S3.SS2.p5.13.m13.2.3.3.2.1.cmml">]</mo></mrow><msub id="S3.SS2.p5.13.m13.2.3.3.3" xref="S3.SS2.p5.13.m13.2.3.3.3.cmml"><mi id="S3.SS2.p5.13.m13.2.3.3.3.2" xref="S3.SS2.p5.13.m13.2.3.3.3.2.cmml">d</mi><mtext id="S3.SS2.p5.13.m13.2.3.3.3.3" xref="S3.SS2.p5.13.m13.2.3.3.3.3a.cmml">disc</mtext></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.13.m13.2b"><apply id="S3.SS2.p5.13.m13.2.3.cmml" xref="S3.SS2.p5.13.m13.2.3"><ci id="S3.SS2.p5.13.m13.2.3.1.cmml" xref="S3.SS2.p5.13.m13.2.3.1">→</ci><apply id="S3.SS2.p5.13.m13.2.3.2.cmml" xref="S3.SS2.p5.13.m13.2.3.2"><csymbol cd="ambiguous" id="S3.SS2.p5.13.m13.2.3.2.1.cmml" xref="S3.SS2.p5.13.m13.2.3.2">subscript</csymbol><ci id="S3.SS2.p5.13.m13.2.3.2.2.cmml" xref="S3.SS2.p5.13.m13.2.3.2.2">𝒳</ci><ci id="S3.SS2.p5.13.m13.2.3.2.3a.cmml" xref="S3.SS2.p5.13.m13.2.3.2.3"><mtext id="S3.SS2.p5.13.m13.2.3.2.3.cmml" mathsize="70%" xref="S3.SS2.p5.13.m13.2.3.2.3">disc</mtext></ci></apply><apply id="S3.SS2.p5.13.m13.2.3.3.cmml" xref="S3.SS2.p5.13.m13.2.3.3"><csymbol cd="ambiguous" id="S3.SS2.p5.13.m13.2.3.3.1.cmml" xref="S3.SS2.p5.13.m13.2.3.3">superscript</csymbol><interval closure="closed" id="S3.SS2.p5.13.m13.2.3.3.2.1.cmml" xref="S3.SS2.p5.13.m13.2.3.3.2.2"><cn id="S3.SS2.p5.13.m13.1.1.cmml" type="integer" xref="S3.SS2.p5.13.m13.1.1">0</cn><cn id="S3.SS2.p5.13.m13.2.2.cmml" type="integer" xref="S3.SS2.p5.13.m13.2.2">1</cn></interval><apply id="S3.SS2.p5.13.m13.2.3.3.3.cmml" xref="S3.SS2.p5.13.m13.2.3.3.3"><csymbol cd="ambiguous" id="S3.SS2.p5.13.m13.2.3.3.3.1.cmml" xref="S3.SS2.p5.13.m13.2.3.3.3">subscript</csymbol><ci id="S3.SS2.p5.13.m13.2.3.3.3.2.cmml" xref="S3.SS2.p5.13.m13.2.3.3.3.2">𝑑</ci><ci id="S3.SS2.p5.13.m13.2.3.3.3.3a.cmml" xref="S3.SS2.p5.13.m13.2.3.3.3.3"><mtext id="S3.SS2.p5.13.m13.2.3.3.3.3.cmml" mathsize="50%" xref="S3.SS2.p5.13.m13.2.3.3.3.3">disc</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.13.m13.2c">\mathcal{X}_{\text{disc}}\to[0,1]^{d_{\text{disc}}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.13.m13.2d">caligraphic_X start_POSTSUBSCRIPT disc end_POSTSUBSCRIPT → [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT disc end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> – such as the min-max scaler – can be selected. For <math alttext="N_{\text{cont}}" class="ltx_Math" display="inline" id="S3.SS2.p5.14.m14.1"><semantics id="S3.SS2.p5.14.m14.1a"><msub id="S3.SS2.p5.14.m14.1.1" xref="S3.SS2.p5.14.m14.1.1.cmml"><mi id="S3.SS2.p5.14.m14.1.1.2" xref="S3.SS2.p5.14.m14.1.1.2.cmml">N</mi><mtext id="S3.SS2.p5.14.m14.1.1.3" xref="S3.SS2.p5.14.m14.1.1.3a.cmml">cont</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.14.m14.1b"><apply id="S3.SS2.p5.14.m14.1.1.cmml" xref="S3.SS2.p5.14.m14.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.14.m14.1.1.1.cmml" xref="S3.SS2.p5.14.m14.1.1">subscript</csymbol><ci id="S3.SS2.p5.14.m14.1.1.2.cmml" xref="S3.SS2.p5.14.m14.1.1.2">𝑁</ci><ci id="S3.SS2.p5.14.m14.1.1.3a.cmml" xref="S3.SS2.p5.14.m14.1.1.3"><mtext id="S3.SS2.p5.14.m14.1.1.3.cmml" mathsize="70%" xref="S3.SS2.p5.14.m14.1.1.3">cont</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.14.m14.1c">N_{\text{cont}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.14.m14.1d">italic_N start_POSTSUBSCRIPT cont end_POSTSUBSCRIPT</annotation></semantics></math>, we start by noting that the problem in the middle panel of Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.F1" title="Figure 1 ‣ 3.2 Instrumental functions ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">1</span></a> occurs due to <math alttext="X_{1}" class="ltx_Math" display="inline" id="S3.SS2.p5.15.m15.1"><semantics id="S3.SS2.p5.15.m15.1a"><msub id="S3.SS2.p5.15.m15.1.1" xref="S3.SS2.p5.15.m15.1.1.cmml"><mi id="S3.SS2.p5.15.m15.1.1.2" xref="S3.SS2.p5.15.m15.1.1.2.cmml">X</mi><mn id="S3.SS2.p5.15.m15.1.1.3" xref="S3.SS2.p5.15.m15.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.15.m15.1b"><apply id="S3.SS2.p5.15.m15.1.1.cmml" xref="S3.SS2.p5.15.m15.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.15.m15.1.1.1.cmml" xref="S3.SS2.p5.15.m15.1.1">subscript</csymbol><ci id="S3.SS2.p5.15.m15.1.1.2.cmml" xref="S3.SS2.p5.15.m15.1.1.2">𝑋</ci><cn id="S3.SS2.p5.15.m15.1.1.3.cmml" type="integer" xref="S3.SS2.p5.15.m15.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.15.m15.1c">X_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.15.m15.1d">italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="X_{2}" class="ltx_Math" display="inline" id="S3.SS2.p5.16.m16.1"><semantics id="S3.SS2.p5.16.m16.1a"><msub id="S3.SS2.p5.16.m16.1.1" xref="S3.SS2.p5.16.m16.1.1.cmml"><mi id="S3.SS2.p5.16.m16.1.1.2" xref="S3.SS2.p5.16.m16.1.1.2.cmml">X</mi><mn id="S3.SS2.p5.16.m16.1.1.3" xref="S3.SS2.p5.16.m16.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.16.m16.1b"><apply id="S3.SS2.p5.16.m16.1.1.cmml" xref="S3.SS2.p5.16.m16.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.16.m16.1.1.1.cmml" xref="S3.SS2.p5.16.m16.1.1">subscript</csymbol><ci id="S3.SS2.p5.16.m16.1.1.2.cmml" xref="S3.SS2.p5.16.m16.1.1.2">𝑋</ci><cn id="S3.SS2.p5.16.m16.1.1.3.cmml" type="integer" xref="S3.SS2.p5.16.m16.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.16.m16.1c">X_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.16.m16.1d">italic_X start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> being correlated. The main idea in constructing <math alttext="N_{\text{cont}}" class="ltx_Math" display="inline" id="S3.SS2.p5.17.m17.1"><semantics id="S3.SS2.p5.17.m17.1a"><msub id="S3.SS2.p5.17.m17.1.1" xref="S3.SS2.p5.17.m17.1.1.cmml"><mi id="S3.SS2.p5.17.m17.1.1.2" xref="S3.SS2.p5.17.m17.1.1.2.cmml">N</mi><mtext id="S3.SS2.p5.17.m17.1.1.3" xref="S3.SS2.p5.17.m17.1.1.3a.cmml">cont</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.17.m17.1b"><apply id="S3.SS2.p5.17.m17.1.1.cmml" xref="S3.SS2.p5.17.m17.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.17.m17.1.1.1.cmml" xref="S3.SS2.p5.17.m17.1.1">subscript</csymbol><ci id="S3.SS2.p5.17.m17.1.1.2.cmml" xref="S3.SS2.p5.17.m17.1.1.2">𝑁</ci><ci id="S3.SS2.p5.17.m17.1.1.3a.cmml" xref="S3.SS2.p5.17.m17.1.1.3"><mtext id="S3.SS2.p5.17.m17.1.1.3.cmml" mathsize="70%" xref="S3.SS2.p5.17.m17.1.1.3">cont</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.17.m17.1c">N_{\text{cont}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.17.m17.1d">italic_N start_POSTSUBSCRIPT cont end_POSTSUBSCRIPT</annotation></semantics></math> will therefore be to first decorrelate the continuous covariates using principal component analysis, and then proceed by transforming the decorrelated variables to <math alttext="[0,1]^{d_{\text{cont}}}" class="ltx_Math" display="inline" id="S3.SS2.p5.18.m18.2"><semantics id="S3.SS2.p5.18.m18.2a"><msup id="S3.SS2.p5.18.m18.2.3" xref="S3.SS2.p5.18.m18.2.3.cmml"><mrow id="S3.SS2.p5.18.m18.2.3.2.2" xref="S3.SS2.p5.18.m18.2.3.2.1.cmml"><mo id="S3.SS2.p5.18.m18.2.3.2.2.1" stretchy="false" xref="S3.SS2.p5.18.m18.2.3.2.1.cmml">[</mo><mn id="S3.SS2.p5.18.m18.1.1" xref="S3.SS2.p5.18.m18.1.1.cmml">0</mn><mo id="S3.SS2.p5.18.m18.2.3.2.2.2" xref="S3.SS2.p5.18.m18.2.3.2.1.cmml">,</mo><mn id="S3.SS2.p5.18.m18.2.2" xref="S3.SS2.p5.18.m18.2.2.cmml">1</mn><mo id="S3.SS2.p5.18.m18.2.3.2.2.3" stretchy="false" xref="S3.SS2.p5.18.m18.2.3.2.1.cmml">]</mo></mrow><msub id="S3.SS2.p5.18.m18.2.3.3" xref="S3.SS2.p5.18.m18.2.3.3.cmml"><mi id="S3.SS2.p5.18.m18.2.3.3.2" xref="S3.SS2.p5.18.m18.2.3.3.2.cmml">d</mi><mtext id="S3.SS2.p5.18.m18.2.3.3.3" xref="S3.SS2.p5.18.m18.2.3.3.3a.cmml">cont</mtext></msub></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.18.m18.2b"><apply id="S3.SS2.p5.18.m18.2.3.cmml" xref="S3.SS2.p5.18.m18.2.3"><csymbol cd="ambiguous" id="S3.SS2.p5.18.m18.2.3.1.cmml" xref="S3.SS2.p5.18.m18.2.3">superscript</csymbol><interval closure="closed" id="S3.SS2.p5.18.m18.2.3.2.1.cmml" xref="S3.SS2.p5.18.m18.2.3.2.2"><cn id="S3.SS2.p5.18.m18.1.1.cmml" type="integer" xref="S3.SS2.p5.18.m18.1.1">0</cn><cn id="S3.SS2.p5.18.m18.2.2.cmml" type="integer" xref="S3.SS2.p5.18.m18.2.2">1</cn></interval><apply id="S3.SS2.p5.18.m18.2.3.3.cmml" xref="S3.SS2.p5.18.m18.2.3.3"><csymbol cd="ambiguous" id="S3.SS2.p5.18.m18.2.3.3.1.cmml" xref="S3.SS2.p5.18.m18.2.3.3">subscript</csymbol><ci id="S3.SS2.p5.18.m18.2.3.3.2.cmml" xref="S3.SS2.p5.18.m18.2.3.3.2">𝑑</ci><ci id="S3.SS2.p5.18.m18.2.3.3.3a.cmml" xref="S3.SS2.p5.18.m18.2.3.3.3"><mtext id="S3.SS2.p5.18.m18.2.3.3.3.cmml" mathsize="50%" xref="S3.SS2.p5.18.m18.2.3.3.3">cont</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.18.m18.2c">[0,1]^{d_{\text{cont}}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.18.m18.2d">[ 0 , 1 ] start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT cont end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> in a sufficiently surjective way.</p> </div> <div class="ltx_para" id="S3.SS2.p6"> <p class="ltx_p" id="S3.SS2.p6.4">Precisely, <math alttext="N_{\text{cont}}" class="ltx_Math" display="inline" id="S3.SS2.p6.1.m1.1"><semantics id="S3.SS2.p6.1.m1.1a"><msub id="S3.SS2.p6.1.m1.1.1" xref="S3.SS2.p6.1.m1.1.1.cmml"><mi id="S3.SS2.p6.1.m1.1.1.2" xref="S3.SS2.p6.1.m1.1.1.2.cmml">N</mi><mtext id="S3.SS2.p6.1.m1.1.1.3" xref="S3.SS2.p6.1.m1.1.1.3a.cmml">cont</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.1.m1.1b"><apply id="S3.SS2.p6.1.m1.1.1.cmml" xref="S3.SS2.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p6.1.m1.1.1.1.cmml" xref="S3.SS2.p6.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p6.1.m1.1.1.2.cmml" xref="S3.SS2.p6.1.m1.1.1.2">𝑁</ci><ci id="S3.SS2.p6.1.m1.1.1.3a.cmml" xref="S3.SS2.p6.1.m1.1.1.3"><mtext id="S3.SS2.p6.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.SS2.p6.1.m1.1.1.3">cont</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.1.m1.1c">N_{\text{cont}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.1.m1.1d">italic_N start_POSTSUBSCRIPT cont end_POSTSUBSCRIPT</annotation></semantics></math> consists of several steps, which are outlined below. An illustration of the results of the transformation is given in the right panel of Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.F1" title="Figure 1 ‣ 3.2 Instrumental functions ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">1</span></a>. Without loss of generality, suppose the first <math alttext="d_{\text{cont}}" class="ltx_Math" display="inline" id="S3.SS2.p6.2.m2.1"><semantics id="S3.SS2.p6.2.m2.1a"><msub id="S3.SS2.p6.2.m2.1.1" xref="S3.SS2.p6.2.m2.1.1.cmml"><mi id="S3.SS2.p6.2.m2.1.1.2" xref="S3.SS2.p6.2.m2.1.1.2.cmml">d</mi><mtext id="S3.SS2.p6.2.m2.1.1.3" xref="S3.SS2.p6.2.m2.1.1.3a.cmml">cont</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.2.m2.1b"><apply id="S3.SS2.p6.2.m2.1.1.cmml" xref="S3.SS2.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p6.2.m2.1.1.1.cmml" xref="S3.SS2.p6.2.m2.1.1">subscript</csymbol><ci id="S3.SS2.p6.2.m2.1.1.2.cmml" xref="S3.SS2.p6.2.m2.1.1.2">𝑑</ci><ci id="S3.SS2.p6.2.m2.1.1.3a.cmml" xref="S3.SS2.p6.2.m2.1.1.3"><mtext id="S3.SS2.p6.2.m2.1.1.3.cmml" mathsize="70%" xref="S3.SS2.p6.2.m2.1.1.3">cont</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.2.m2.1c">d_{\text{cont}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.2.m2.1d">italic_d start_POSTSUBSCRIPT cont end_POSTSUBSCRIPT</annotation></semantics></math> elements of <math alttext="\tilde{X}" class="ltx_Math" display="inline" id="S3.SS2.p6.3.m3.1"><semantics id="S3.SS2.p6.3.m3.1a"><mover accent="true" id="S3.SS2.p6.3.m3.1.1" xref="S3.SS2.p6.3.m3.1.1.cmml"><mi id="S3.SS2.p6.3.m3.1.1.2" xref="S3.SS2.p6.3.m3.1.1.2.cmml">X</mi><mo id="S3.SS2.p6.3.m3.1.1.1" xref="S3.SS2.p6.3.m3.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.3.m3.1b"><apply id="S3.SS2.p6.3.m3.1.1.cmml" xref="S3.SS2.p6.3.m3.1.1"><ci id="S3.SS2.p6.3.m3.1.1.1.cmml" xref="S3.SS2.p6.3.m3.1.1.1">~</ci><ci id="S3.SS2.p6.3.m3.1.1.2.cmml" xref="S3.SS2.p6.3.m3.1.1.2">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.3.m3.1c">\tilde{X}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.3.m3.1d">over~ start_ARG italic_X end_ARG</annotation></semantics></math> correspond to the continuous covariates. Furthermore, denote this subvector as <math alttext="X^{\text{cont}}" class="ltx_Math" display="inline" id="S3.SS2.p6.4.m4.1"><semantics id="S3.SS2.p6.4.m4.1a"><msup id="S3.SS2.p6.4.m4.1.1" xref="S3.SS2.p6.4.m4.1.1.cmml"><mi id="S3.SS2.p6.4.m4.1.1.2" xref="S3.SS2.p6.4.m4.1.1.2.cmml">X</mi><mtext id="S3.SS2.p6.4.m4.1.1.3" xref="S3.SS2.p6.4.m4.1.1.3a.cmml">cont</mtext></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.4.m4.1b"><apply id="S3.SS2.p6.4.m4.1.1.cmml" xref="S3.SS2.p6.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS2.p6.4.m4.1.1.1.cmml" xref="S3.SS2.p6.4.m4.1.1">superscript</csymbol><ci id="S3.SS2.p6.4.m4.1.1.2.cmml" xref="S3.SS2.p6.4.m4.1.1.2">𝑋</ci><ci id="S3.SS2.p6.4.m4.1.1.3a.cmml" xref="S3.SS2.p6.4.m4.1.1.3"><mtext id="S3.SS2.p6.4.m4.1.1.3.cmml" mathsize="70%" xref="S3.SS2.p6.4.m4.1.1.3">cont</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.4.m4.1c">X^{\text{cont}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.4.m4.1d">italic_X start_POSTSUPERSCRIPT cont end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> <ol class="ltx_enumerate" id="S3.I4"> <li class="ltx_item" id="S3.I4.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S3.I4.i1.p1"> <p class="ltx_p" id="S3.I4.i1.p1.2">Perform a principle component analysis (PCA) of the continuous covariates <math alttext="\{x_{i}^{\text{cont}}\}_{i=1}^{n}" class="ltx_Math" display="inline" id="S3.I4.i1.p1.1.m1.1"><semantics id="S3.I4.i1.p1.1.m1.1a"><msubsup id="S3.I4.i1.p1.1.m1.1.1" xref="S3.I4.i1.p1.1.m1.1.1.cmml"><mrow id="S3.I4.i1.p1.1.m1.1.1.1.1.1" xref="S3.I4.i1.p1.1.m1.1.1.1.1.2.cmml"><mo id="S3.I4.i1.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.I4.i1.p1.1.m1.1.1.1.1.2.cmml">{</mo><msubsup id="S3.I4.i1.p1.1.m1.1.1.1.1.1.1" xref="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.2.2" xref="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.2.2.cmml">x</mi><mi id="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.2.3" xref="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.2.3.cmml">i</mi><mtext id="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.3" xref="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.3a.cmml">cont</mtext></msubsup><mo id="S3.I4.i1.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.I4.i1.p1.1.m1.1.1.1.1.2.cmml">}</mo></mrow><mrow id="S3.I4.i1.p1.1.m1.1.1.1.3" xref="S3.I4.i1.p1.1.m1.1.1.1.3.cmml"><mi id="S3.I4.i1.p1.1.m1.1.1.1.3.2" xref="S3.I4.i1.p1.1.m1.1.1.1.3.2.cmml">i</mi><mo id="S3.I4.i1.p1.1.m1.1.1.1.3.1" xref="S3.I4.i1.p1.1.m1.1.1.1.3.1.cmml">=</mo><mn id="S3.I4.i1.p1.1.m1.1.1.1.3.3" xref="S3.I4.i1.p1.1.m1.1.1.1.3.3.cmml">1</mn></mrow><mi id="S3.I4.i1.p1.1.m1.1.1.3" xref="S3.I4.i1.p1.1.m1.1.1.3.cmml">n</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.I4.i1.p1.1.m1.1b"><apply id="S3.I4.i1.p1.1.m1.1.1.cmml" xref="S3.I4.i1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I4.i1.p1.1.m1.1.1.2.cmml" xref="S3.I4.i1.p1.1.m1.1.1">superscript</csymbol><apply id="S3.I4.i1.p1.1.m1.1.1.1.cmml" xref="S3.I4.i1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I4.i1.p1.1.m1.1.1.1.2.cmml" xref="S3.I4.i1.p1.1.m1.1.1">subscript</csymbol><set id="S3.I4.i1.p1.1.m1.1.1.1.1.2.cmml" xref="S3.I4.i1.p1.1.m1.1.1.1.1.1"><apply id="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S3.I4.i1.p1.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.I4.i1.p1.1.m1.1.1.1.1.1.1">superscript</csymbol><apply id="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.I4.i1.p1.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.2.1.cmml" xref="S3.I4.i1.p1.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.2.2.cmml" xref="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.2.2">𝑥</ci><ci id="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.2.3.cmml" xref="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.2.3">𝑖</ci></apply><ci id="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.3a.cmml" xref="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.3"><mtext id="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.3.cmml" mathsize="70%" xref="S3.I4.i1.p1.1.m1.1.1.1.1.1.1.3">cont</mtext></ci></apply></set><apply id="S3.I4.i1.p1.1.m1.1.1.1.3.cmml" xref="S3.I4.i1.p1.1.m1.1.1.1.3"><eq id="S3.I4.i1.p1.1.m1.1.1.1.3.1.cmml" xref="S3.I4.i1.p1.1.m1.1.1.1.3.1"></eq><ci id="S3.I4.i1.p1.1.m1.1.1.1.3.2.cmml" xref="S3.I4.i1.p1.1.m1.1.1.1.3.2">𝑖</ci><cn id="S3.I4.i1.p1.1.m1.1.1.1.3.3.cmml" type="integer" xref="S3.I4.i1.p1.1.m1.1.1.1.3.3">1</cn></apply></apply><ci id="S3.I4.i1.p1.1.m1.1.1.3.cmml" xref="S3.I4.i1.p1.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i1.p1.1.m1.1c">\{x_{i}^{\text{cont}}\}_{i=1}^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i1.p1.1.m1.1d">{ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT cont end_POSTSUPERSCRIPT } start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. Each observation is transformed to its score <math alttext="s_{i}" class="ltx_Math" display="inline" id="S3.I4.i1.p1.2.m2.1"><semantics id="S3.I4.i1.p1.2.m2.1a"><msub id="S3.I4.i1.p1.2.m2.1.1" xref="S3.I4.i1.p1.2.m2.1.1.cmml"><mi id="S3.I4.i1.p1.2.m2.1.1.2" xref="S3.I4.i1.p1.2.m2.1.1.2.cmml">s</mi><mi id="S3.I4.i1.p1.2.m2.1.1.3" xref="S3.I4.i1.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I4.i1.p1.2.m2.1b"><apply id="S3.I4.i1.p1.2.m2.1.1.cmml" xref="S3.I4.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I4.i1.p1.2.m2.1.1.1.cmml" xref="S3.I4.i1.p1.2.m2.1.1">subscript</csymbol><ci id="S3.I4.i1.p1.2.m2.1.1.2.cmml" xref="S3.I4.i1.p1.2.m2.1.1.2">𝑠</ci><ci id="S3.I4.i1.p1.2.m2.1.1.3.cmml" xref="S3.I4.i1.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i1.p1.2.m2.1c">s_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i1.p1.2.m2.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> on the principal components.</p> </div> </li> <li class="ltx_item" id="S3.I4.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S3.I4.i2.p1"> <p class="ltx_p" id="S3.I4.i2.p1.4">Scale these scores elementwise, so that <math alttext="\max_{i}s_{ij}-\min_{i}s_{ij}=2" class="ltx_Math" display="inline" id="S3.I4.i2.p1.1.m1.1"><semantics id="S3.I4.i2.p1.1.m1.1a"><mrow id="S3.I4.i2.p1.1.m1.1.1" xref="S3.I4.i2.p1.1.m1.1.1.cmml"><mrow id="S3.I4.i2.p1.1.m1.1.1.2" xref="S3.I4.i2.p1.1.m1.1.1.2.cmml"><mrow id="S3.I4.i2.p1.1.m1.1.1.2.2" xref="S3.I4.i2.p1.1.m1.1.1.2.2.cmml"><msub id="S3.I4.i2.p1.1.m1.1.1.2.2.1" xref="S3.I4.i2.p1.1.m1.1.1.2.2.1.cmml"><mi id="S3.I4.i2.p1.1.m1.1.1.2.2.1.2" xref="S3.I4.i2.p1.1.m1.1.1.2.2.1.2.cmml">max</mi><mi id="S3.I4.i2.p1.1.m1.1.1.2.2.1.3" xref="S3.I4.i2.p1.1.m1.1.1.2.2.1.3.cmml">i</mi></msub><mo id="S3.I4.i2.p1.1.m1.1.1.2.2a" lspace="0.167em" 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xref="S3.I4.i2.p1.1.m1.1.1.2.2.2.3.1"></times><ci id="S3.I4.i2.p1.1.m1.1.1.2.2.2.3.2.cmml" xref="S3.I4.i2.p1.1.m1.1.1.2.2.2.3.2">𝑖</ci><ci id="S3.I4.i2.p1.1.m1.1.1.2.2.2.3.3.cmml" xref="S3.I4.i2.p1.1.m1.1.1.2.2.2.3.3">𝑗</ci></apply></apply></apply><apply id="S3.I4.i2.p1.1.m1.1.1.2.3.cmml" xref="S3.I4.i2.p1.1.m1.1.1.2.3"><apply id="S3.I4.i2.p1.1.m1.1.1.2.3.1.cmml" xref="S3.I4.i2.p1.1.m1.1.1.2.3.1"><csymbol cd="ambiguous" id="S3.I4.i2.p1.1.m1.1.1.2.3.1.1.cmml" xref="S3.I4.i2.p1.1.m1.1.1.2.3.1">subscript</csymbol><min id="S3.I4.i2.p1.1.m1.1.1.2.3.1.2.cmml" xref="S3.I4.i2.p1.1.m1.1.1.2.3.1.2"></min><ci id="S3.I4.i2.p1.1.m1.1.1.2.3.1.3.cmml" xref="S3.I4.i2.p1.1.m1.1.1.2.3.1.3">𝑖</ci></apply><apply id="S3.I4.i2.p1.1.m1.1.1.2.3.2.cmml" xref="S3.I4.i2.p1.1.m1.1.1.2.3.2"><csymbol cd="ambiguous" id="S3.I4.i2.p1.1.m1.1.1.2.3.2.1.cmml" xref="S3.I4.i2.p1.1.m1.1.1.2.3.2">subscript</csymbol><ci id="S3.I4.i2.p1.1.m1.1.1.2.3.2.2.cmml" xref="S3.I4.i2.p1.1.m1.1.1.2.3.2.2">𝑠</ci><apply id="S3.I4.i2.p1.1.m1.1.1.2.3.2.3.cmml" xref="S3.I4.i2.p1.1.m1.1.1.2.3.2.3"><times id="S3.I4.i2.p1.1.m1.1.1.2.3.2.3.1.cmml" xref="S3.I4.i2.p1.1.m1.1.1.2.3.2.3.1"></times><ci id="S3.I4.i2.p1.1.m1.1.1.2.3.2.3.2.cmml" xref="S3.I4.i2.p1.1.m1.1.1.2.3.2.3.2">𝑖</ci><ci id="S3.I4.i2.p1.1.m1.1.1.2.3.2.3.3.cmml" xref="S3.I4.i2.p1.1.m1.1.1.2.3.2.3.3">𝑗</ci></apply></apply></apply></apply><cn id="S3.I4.i2.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.I4.i2.p1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.1.m1.1c">\max_{i}s_{ij}-\min_{i}s_{ij}=2</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.1.m1.1d">roman_max start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT - roman_min start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = 2</annotation></semantics></math>, for all <math alttext="j\in\{1,\dots,d_{\text{cont}}\}" class="ltx_Math" display="inline" id="S3.I4.i2.p1.2.m2.3"><semantics id="S3.I4.i2.p1.2.m2.3a"><mrow id="S3.I4.i2.p1.2.m2.3.3" xref="S3.I4.i2.p1.2.m2.3.3.cmml"><mi id="S3.I4.i2.p1.2.m2.3.3.3" xref="S3.I4.i2.p1.2.m2.3.3.3.cmml">j</mi><mo id="S3.I4.i2.p1.2.m2.3.3.2" xref="S3.I4.i2.p1.2.m2.3.3.2.cmml">∈</mo><mrow id="S3.I4.i2.p1.2.m2.3.3.1.1" xref="S3.I4.i2.p1.2.m2.3.3.1.2.cmml"><mo id="S3.I4.i2.p1.2.m2.3.3.1.1.2" stretchy="false" xref="S3.I4.i2.p1.2.m2.3.3.1.2.cmml">{</mo><mn id="S3.I4.i2.p1.2.m2.1.1" xref="S3.I4.i2.p1.2.m2.1.1.cmml">1</mn><mo id="S3.I4.i2.p1.2.m2.3.3.1.1.3" xref="S3.I4.i2.p1.2.m2.3.3.1.2.cmml">,</mo><mi id="S3.I4.i2.p1.2.m2.2.2" mathvariant="normal" xref="S3.I4.i2.p1.2.m2.2.2.cmml">…</mi><mo id="S3.I4.i2.p1.2.m2.3.3.1.1.4" xref="S3.I4.i2.p1.2.m2.3.3.1.2.cmml">,</mo><msub id="S3.I4.i2.p1.2.m2.3.3.1.1.1" xref="S3.I4.i2.p1.2.m2.3.3.1.1.1.cmml"><mi id="S3.I4.i2.p1.2.m2.3.3.1.1.1.2" xref="S3.I4.i2.p1.2.m2.3.3.1.1.1.2.cmml">d</mi><mtext id="S3.I4.i2.p1.2.m2.3.3.1.1.1.3" xref="S3.I4.i2.p1.2.m2.3.3.1.1.1.3a.cmml">cont</mtext></msub><mo id="S3.I4.i2.p1.2.m2.3.3.1.1.5" stretchy="false" xref="S3.I4.i2.p1.2.m2.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.2.m2.3b"><apply id="S3.I4.i2.p1.2.m2.3.3.cmml" xref="S3.I4.i2.p1.2.m2.3.3"><in id="S3.I4.i2.p1.2.m2.3.3.2.cmml" xref="S3.I4.i2.p1.2.m2.3.3.2"></in><ci id="S3.I4.i2.p1.2.m2.3.3.3.cmml" xref="S3.I4.i2.p1.2.m2.3.3.3">𝑗</ci><set id="S3.I4.i2.p1.2.m2.3.3.1.2.cmml" xref="S3.I4.i2.p1.2.m2.3.3.1.1"><cn id="S3.I4.i2.p1.2.m2.1.1.cmml" type="integer" xref="S3.I4.i2.p1.2.m2.1.1">1</cn><ci id="S3.I4.i2.p1.2.m2.2.2.cmml" xref="S3.I4.i2.p1.2.m2.2.2">…</ci><apply id="S3.I4.i2.p1.2.m2.3.3.1.1.1.cmml" xref="S3.I4.i2.p1.2.m2.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.I4.i2.p1.2.m2.3.3.1.1.1.1.cmml" xref="S3.I4.i2.p1.2.m2.3.3.1.1.1">subscript</csymbol><ci id="S3.I4.i2.p1.2.m2.3.3.1.1.1.2.cmml" xref="S3.I4.i2.p1.2.m2.3.3.1.1.1.2">𝑑</ci><ci id="S3.I4.i2.p1.2.m2.3.3.1.1.1.3a.cmml" xref="S3.I4.i2.p1.2.m2.3.3.1.1.1.3"><mtext id="S3.I4.i2.p1.2.m2.3.3.1.1.1.3.cmml" mathsize="70%" xref="S3.I4.i2.p1.2.m2.3.3.1.1.1.3">cont</mtext></ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.2.m2.3c">j\in\{1,\dots,d_{\text{cont}}\}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.2.m2.3d">italic_j ∈ { 1 , … , italic_d start_POSTSUBSCRIPT cont end_POSTSUBSCRIPT }</annotation></semantics></math>. Shift the scaled scores to <math alttext="[-1,1]^{d_{\text{cont}}}" class="ltx_Math" display="inline" id="S3.I4.i2.p1.3.m3.2"><semantics id="S3.I4.i2.p1.3.m3.2a"><msup id="S3.I4.i2.p1.3.m3.2.2" xref="S3.I4.i2.p1.3.m3.2.2.cmml"><mrow id="S3.I4.i2.p1.3.m3.2.2.1.1" xref="S3.I4.i2.p1.3.m3.2.2.1.2.cmml"><mo id="S3.I4.i2.p1.3.m3.2.2.1.1.2" stretchy="false" xref="S3.I4.i2.p1.3.m3.2.2.1.2.cmml">[</mo><mrow id="S3.I4.i2.p1.3.m3.2.2.1.1.1" xref="S3.I4.i2.p1.3.m3.2.2.1.1.1.cmml"><mo id="S3.I4.i2.p1.3.m3.2.2.1.1.1a" xref="S3.I4.i2.p1.3.m3.2.2.1.1.1.cmml">−</mo><mn id="S3.I4.i2.p1.3.m3.2.2.1.1.1.2" xref="S3.I4.i2.p1.3.m3.2.2.1.1.1.2.cmml">1</mn></mrow><mo id="S3.I4.i2.p1.3.m3.2.2.1.1.3" xref="S3.I4.i2.p1.3.m3.2.2.1.2.cmml">,</mo><mn id="S3.I4.i2.p1.3.m3.1.1" xref="S3.I4.i2.p1.3.m3.1.1.cmml">1</mn><mo id="S3.I4.i2.p1.3.m3.2.2.1.1.4" stretchy="false" xref="S3.I4.i2.p1.3.m3.2.2.1.2.cmml">]</mo></mrow><msub id="S3.I4.i2.p1.3.m3.2.2.3" xref="S3.I4.i2.p1.3.m3.2.2.3.cmml"><mi id="S3.I4.i2.p1.3.m3.2.2.3.2" xref="S3.I4.i2.p1.3.m3.2.2.3.2.cmml">d</mi><mtext id="S3.I4.i2.p1.3.m3.2.2.3.3" xref="S3.I4.i2.p1.3.m3.2.2.3.3a.cmml">cont</mtext></msub></msup><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.3.m3.2b"><apply id="S3.I4.i2.p1.3.m3.2.2.cmml" xref="S3.I4.i2.p1.3.m3.2.2"><csymbol cd="ambiguous" id="S3.I4.i2.p1.3.m3.2.2.2.cmml" xref="S3.I4.i2.p1.3.m3.2.2">superscript</csymbol><interval closure="closed" id="S3.I4.i2.p1.3.m3.2.2.1.2.cmml" xref="S3.I4.i2.p1.3.m3.2.2.1.1"><apply id="S3.I4.i2.p1.3.m3.2.2.1.1.1.cmml" xref="S3.I4.i2.p1.3.m3.2.2.1.1.1"><minus id="S3.I4.i2.p1.3.m3.2.2.1.1.1.1.cmml" xref="S3.I4.i2.p1.3.m3.2.2.1.1.1"></minus><cn id="S3.I4.i2.p1.3.m3.2.2.1.1.1.2.cmml" type="integer" xref="S3.I4.i2.p1.3.m3.2.2.1.1.1.2">1</cn></apply><cn id="S3.I4.i2.p1.3.m3.1.1.cmml" type="integer" xref="S3.I4.i2.p1.3.m3.1.1">1</cn></interval><apply id="S3.I4.i2.p1.3.m3.2.2.3.cmml" xref="S3.I4.i2.p1.3.m3.2.2.3"><csymbol cd="ambiguous" id="S3.I4.i2.p1.3.m3.2.2.3.1.cmml" xref="S3.I4.i2.p1.3.m3.2.2.3">subscript</csymbol><ci id="S3.I4.i2.p1.3.m3.2.2.3.2.cmml" xref="S3.I4.i2.p1.3.m3.2.2.3.2">𝑑</ci><ci id="S3.I4.i2.p1.3.m3.2.2.3.3a.cmml" xref="S3.I4.i2.p1.3.m3.2.2.3.3"><mtext id="S3.I4.i2.p1.3.m3.2.2.3.3.cmml" mathsize="50%" xref="S3.I4.i2.p1.3.m3.2.2.3.3">cont</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.3.m3.2c">[-1,1]^{d_{\text{cont}}}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.3.m3.2d">[ - 1 , 1 ] start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT cont end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, and denote the results as <math alttext="s_{i}^{*}" class="ltx_Math" display="inline" id="S3.I4.i2.p1.4.m4.1"><semantics id="S3.I4.i2.p1.4.m4.1a"><msubsup id="S3.I4.i2.p1.4.m4.1.1" xref="S3.I4.i2.p1.4.m4.1.1.cmml"><mi id="S3.I4.i2.p1.4.m4.1.1.2.2" xref="S3.I4.i2.p1.4.m4.1.1.2.2.cmml">s</mi><mi id="S3.I4.i2.p1.4.m4.1.1.2.3" xref="S3.I4.i2.p1.4.m4.1.1.2.3.cmml">i</mi><mo id="S3.I4.i2.p1.4.m4.1.1.3" xref="S3.I4.i2.p1.4.m4.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.4.m4.1b"><apply id="S3.I4.i2.p1.4.m4.1.1.cmml" xref="S3.I4.i2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.I4.i2.p1.4.m4.1.1.1.cmml" xref="S3.I4.i2.p1.4.m4.1.1">superscript</csymbol><apply id="S3.I4.i2.p1.4.m4.1.1.2.cmml" xref="S3.I4.i2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.I4.i2.p1.4.m4.1.1.2.1.cmml" xref="S3.I4.i2.p1.4.m4.1.1">subscript</csymbol><ci id="S3.I4.i2.p1.4.m4.1.1.2.2.cmml" xref="S3.I4.i2.p1.4.m4.1.1.2.2">𝑠</ci><ci id="S3.I4.i2.p1.4.m4.1.1.2.3.cmml" xref="S3.I4.i2.p1.4.m4.1.1.2.3">𝑖</ci></apply><times id="S3.I4.i2.p1.4.m4.1.1.3.cmml" xref="S3.I4.i2.p1.4.m4.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.4.m4.1c">s_{i}^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.4.m4.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S3.I4.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S3.I4.i3.p1"> <p class="ltx_p" id="S3.I4.i3.p1.7">Typically, the points <math alttext="s_{i}^{*}" class="ltx_Math" display="inline" id="S3.I4.i3.p1.1.m1.1"><semantics id="S3.I4.i3.p1.1.m1.1a"><msubsup id="S3.I4.i3.p1.1.m1.1.1" xref="S3.I4.i3.p1.1.m1.1.1.cmml"><mi id="S3.I4.i3.p1.1.m1.1.1.2.2" xref="S3.I4.i3.p1.1.m1.1.1.2.2.cmml">s</mi><mi id="S3.I4.i3.p1.1.m1.1.1.2.3" xref="S3.I4.i3.p1.1.m1.1.1.2.3.cmml">i</mi><mo id="S3.I4.i3.p1.1.m1.1.1.3" xref="S3.I4.i3.p1.1.m1.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.I4.i3.p1.1.m1.1b"><apply id="S3.I4.i3.p1.1.m1.1.1.cmml" xref="S3.I4.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I4.i3.p1.1.m1.1.1.1.cmml" xref="S3.I4.i3.p1.1.m1.1.1">superscript</csymbol><apply id="S3.I4.i3.p1.1.m1.1.1.2.cmml" xref="S3.I4.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I4.i3.p1.1.m1.1.1.2.1.cmml" xref="S3.I4.i3.p1.1.m1.1.1">subscript</csymbol><ci id="S3.I4.i3.p1.1.m1.1.1.2.2.cmml" xref="S3.I4.i3.p1.1.m1.1.1.2.2">𝑠</ci><ci id="S3.I4.i3.p1.1.m1.1.1.2.3.cmml" xref="S3.I4.i3.p1.1.m1.1.1.2.3">𝑖</ci></apply><times id="S3.I4.i3.p1.1.m1.1.1.3.cmml" xref="S3.I4.i3.p1.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i3.p1.1.m1.1c">s_{i}^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i3.p1.1.m1.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> will be concentrated inside the inscribing sphere of <math alttext="[-1,1]^{d_{\text{cont}}}" class="ltx_Math" display="inline" id="S3.I4.i3.p1.2.m2.2"><semantics id="S3.I4.i3.p1.2.m2.2a"><msup id="S3.I4.i3.p1.2.m2.2.2" xref="S3.I4.i3.p1.2.m2.2.2.cmml"><mrow id="S3.I4.i3.p1.2.m2.2.2.1.1" xref="S3.I4.i3.p1.2.m2.2.2.1.2.cmml"><mo id="S3.I4.i3.p1.2.m2.2.2.1.1.2" stretchy="false" xref="S3.I4.i3.p1.2.m2.2.2.1.2.cmml">[</mo><mrow id="S3.I4.i3.p1.2.m2.2.2.1.1.1" xref="S3.I4.i3.p1.2.m2.2.2.1.1.1.cmml"><mo id="S3.I4.i3.p1.2.m2.2.2.1.1.1a" xref="S3.I4.i3.p1.2.m2.2.2.1.1.1.cmml">−</mo><mn id="S3.I4.i3.p1.2.m2.2.2.1.1.1.2" xref="S3.I4.i3.p1.2.m2.2.2.1.1.1.2.cmml">1</mn></mrow><mo id="S3.I4.i3.p1.2.m2.2.2.1.1.3" xref="S3.I4.i3.p1.2.m2.2.2.1.2.cmml">,</mo><mn id="S3.I4.i3.p1.2.m2.1.1" xref="S3.I4.i3.p1.2.m2.1.1.cmml">1</mn><mo id="S3.I4.i3.p1.2.m2.2.2.1.1.4" stretchy="false" xref="S3.I4.i3.p1.2.m2.2.2.1.2.cmml">]</mo></mrow><msub id="S3.I4.i3.p1.2.m2.2.2.3" xref="S3.I4.i3.p1.2.m2.2.2.3.cmml"><mi id="S3.I4.i3.p1.2.m2.2.2.3.2" xref="S3.I4.i3.p1.2.m2.2.2.3.2.cmml">d</mi><mtext id="S3.I4.i3.p1.2.m2.2.2.3.3" xref="S3.I4.i3.p1.2.m2.2.2.3.3a.cmml">cont</mtext></msub></msup><annotation-xml encoding="MathML-Content" id="S3.I4.i3.p1.2.m2.2b"><apply id="S3.I4.i3.p1.2.m2.2.2.cmml" xref="S3.I4.i3.p1.2.m2.2.2"><csymbol cd="ambiguous" id="S3.I4.i3.p1.2.m2.2.2.2.cmml" xref="S3.I4.i3.p1.2.m2.2.2">superscript</csymbol><interval closure="closed" id="S3.I4.i3.p1.2.m2.2.2.1.2.cmml" xref="S3.I4.i3.p1.2.m2.2.2.1.1"><apply id="S3.I4.i3.p1.2.m2.2.2.1.1.1.cmml" xref="S3.I4.i3.p1.2.m2.2.2.1.1.1"><minus id="S3.I4.i3.p1.2.m2.2.2.1.1.1.1.cmml" xref="S3.I4.i3.p1.2.m2.2.2.1.1.1"></minus><cn id="S3.I4.i3.p1.2.m2.2.2.1.1.1.2.cmml" type="integer" xref="S3.I4.i3.p1.2.m2.2.2.1.1.1.2">1</cn></apply><cn id="S3.I4.i3.p1.2.m2.1.1.cmml" type="integer" xref="S3.I4.i3.p1.2.m2.1.1">1</cn></interval><apply id="S3.I4.i3.p1.2.m2.2.2.3.cmml" xref="S3.I4.i3.p1.2.m2.2.2.3"><csymbol cd="ambiguous" id="S3.I4.i3.p1.2.m2.2.2.3.1.cmml" xref="S3.I4.i3.p1.2.m2.2.2.3">subscript</csymbol><ci id="S3.I4.i3.p1.2.m2.2.2.3.2.cmml" xref="S3.I4.i3.p1.2.m2.2.2.3.2">𝑑</ci><ci id="S3.I4.i3.p1.2.m2.2.2.3.3a.cmml" xref="S3.I4.i3.p1.2.m2.2.2.3.3"><mtext id="S3.I4.i3.p1.2.m2.2.2.3.3.cmml" mathsize="50%" xref="S3.I4.i3.p1.2.m2.2.2.3.3">cont</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i3.p1.2.m2.2c">[-1,1]^{d_{\text{cont}}}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i3.p1.2.m2.2d">[ - 1 , 1 ] start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT cont end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, leading to a lower concentration of points near the corners of <math alttext="[-1,1]^{d_{\text{cont}}}" class="ltx_Math" display="inline" id="S3.I4.i3.p1.3.m3.2"><semantics id="S3.I4.i3.p1.3.m3.2a"><msup id="S3.I4.i3.p1.3.m3.2.2" xref="S3.I4.i3.p1.3.m3.2.2.cmml"><mrow id="S3.I4.i3.p1.3.m3.2.2.1.1" xref="S3.I4.i3.p1.3.m3.2.2.1.2.cmml"><mo id="S3.I4.i3.p1.3.m3.2.2.1.1.2" stretchy="false" xref="S3.I4.i3.p1.3.m3.2.2.1.2.cmml">[</mo><mrow id="S3.I4.i3.p1.3.m3.2.2.1.1.1" xref="S3.I4.i3.p1.3.m3.2.2.1.1.1.cmml"><mo id="S3.I4.i3.p1.3.m3.2.2.1.1.1a" xref="S3.I4.i3.p1.3.m3.2.2.1.1.1.cmml">−</mo><mn id="S3.I4.i3.p1.3.m3.2.2.1.1.1.2" xref="S3.I4.i3.p1.3.m3.2.2.1.1.1.2.cmml">1</mn></mrow><mo id="S3.I4.i3.p1.3.m3.2.2.1.1.3" xref="S3.I4.i3.p1.3.m3.2.2.1.2.cmml">,</mo><mn id="S3.I4.i3.p1.3.m3.1.1" xref="S3.I4.i3.p1.3.m3.1.1.cmml">1</mn><mo id="S3.I4.i3.p1.3.m3.2.2.1.1.4" stretchy="false" xref="S3.I4.i3.p1.3.m3.2.2.1.2.cmml">]</mo></mrow><msub id="S3.I4.i3.p1.3.m3.2.2.3" xref="S3.I4.i3.p1.3.m3.2.2.3.cmml"><mi id="S3.I4.i3.p1.3.m3.2.2.3.2" xref="S3.I4.i3.p1.3.m3.2.2.3.2.cmml">d</mi><mtext id="S3.I4.i3.p1.3.m3.2.2.3.3" xref="S3.I4.i3.p1.3.m3.2.2.3.3a.cmml">cont</mtext></msub></msup><annotation-xml encoding="MathML-Content" id="S3.I4.i3.p1.3.m3.2b"><apply id="S3.I4.i3.p1.3.m3.2.2.cmml" xref="S3.I4.i3.p1.3.m3.2.2"><csymbol cd="ambiguous" id="S3.I4.i3.p1.3.m3.2.2.2.cmml" xref="S3.I4.i3.p1.3.m3.2.2">superscript</csymbol><interval closure="closed" id="S3.I4.i3.p1.3.m3.2.2.1.2.cmml" xref="S3.I4.i3.p1.3.m3.2.2.1.1"><apply id="S3.I4.i3.p1.3.m3.2.2.1.1.1.cmml" xref="S3.I4.i3.p1.3.m3.2.2.1.1.1"><minus id="S3.I4.i3.p1.3.m3.2.2.1.1.1.1.cmml" xref="S3.I4.i3.p1.3.m3.2.2.1.1.1"></minus><cn id="S3.I4.i3.p1.3.m3.2.2.1.1.1.2.cmml" type="integer" xref="S3.I4.i3.p1.3.m3.2.2.1.1.1.2">1</cn></apply><cn id="S3.I4.i3.p1.3.m3.1.1.cmml" type="integer" xref="S3.I4.i3.p1.3.m3.1.1">1</cn></interval><apply id="S3.I4.i3.p1.3.m3.2.2.3.cmml" xref="S3.I4.i3.p1.3.m3.2.2.3"><csymbol cd="ambiguous" id="S3.I4.i3.p1.3.m3.2.2.3.1.cmml" xref="S3.I4.i3.p1.3.m3.2.2.3">subscript</csymbol><ci id="S3.I4.i3.p1.3.m3.2.2.3.2.cmml" xref="S3.I4.i3.p1.3.m3.2.2.3.2">𝑑</ci><ci id="S3.I4.i3.p1.3.m3.2.2.3.3a.cmml" xref="S3.I4.i3.p1.3.m3.2.2.3.3"><mtext id="S3.I4.i3.p1.3.m3.2.2.3.3.cmml" mathsize="50%" xref="S3.I4.i3.p1.3.m3.2.2.3.3">cont</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i3.p1.3.m3.2c">[-1,1]^{d_{\text{cont}}}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i3.p1.3.m3.2d">[ - 1 , 1 ] start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT cont end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. To resolve this issue, we apply a function that has the effect of spreading out points in the unit sphere more evenly inside <math alttext="[-1,1]^{d_{\text{cont}}}" class="ltx_Math" display="inline" id="S3.I4.i3.p1.4.m4.2"><semantics id="S3.I4.i3.p1.4.m4.2a"><msup id="S3.I4.i3.p1.4.m4.2.2" xref="S3.I4.i3.p1.4.m4.2.2.cmml"><mrow id="S3.I4.i3.p1.4.m4.2.2.1.1" xref="S3.I4.i3.p1.4.m4.2.2.1.2.cmml"><mo id="S3.I4.i3.p1.4.m4.2.2.1.1.2" stretchy="false" xref="S3.I4.i3.p1.4.m4.2.2.1.2.cmml">[</mo><mrow id="S3.I4.i3.p1.4.m4.2.2.1.1.1" xref="S3.I4.i3.p1.4.m4.2.2.1.1.1.cmml"><mo id="S3.I4.i3.p1.4.m4.2.2.1.1.1a" xref="S3.I4.i3.p1.4.m4.2.2.1.1.1.cmml">−</mo><mn id="S3.I4.i3.p1.4.m4.2.2.1.1.1.2" xref="S3.I4.i3.p1.4.m4.2.2.1.1.1.2.cmml">1</mn></mrow><mo id="S3.I4.i3.p1.4.m4.2.2.1.1.3" xref="S3.I4.i3.p1.4.m4.2.2.1.2.cmml">,</mo><mn id="S3.I4.i3.p1.4.m4.1.1" xref="S3.I4.i3.p1.4.m4.1.1.cmml">1</mn><mo id="S3.I4.i3.p1.4.m4.2.2.1.1.4" stretchy="false" xref="S3.I4.i3.p1.4.m4.2.2.1.2.cmml">]</mo></mrow><msub id="S3.I4.i3.p1.4.m4.2.2.3" xref="S3.I4.i3.p1.4.m4.2.2.3.cmml"><mi id="S3.I4.i3.p1.4.m4.2.2.3.2" xref="S3.I4.i3.p1.4.m4.2.2.3.2.cmml">d</mi><mtext id="S3.I4.i3.p1.4.m4.2.2.3.3" xref="S3.I4.i3.p1.4.m4.2.2.3.3a.cmml">cont</mtext></msub></msup><annotation-xml encoding="MathML-Content" id="S3.I4.i3.p1.4.m4.2b"><apply id="S3.I4.i3.p1.4.m4.2.2.cmml" xref="S3.I4.i3.p1.4.m4.2.2"><csymbol cd="ambiguous" id="S3.I4.i3.p1.4.m4.2.2.2.cmml" xref="S3.I4.i3.p1.4.m4.2.2">superscript</csymbol><interval closure="closed" id="S3.I4.i3.p1.4.m4.2.2.1.2.cmml" xref="S3.I4.i3.p1.4.m4.2.2.1.1"><apply id="S3.I4.i3.p1.4.m4.2.2.1.1.1.cmml" xref="S3.I4.i3.p1.4.m4.2.2.1.1.1"><minus id="S3.I4.i3.p1.4.m4.2.2.1.1.1.1.cmml" xref="S3.I4.i3.p1.4.m4.2.2.1.1.1"></minus><cn id="S3.I4.i3.p1.4.m4.2.2.1.1.1.2.cmml" type="integer" xref="S3.I4.i3.p1.4.m4.2.2.1.1.1.2">1</cn></apply><cn id="S3.I4.i3.p1.4.m4.1.1.cmml" type="integer" xref="S3.I4.i3.p1.4.m4.1.1">1</cn></interval><apply id="S3.I4.i3.p1.4.m4.2.2.3.cmml" xref="S3.I4.i3.p1.4.m4.2.2.3"><csymbol cd="ambiguous" id="S3.I4.i3.p1.4.m4.2.2.3.1.cmml" xref="S3.I4.i3.p1.4.m4.2.2.3">subscript</csymbol><ci id="S3.I4.i3.p1.4.m4.2.2.3.2.cmml" xref="S3.I4.i3.p1.4.m4.2.2.3.2">𝑑</ci><ci id="S3.I4.i3.p1.4.m4.2.2.3.3a.cmml" xref="S3.I4.i3.p1.4.m4.2.2.3.3"><mtext id="S3.I4.i3.p1.4.m4.2.2.3.3.cmml" mathsize="50%" xref="S3.I4.i3.p1.4.m4.2.2.3.3">cont</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i3.p1.4.m4.2c">[-1,1]^{d_{\text{cont}}}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i3.p1.4.m4.2d">[ - 1 , 1 ] start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT cont end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. In our implementation, we use <math alttext="\sin(0.5\pi x)" class="ltx_Math" display="inline" id="S3.I4.i3.p1.5.m5.2"><semantics id="S3.I4.i3.p1.5.m5.2a"><mrow id="S3.I4.i3.p1.5.m5.2.2.1" xref="S3.I4.i3.p1.5.m5.2.2.2.cmml"><mi id="S3.I4.i3.p1.5.m5.1.1" xref="S3.I4.i3.p1.5.m5.1.1.cmml">sin</mi><mo id="S3.I4.i3.p1.5.m5.2.2.1a" xref="S3.I4.i3.p1.5.m5.2.2.2.cmml">⁡</mo><mrow id="S3.I4.i3.p1.5.m5.2.2.1.1" xref="S3.I4.i3.p1.5.m5.2.2.2.cmml"><mo id="S3.I4.i3.p1.5.m5.2.2.1.1.2" stretchy="false" xref="S3.I4.i3.p1.5.m5.2.2.2.cmml">(</mo><mrow id="S3.I4.i3.p1.5.m5.2.2.1.1.1" xref="S3.I4.i3.p1.5.m5.2.2.1.1.1.cmml"><mn id="S3.I4.i3.p1.5.m5.2.2.1.1.1.2" xref="S3.I4.i3.p1.5.m5.2.2.1.1.1.2.cmml">0.5</mn><mo id="S3.I4.i3.p1.5.m5.2.2.1.1.1.1" xref="S3.I4.i3.p1.5.m5.2.2.1.1.1.1.cmml">⁢</mo><mi id="S3.I4.i3.p1.5.m5.2.2.1.1.1.3" xref="S3.I4.i3.p1.5.m5.2.2.1.1.1.3.cmml">π</mi><mo id="S3.I4.i3.p1.5.m5.2.2.1.1.1.1a" xref="S3.I4.i3.p1.5.m5.2.2.1.1.1.1.cmml">⁢</mo><mi id="S3.I4.i3.p1.5.m5.2.2.1.1.1.4" xref="S3.I4.i3.p1.5.m5.2.2.1.1.1.4.cmml">x</mi></mrow><mo id="S3.I4.i3.p1.5.m5.2.2.1.1.3" stretchy="false" xref="S3.I4.i3.p1.5.m5.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i3.p1.5.m5.2b"><apply id="S3.I4.i3.p1.5.m5.2.2.2.cmml" xref="S3.I4.i3.p1.5.m5.2.2.1"><sin id="S3.I4.i3.p1.5.m5.1.1.cmml" xref="S3.I4.i3.p1.5.m5.1.1"></sin><apply id="S3.I4.i3.p1.5.m5.2.2.1.1.1.cmml" xref="S3.I4.i3.p1.5.m5.2.2.1.1.1"><times id="S3.I4.i3.p1.5.m5.2.2.1.1.1.1.cmml" xref="S3.I4.i3.p1.5.m5.2.2.1.1.1.1"></times><cn id="S3.I4.i3.p1.5.m5.2.2.1.1.1.2.cmml" type="float" xref="S3.I4.i3.p1.5.m5.2.2.1.1.1.2">0.5</cn><ci id="S3.I4.i3.p1.5.m5.2.2.1.1.1.3.cmml" xref="S3.I4.i3.p1.5.m5.2.2.1.1.1.3">𝜋</ci><ci id="S3.I4.i3.p1.5.m5.2.2.1.1.1.4.cmml" xref="S3.I4.i3.p1.5.m5.2.2.1.1.1.4">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i3.p1.5.m5.2c">\sin(0.5\pi x)</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i3.p1.5.m5.2d">roman_sin ( 0.5 italic_π italic_x )</annotation></semantics></math> component-wise, but other options like <math alttext="\text{arctan}(bx)/\text{arctan}(b)" class="ltx_Math" display="inline" id="S3.I4.i3.p1.6.m6.2"><semantics id="S3.I4.i3.p1.6.m6.2a"><mrow id="S3.I4.i3.p1.6.m6.2.2" xref="S3.I4.i3.p1.6.m6.2.2.cmml"><mrow id="S3.I4.i3.p1.6.m6.2.2.1" xref="S3.I4.i3.p1.6.m6.2.2.1.cmml"><mrow id="S3.I4.i3.p1.6.m6.2.2.1.1" xref="S3.I4.i3.p1.6.m6.2.2.1.1.cmml"><mtext id="S3.I4.i3.p1.6.m6.2.2.1.1.3" xref="S3.I4.i3.p1.6.m6.2.2.1.1.3a.cmml">arctan</mtext><mo id="S3.I4.i3.p1.6.m6.2.2.1.1.2" xref="S3.I4.i3.p1.6.m6.2.2.1.1.2.cmml">⁢</mo><mrow id="S3.I4.i3.p1.6.m6.2.2.1.1.1.1" xref="S3.I4.i3.p1.6.m6.2.2.1.1.1.1.1.cmml"><mo id="S3.I4.i3.p1.6.m6.2.2.1.1.1.1.2" stretchy="false" xref="S3.I4.i3.p1.6.m6.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S3.I4.i3.p1.6.m6.2.2.1.1.1.1.1" xref="S3.I4.i3.p1.6.m6.2.2.1.1.1.1.1.cmml"><mi id="S3.I4.i3.p1.6.m6.2.2.1.1.1.1.1.2" 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xref="S3.I4.i3.p1.6.m6.2.2.1.1.1.1.1.3">𝑥</ci></apply></apply><ci id="S3.I4.i3.p1.6.m6.2.2.1.3a.cmml" xref="S3.I4.i3.p1.6.m6.2.2.1.3"><mtext id="S3.I4.i3.p1.6.m6.2.2.1.3.cmml" xref="S3.I4.i3.p1.6.m6.2.2.1.3">arctan</mtext></ci></apply><ci id="S3.I4.i3.p1.6.m6.1.1.cmml" xref="S3.I4.i3.p1.6.m6.1.1">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i3.p1.6.m6.2c">\text{arctan}(bx)/\text{arctan}(b)</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i3.p1.6.m6.2d">arctan ( italic_b italic_x ) / arctan ( italic_b )</annotation></semantics></math> for <math alttext="b>0" class="ltx_Math" display="inline" id="S3.I4.i3.p1.7.m7.1"><semantics id="S3.I4.i3.p1.7.m7.1a"><mrow id="S3.I4.i3.p1.7.m7.1.1" xref="S3.I4.i3.p1.7.m7.1.1.cmml"><mi id="S3.I4.i3.p1.7.m7.1.1.2" xref="S3.I4.i3.p1.7.m7.1.1.2.cmml">b</mi><mo id="S3.I4.i3.p1.7.m7.1.1.1" xref="S3.I4.i3.p1.7.m7.1.1.1.cmml">></mo><mn id="S3.I4.i3.p1.7.m7.1.1.3" xref="S3.I4.i3.p1.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i3.p1.7.m7.1b"><apply id="S3.I4.i3.p1.7.m7.1.1.cmml" xref="S3.I4.i3.p1.7.m7.1.1"><gt id="S3.I4.i3.p1.7.m7.1.1.1.cmml" xref="S3.I4.i3.p1.7.m7.1.1.1"></gt><ci id="S3.I4.i3.p1.7.m7.1.1.2.cmml" xref="S3.I4.i3.p1.7.m7.1.1.2">𝑏</ci><cn id="S3.I4.i3.p1.7.m7.1.1.3.cmml" type="integer" xref="S3.I4.i3.p1.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i3.p1.7.m7.1c">b>0</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i3.p1.7.m7.1d">italic_b > 0</annotation></semantics></math> or the ones discussed in <cite class="ltx_cite ltx_citemacro_cite">Andrews and Shi, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib1" title="">2013</a>)</cite> would also be possible.</p> </div> </li> <li class="ltx_item" id="S3.I4.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">4.</span> <div class="ltx_para" id="S3.I4.i4.p1"> <p class="ltx_p" id="S3.I4.i4.p1.1">Finally, the points resulting from the previous step are scaled and shifted to <math alttext="[0,1]^{d_{\text{cont}}}" class="ltx_Math" display="inline" id="S3.I4.i4.p1.1.m1.2"><semantics id="S3.I4.i4.p1.1.m1.2a"><msup id="S3.I4.i4.p1.1.m1.2.3" xref="S3.I4.i4.p1.1.m1.2.3.cmml"><mrow id="S3.I4.i4.p1.1.m1.2.3.2.2" xref="S3.I4.i4.p1.1.m1.2.3.2.1.cmml"><mo id="S3.I4.i4.p1.1.m1.2.3.2.2.1" stretchy="false" xref="S3.I4.i4.p1.1.m1.2.3.2.1.cmml">[</mo><mn id="S3.I4.i4.p1.1.m1.1.1" xref="S3.I4.i4.p1.1.m1.1.1.cmml">0</mn><mo id="S3.I4.i4.p1.1.m1.2.3.2.2.2" xref="S3.I4.i4.p1.1.m1.2.3.2.1.cmml">,</mo><mn id="S3.I4.i4.p1.1.m1.2.2" xref="S3.I4.i4.p1.1.m1.2.2.cmml">1</mn><mo id="S3.I4.i4.p1.1.m1.2.3.2.2.3" stretchy="false" xref="S3.I4.i4.p1.1.m1.2.3.2.1.cmml">]</mo></mrow><msub id="S3.I4.i4.p1.1.m1.2.3.3" xref="S3.I4.i4.p1.1.m1.2.3.3.cmml"><mi id="S3.I4.i4.p1.1.m1.2.3.3.2" xref="S3.I4.i4.p1.1.m1.2.3.3.2.cmml">d</mi><mtext id="S3.I4.i4.p1.1.m1.2.3.3.3" xref="S3.I4.i4.p1.1.m1.2.3.3.3a.cmml">cont</mtext></msub></msup><annotation-xml encoding="MathML-Content" id="S3.I4.i4.p1.1.m1.2b"><apply id="S3.I4.i4.p1.1.m1.2.3.cmml" xref="S3.I4.i4.p1.1.m1.2.3"><csymbol cd="ambiguous" id="S3.I4.i4.p1.1.m1.2.3.1.cmml" xref="S3.I4.i4.p1.1.m1.2.3">superscript</csymbol><interval closure="closed" id="S3.I4.i4.p1.1.m1.2.3.2.1.cmml" xref="S3.I4.i4.p1.1.m1.2.3.2.2"><cn id="S3.I4.i4.p1.1.m1.1.1.cmml" type="integer" xref="S3.I4.i4.p1.1.m1.1.1">0</cn><cn id="S3.I4.i4.p1.1.m1.2.2.cmml" type="integer" xref="S3.I4.i4.p1.1.m1.2.2">1</cn></interval><apply id="S3.I4.i4.p1.1.m1.2.3.3.cmml" xref="S3.I4.i4.p1.1.m1.2.3.3"><csymbol cd="ambiguous" id="S3.I4.i4.p1.1.m1.2.3.3.1.cmml" xref="S3.I4.i4.p1.1.m1.2.3.3">subscript</csymbol><ci id="S3.I4.i4.p1.1.m1.2.3.3.2.cmml" xref="S3.I4.i4.p1.1.m1.2.3.3.2">𝑑</ci><ci id="S3.I4.i4.p1.1.m1.2.3.3.3a.cmml" xref="S3.I4.i4.p1.1.m1.2.3.3.3"><mtext id="S3.I4.i4.p1.1.m1.2.3.3.3.cmml" mathsize="50%" xref="S3.I4.i4.p1.1.m1.2.3.3.3">cont</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i4.p1.1.m1.2c">[0,1]^{d_{\text{cont}}}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i4.p1.1.m1.2d">[ 0 , 1 ] start_POSTSUPERSCRIPT italic_d start_POSTSUBSCRIPT cont end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S3.SS2.p6.5">Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Instrumental_functions</span> provides several remarks on handling dependent categorical covariates and on the dimensionality of <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S3.SS2.p6.5.m1.1"><semantics id="S3.SS2.p6.5.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p6.5.m1.1.1" xref="S3.SS2.p6.5.m1.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.5.m1.1b"><ci id="S3.SS2.p6.5.m1.1.1.cmml" 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Middle panel: the transformed space using a component-wise min-max scaler. The gray, dashed grid depicts the supports of the considered instrumental functions. The shaded rectangles highlight the supports of two instrumental functions that fall entirely outside the transformed covariate space. Right panel: the transformed space using the PCA-based transformation.</span></figcaption> </figure> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.3 </span>Time-independent effects of covariates</h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.1">In practice, it is often assumed that covariate effects do not change over time, as is for example the case in the traditional Cox proportional hazards model. It is therefore of great interest to study the proposed methodology under this additional information and, more specifically, to investigate if narrower bounds on the covariate effects can be obtained.</p> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.6">A straightforward modification of the methodology to a setting that includes the assumption of a time-independent covariate effect <math alttext="\beta_{k}" class="ltx_Math" display="inline" id="S3.SS3.p2.1.m1.1"><semantics id="S3.SS3.p2.1.m1.1a"><msub id="S3.SS3.p2.1.m1.1.1" xref="S3.SS3.p2.1.m1.1.1.cmml"><mi id="S3.SS3.p2.1.m1.1.1.2" xref="S3.SS3.p2.1.m1.1.1.2.cmml">β</mi><mi id="S3.SS3.p2.1.m1.1.1.3" xref="S3.SS3.p2.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.1.m1.1b"><apply id="S3.SS3.p2.1.m1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.1.m1.1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.p2.1.m1.1.1.2.cmml" xref="S3.SS3.p2.1.m1.1.1.2">𝛽</ci><ci id="S3.SS3.p2.1.m1.1.1.3.cmml" xref="S3.SS3.p2.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.1.m1.1c">\beta_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.1.m1.1d">italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> can be made by simply estimating the identified intervals of <math alttext="\beta_{k}" class="ltx_Math" display="inline" id="S3.SS3.p2.2.m2.1"><semantics id="S3.SS3.p2.2.m2.1a"><msub id="S3.SS3.p2.2.m2.1.1" xref="S3.SS3.p2.2.m2.1.1.cmml"><mi id="S3.SS3.p2.2.m2.1.1.2" xref="S3.SS3.p2.2.m2.1.1.2.cmml">β</mi><mi id="S3.SS3.p2.2.m2.1.1.3" xref="S3.SS3.p2.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.2.m2.1b"><apply id="S3.SS3.p2.2.m2.1.1.cmml" xref="S3.SS3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.2.m2.1.1.1.cmml" xref="S3.SS3.p2.2.m2.1.1">subscript</csymbol><ci id="S3.SS3.p2.2.m2.1.1.2.cmml" xref="S3.SS3.p2.2.m2.1.1.2">𝛽</ci><ci id="S3.SS3.p2.2.m2.1.1.3.cmml" xref="S3.SS3.p2.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.2.m2.1c">\beta_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.2.m2.1d">italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> at multiple time points and later on combining all estimated intervals by means of intersection or majority vote. Specifically, suppose <math alttext="A" class="ltx_Math" display="inline" id="S3.SS3.p2.3.m3.1"><semantics id="S3.SS3.p2.3.m3.1a"><mi id="S3.SS3.p2.3.m3.1.1" xref="S3.SS3.p2.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.3.m3.1b"><ci id="S3.SS3.p2.3.m3.1.1.cmml" xref="S3.SS3.p2.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.3.m3.1d">italic_A</annotation></semantics></math> identified intervals <math alttext="\hat{\mathcal{B}}_{I,k}^{1},\dots,\hat{\mathcal{B}}_{I,k}^{A}" class="ltx_Math" display="inline" id="S3.SS3.p2.4.m4.7"><semantics id="S3.SS3.p2.4.m4.7a"><mrow id="S3.SS3.p2.4.m4.7.7.2" xref="S3.SS3.p2.4.m4.7.7.3.cmml"><msubsup id="S3.SS3.p2.4.m4.6.6.1.1" xref="S3.SS3.p2.4.m4.6.6.1.1.cmml"><mover accent="true" id="S3.SS3.p2.4.m4.6.6.1.1.2.2" xref="S3.SS3.p2.4.m4.6.6.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" 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start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , … , over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT</annotation></semantics></math> are estimated at different time points <math alttext="t_{1},\dots,t_{A}" class="ltx_Math" display="inline" id="S3.SS3.p2.5.m5.3"><semantics id="S3.SS3.p2.5.m5.3a"><mrow id="S3.SS3.p2.5.m5.3.3.2" xref="S3.SS3.p2.5.m5.3.3.3.cmml"><msub id="S3.SS3.p2.5.m5.2.2.1.1" xref="S3.SS3.p2.5.m5.2.2.1.1.cmml"><mi id="S3.SS3.p2.5.m5.2.2.1.1.2" xref="S3.SS3.p2.5.m5.2.2.1.1.2.cmml">t</mi><mn id="S3.SS3.p2.5.m5.2.2.1.1.3" xref="S3.SS3.p2.5.m5.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.SS3.p2.5.m5.3.3.2.3" xref="S3.SS3.p2.5.m5.3.3.3.cmml">,</mo><mi id="S3.SS3.p2.5.m5.1.1" mathvariant="normal" xref="S3.SS3.p2.5.m5.1.1.cmml">…</mi><mo id="S3.SS3.p2.5.m5.3.3.2.4" xref="S3.SS3.p2.5.m5.3.3.3.cmml">,</mo><msub id="S3.SS3.p2.5.m5.3.3.2.2" xref="S3.SS3.p2.5.m5.3.3.2.2.cmml"><mi id="S3.SS3.p2.5.m5.3.3.2.2.2" xref="S3.SS3.p2.5.m5.3.3.2.2.2.cmml">t</mi><mi id="S3.SS3.p2.5.m5.3.3.2.2.3" xref="S3.SS3.p2.5.m5.3.3.2.2.3.cmml">A</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.5.m5.3b"><list id="S3.SS3.p2.5.m5.3.3.3.cmml" xref="S3.SS3.p2.5.m5.3.3.2"><apply id="S3.SS3.p2.5.m5.2.2.1.1.cmml" xref="S3.SS3.p2.5.m5.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.5.m5.2.2.1.1.1.cmml" xref="S3.SS3.p2.5.m5.2.2.1.1">subscript</csymbol><ci id="S3.SS3.p2.5.m5.2.2.1.1.2.cmml" xref="S3.SS3.p2.5.m5.2.2.1.1.2">𝑡</ci><cn id="S3.SS3.p2.5.m5.2.2.1.1.3.cmml" type="integer" xref="S3.SS3.p2.5.m5.2.2.1.1.3">1</cn></apply><ci id="S3.SS3.p2.5.m5.1.1.cmml" xref="S3.SS3.p2.5.m5.1.1">…</ci><apply id="S3.SS3.p2.5.m5.3.3.2.2.cmml" xref="S3.SS3.p2.5.m5.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS3.p2.5.m5.3.3.2.2.1.cmml" xref="S3.SS3.p2.5.m5.3.3.2.2">subscript</csymbol><ci id="S3.SS3.p2.5.m5.3.3.2.2.2.cmml" xref="S3.SS3.p2.5.m5.3.3.2.2.2">𝑡</ci><ci id="S3.SS3.p2.5.m5.3.3.2.2.3.cmml" xref="S3.SS3.p2.5.m5.3.3.2.2.3">𝐴</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.5.m5.3c">t_{1},\dots,t_{A}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.5.m5.3d">italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_t start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math>. If it is assumed that <math alttext="\beta_{k}" class="ltx_Math" display="inline" id="S3.SS3.p2.6.m6.1"><semantics id="S3.SS3.p2.6.m6.1a"><msub id="S3.SS3.p2.6.m6.1.1" xref="S3.SS3.p2.6.m6.1.1.cmml"><mi id="S3.SS3.p2.6.m6.1.1.2" xref="S3.SS3.p2.6.m6.1.1.2.cmml">β</mi><mi id="S3.SS3.p2.6.m6.1.1.3" xref="S3.SS3.p2.6.m6.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.6.m6.1b"><apply id="S3.SS3.p2.6.m6.1.1.cmml" xref="S3.SS3.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.6.m6.1.1.1.cmml" xref="S3.SS3.p2.6.m6.1.1">subscript</csymbol><ci id="S3.SS3.p2.6.m6.1.1.2.cmml" xref="S3.SS3.p2.6.m6.1.1.2">𝛽</ci><ci id="S3.SS3.p2.6.m6.1.1.3.cmml" xref="S3.SS3.p2.6.m6.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.6.m6.1c">\beta_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.6.m6.1d">italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is independent of time, each of these identified intervals pertains to precisely the same quantity, and hence it is sensible to combine them via intersection:</p> <table class="ltx_equation ltx_eqn_table" id="S3.E11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\hat{\mathcal{B}}_{I,k}=\bigcap_{a=1}^{A}\hat{\mathcal{B}}_{I,k}^{a}." class="ltx_Math" display="block" id="S3.E11.m1.5"><semantics id="S3.E11.m1.5a"><mrow id="S3.E11.m1.5.5.1" xref="S3.E11.m1.5.5.1.1.cmml"><mrow id="S3.E11.m1.5.5.1.1" xref="S3.E11.m1.5.5.1.1.cmml"><msub id="S3.E11.m1.5.5.1.1.2" xref="S3.E11.m1.5.5.1.1.2.cmml"><mover accent="true" id="S3.E11.m1.5.5.1.1.2.2" xref="S3.E11.m1.5.5.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E11.m1.5.5.1.1.2.2.2" xref="S3.E11.m1.5.5.1.1.2.2.2.cmml">ℬ</mi><mo id="S3.E11.m1.5.5.1.1.2.2.1" xref="S3.E11.m1.5.5.1.1.2.2.1.cmml">^</mo></mover><mrow id="S3.E11.m1.2.2.2.4" xref="S3.E11.m1.2.2.2.3.cmml"><mi id="S3.E11.m1.1.1.1.1" xref="S3.E11.m1.1.1.1.1.cmml">I</mi><mo id="S3.E11.m1.2.2.2.4.1" xref="S3.E11.m1.2.2.2.3.cmml">,</mo><mi id="S3.E11.m1.2.2.2.2" xref="S3.E11.m1.2.2.2.2.cmml">k</mi></mrow></msub><mo id="S3.E11.m1.5.5.1.1.1" rspace="0.111em" xref="S3.E11.m1.5.5.1.1.1.cmml">=</mo><mrow id="S3.E11.m1.5.5.1.1.3" xref="S3.E11.m1.5.5.1.1.3.cmml"><munderover id="S3.E11.m1.5.5.1.1.3.1" xref="S3.E11.m1.5.5.1.1.3.1.cmml"><mo id="S3.E11.m1.5.5.1.1.3.1.2.2" movablelimits="false" xref="S3.E11.m1.5.5.1.1.3.1.2.2.cmml">⋂</mo><mrow id="S3.E11.m1.5.5.1.1.3.1.2.3" xref="S3.E11.m1.5.5.1.1.3.1.2.3.cmml"><mi id="S3.E11.m1.5.5.1.1.3.1.2.3.2" xref="S3.E11.m1.5.5.1.1.3.1.2.3.2.cmml">a</mi><mo id="S3.E11.m1.5.5.1.1.3.1.2.3.1" xref="S3.E11.m1.5.5.1.1.3.1.2.3.1.cmml">=</mo><mn id="S3.E11.m1.5.5.1.1.3.1.2.3.3" xref="S3.E11.m1.5.5.1.1.3.1.2.3.3.cmml">1</mn></mrow><mi id="S3.E11.m1.5.5.1.1.3.1.3" xref="S3.E11.m1.5.5.1.1.3.1.3.cmml">A</mi></munderover><msubsup id="S3.E11.m1.5.5.1.1.3.2" xref="S3.E11.m1.5.5.1.1.3.2.cmml"><mover accent="true" id="S3.E11.m1.5.5.1.1.3.2.2.2" xref="S3.E11.m1.5.5.1.1.3.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E11.m1.5.5.1.1.3.2.2.2.2" xref="S3.E11.m1.5.5.1.1.3.2.2.2.2.cmml">ℬ</mi><mo id="S3.E11.m1.5.5.1.1.3.2.2.2.1" xref="S3.E11.m1.5.5.1.1.3.2.2.2.1.cmml">^</mo></mover><mrow id="S3.E11.m1.4.4.2.4" xref="S3.E11.m1.4.4.2.3.cmml"><mi id="S3.E11.m1.3.3.1.1" xref="S3.E11.m1.3.3.1.1.cmml">I</mi><mo id="S3.E11.m1.4.4.2.4.1" xref="S3.E11.m1.4.4.2.3.cmml">,</mo><mi id="S3.E11.m1.4.4.2.2" xref="S3.E11.m1.4.4.2.2.cmml">k</mi></mrow><mi id="S3.E11.m1.5.5.1.1.3.2.3" xref="S3.E11.m1.5.5.1.1.3.2.3.cmml">a</mi></msubsup></mrow></mrow><mo id="S3.E11.m1.5.5.1.2" lspace="0em" xref="S3.E11.m1.5.5.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E11.m1.5b"><apply id="S3.E11.m1.5.5.1.1.cmml" xref="S3.E11.m1.5.5.1"><eq id="S3.E11.m1.5.5.1.1.1.cmml" xref="S3.E11.m1.5.5.1.1.1"></eq><apply 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xref="S3.E11.m1.5.5.1.1.3.2.2.2"><ci id="S3.E11.m1.5.5.1.1.3.2.2.2.1.cmml" xref="S3.E11.m1.5.5.1.1.3.2.2.2.1">^</ci><ci id="S3.E11.m1.5.5.1.1.3.2.2.2.2.cmml" xref="S3.E11.m1.5.5.1.1.3.2.2.2.2">ℬ</ci></apply><list id="S3.E11.m1.4.4.2.3.cmml" xref="S3.E11.m1.4.4.2.4"><ci id="S3.E11.m1.3.3.1.1.cmml" xref="S3.E11.m1.3.3.1.1">𝐼</ci><ci id="S3.E11.m1.4.4.2.2.cmml" xref="S3.E11.m1.4.4.2.2">𝑘</ci></list></apply><ci id="S3.E11.m1.5.5.1.1.3.2.3.cmml" xref="S3.E11.m1.5.5.1.1.3.2.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E11.m1.5c">\hat{\mathcal{B}}_{I,k}=\bigcap_{a=1}^{A}\hat{\mathcal{B}}_{I,k}^{a}.</annotation><annotation encoding="application/x-llamapun" id="S3.E11.m1.5d">over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT = ⋂ start_POSTSUBSCRIPT italic_a = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.p2.12">To obtain a correct level <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS3.p2.7.m1.1"><semantics id="S3.SS3.p2.7.m1.1a"><mi id="S3.SS3.p2.7.m1.1.1" xref="S3.SS3.p2.7.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.7.m1.1b"><ci id="S3.SS3.p2.7.m1.1.1.cmml" xref="S3.SS3.p2.7.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.7.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.7.m1.1d">italic_α</annotation></semantics></math> of <math alttext="\hat{\mathcal{B}}_{I,k}" class="ltx_Math" display="inline" id="S3.SS3.p2.8.m2.2"><semantics id="S3.SS3.p2.8.m2.2a"><msub id="S3.SS3.p2.8.m2.2.3" xref="S3.SS3.p2.8.m2.2.3.cmml"><mover accent="true" id="S3.SS3.p2.8.m2.2.3.2" xref="S3.SS3.p2.8.m2.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.8.m2.2.3.2.2" xref="S3.SS3.p2.8.m2.2.3.2.2.cmml">ℬ</mi><mo id="S3.SS3.p2.8.m2.2.3.2.1" xref="S3.SS3.p2.8.m2.2.3.2.1.cmml">^</mo></mover><mrow id="S3.SS3.p2.8.m2.2.2.2.4" xref="S3.SS3.p2.8.m2.2.2.2.3.cmml"><mi id="S3.SS3.p2.8.m2.1.1.1.1" xref="S3.SS3.p2.8.m2.1.1.1.1.cmml">I</mi><mo id="S3.SS3.p2.8.m2.2.2.2.4.1" xref="S3.SS3.p2.8.m2.2.2.2.3.cmml">,</mo><mi id="S3.SS3.p2.8.m2.2.2.2.2" xref="S3.SS3.p2.8.m2.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.8.m2.2b"><apply id="S3.SS3.p2.8.m2.2.3.cmml" xref="S3.SS3.p2.8.m2.2.3"><csymbol cd="ambiguous" id="S3.SS3.p2.8.m2.2.3.1.cmml" xref="S3.SS3.p2.8.m2.2.3">subscript</csymbol><apply id="S3.SS3.p2.8.m2.2.3.2.cmml" xref="S3.SS3.p2.8.m2.2.3.2"><ci id="S3.SS3.p2.8.m2.2.3.2.1.cmml" xref="S3.SS3.p2.8.m2.2.3.2.1">^</ci><ci id="S3.SS3.p2.8.m2.2.3.2.2.cmml" xref="S3.SS3.p2.8.m2.2.3.2.2">ℬ</ci></apply><list id="S3.SS3.p2.8.m2.2.2.2.3.cmml" xref="S3.SS3.p2.8.m2.2.2.2.4"><ci id="S3.SS3.p2.8.m2.1.1.1.1.cmml" xref="S3.SS3.p2.8.m2.1.1.1.1">𝐼</ci><ci id="S3.SS3.p2.8.m2.2.2.2.2.cmml" xref="S3.SS3.p2.8.m2.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.8.m2.2c">\hat{\mathcal{B}}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.8.m2.2d">over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, it is required that each separate identified interval is estimated with Bonferroni corrected confidence level <math alttext="1-\alpha/A" class="ltx_Math" display="inline" id="S3.SS3.p2.9.m3.1"><semantics id="S3.SS3.p2.9.m3.1a"><mrow id="S3.SS3.p2.9.m3.1.1" xref="S3.SS3.p2.9.m3.1.1.cmml"><mn id="S3.SS3.p2.9.m3.1.1.2" xref="S3.SS3.p2.9.m3.1.1.2.cmml">1</mn><mo id="S3.SS3.p2.9.m3.1.1.1" xref="S3.SS3.p2.9.m3.1.1.1.cmml">−</mo><mrow id="S3.SS3.p2.9.m3.1.1.3" xref="S3.SS3.p2.9.m3.1.1.3.cmml"><mi id="S3.SS3.p2.9.m3.1.1.3.2" xref="S3.SS3.p2.9.m3.1.1.3.2.cmml">α</mi><mo id="S3.SS3.p2.9.m3.1.1.3.1" xref="S3.SS3.p2.9.m3.1.1.3.1.cmml">/</mo><mi id="S3.SS3.p2.9.m3.1.1.3.3" xref="S3.SS3.p2.9.m3.1.1.3.3.cmml">A</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.9.m3.1b"><apply id="S3.SS3.p2.9.m3.1.1.cmml" xref="S3.SS3.p2.9.m3.1.1"><minus id="S3.SS3.p2.9.m3.1.1.1.cmml" xref="S3.SS3.p2.9.m3.1.1.1"></minus><cn id="S3.SS3.p2.9.m3.1.1.2.cmml" type="integer" xref="S3.SS3.p2.9.m3.1.1.2">1</cn><apply id="S3.SS3.p2.9.m3.1.1.3.cmml" xref="S3.SS3.p2.9.m3.1.1.3"><divide id="S3.SS3.p2.9.m3.1.1.3.1.cmml" xref="S3.SS3.p2.9.m3.1.1.3.1"></divide><ci id="S3.SS3.p2.9.m3.1.1.3.2.cmml" xref="S3.SS3.p2.9.m3.1.1.3.2">𝛼</ci><ci id="S3.SS3.p2.9.m3.1.1.3.3.cmml" xref="S3.SS3.p2.9.m3.1.1.3.3">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.9.m3.1c">1-\alpha/A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.9.m3.1d">1 - italic_α / italic_A</annotation></semantics></math>. When many time points are selected – i.e., when <math alttext="A" class="ltx_Math" display="inline" id="S3.SS3.p2.10.m4.1"><semantics id="S3.SS3.p2.10.m4.1a"><mi id="S3.SS3.p2.10.m4.1.1" xref="S3.SS3.p2.10.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.10.m4.1b"><ci id="S3.SS3.p2.10.m4.1.1.cmml" xref="S3.SS3.p2.10.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.10.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.10.m4.1d">italic_A</annotation></semantics></math> is large – the level of each identified interval will be very small, so that the identified intervals <math alttext="\hat{\mathcal{B}}_{I,k}^{a}" class="ltx_Math" display="inline" id="S3.SS3.p2.11.m5.2"><semantics id="S3.SS3.p2.11.m5.2a"><msubsup id="S3.SS3.p2.11.m5.2.3" xref="S3.SS3.p2.11.m5.2.3.cmml"><mover accent="true" id="S3.SS3.p2.11.m5.2.3.2.2" xref="S3.SS3.p2.11.m5.2.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.11.m5.2.3.2.2.2" xref="S3.SS3.p2.11.m5.2.3.2.2.2.cmml">ℬ</mi><mo id="S3.SS3.p2.11.m5.2.3.2.2.1" xref="S3.SS3.p2.11.m5.2.3.2.2.1.cmml">^</mo></mover><mrow id="S3.SS3.p2.11.m5.2.2.2.4" xref="S3.SS3.p2.11.m5.2.2.2.3.cmml"><mi id="S3.SS3.p2.11.m5.1.1.1.1" xref="S3.SS3.p2.11.m5.1.1.1.1.cmml">I</mi><mo id="S3.SS3.p2.11.m5.2.2.2.4.1" xref="S3.SS3.p2.11.m5.2.2.2.3.cmml">,</mo><mi id="S3.SS3.p2.11.m5.2.2.2.2" xref="S3.SS3.p2.11.m5.2.2.2.2.cmml">k</mi></mrow><mi id="S3.SS3.p2.11.m5.2.3.3" xref="S3.SS3.p2.11.m5.2.3.3.cmml">a</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.11.m5.2b"><apply id="S3.SS3.p2.11.m5.2.3.cmml" xref="S3.SS3.p2.11.m5.2.3"><csymbol cd="ambiguous" id="S3.SS3.p2.11.m5.2.3.1.cmml" xref="S3.SS3.p2.11.m5.2.3">superscript</csymbol><apply id="S3.SS3.p2.11.m5.2.3.2.cmml" xref="S3.SS3.p2.11.m5.2.3"><csymbol cd="ambiguous" id="S3.SS3.p2.11.m5.2.3.2.1.cmml" xref="S3.SS3.p2.11.m5.2.3">subscript</csymbol><apply id="S3.SS3.p2.11.m5.2.3.2.2.cmml" xref="S3.SS3.p2.11.m5.2.3.2.2"><ci id="S3.SS3.p2.11.m5.2.3.2.2.1.cmml" xref="S3.SS3.p2.11.m5.2.3.2.2.1">^</ci><ci id="S3.SS3.p2.11.m5.2.3.2.2.2.cmml" xref="S3.SS3.p2.11.m5.2.3.2.2.2">ℬ</ci></apply><list id="S3.SS3.p2.11.m5.2.2.2.3.cmml" xref="S3.SS3.p2.11.m5.2.2.2.4"><ci id="S3.SS3.p2.11.m5.1.1.1.1.cmml" xref="S3.SS3.p2.11.m5.1.1.1.1">𝐼</ci><ci id="S3.SS3.p2.11.m5.2.2.2.2.cmml" xref="S3.SS3.p2.11.m5.2.2.2.2">𝑘</ci></list></apply><ci id="S3.SS3.p2.11.m5.2.3.3.cmml" xref="S3.SS3.p2.11.m5.2.3.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.11.m5.2c">\hat{\mathcal{B}}_{I,k}^{a}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.11.m5.2d">over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT</annotation></semantics></math> might become too wide. In this case, one might use the majority voting rule with threshold <math alttext="1/2" class="ltx_Math" display="inline" id="S3.SS3.p2.12.m6.1"><semantics id="S3.SS3.p2.12.m6.1a"><mrow id="S3.SS3.p2.12.m6.1.1" xref="S3.SS3.p2.12.m6.1.1.cmml"><mn id="S3.SS3.p2.12.m6.1.1.2" xref="S3.SS3.p2.12.m6.1.1.2.cmml">1</mn><mo id="S3.SS3.p2.12.m6.1.1.1" xref="S3.SS3.p2.12.m6.1.1.1.cmml">/</mo><mn id="S3.SS3.p2.12.m6.1.1.3" xref="S3.SS3.p2.12.m6.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.12.m6.1b"><apply id="S3.SS3.p2.12.m6.1.1.cmml" xref="S3.SS3.p2.12.m6.1.1"><divide id="S3.SS3.p2.12.m6.1.1.1.cmml" xref="S3.SS3.p2.12.m6.1.1.1"></divide><cn id="S3.SS3.p2.12.m6.1.1.2.cmml" type="integer" xref="S3.SS3.p2.12.m6.1.1.2">1</cn><cn id="S3.SS3.p2.12.m6.1.1.3.cmml" type="integer" xref="S3.SS3.p2.12.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.12.m6.1c">1/2</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.12.m6.1d">1 / 2</annotation></semantics></math> as described in <cite class="ltx_cite ltx_citemacro_cite">Gasparin and Ramdas, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib17" title="">2024</a>)</cite>:</p> <table class="ltx_equation ltx_eqn_table" id="S3.E12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\hat{\mathcal{B}}_{I,k}=\left\{\beta_{k}\in\mathcal{B}_{k}\mid\frac{1}{A}\sum_% {a=1}^{A}\mathbbm{1}(\beta_{k}\in\hat{\mathcal{B}}_{I,k}^{a})>\frac{1}{2}% \right\}," class="ltx_Math" display="block" id="S3.E12.m1.5"><semantics id="S3.E12.m1.5a"><mrow id="S3.E12.m1.5.5.1" xref="S3.E12.m1.5.5.1.1.cmml"><mrow id="S3.E12.m1.5.5.1.1" xref="S3.E12.m1.5.5.1.1.cmml"><msub id="S3.E12.m1.5.5.1.1.4" xref="S3.E12.m1.5.5.1.1.4.cmml"><mover accent="true" id="S3.E12.m1.5.5.1.1.4.2" xref="S3.E12.m1.5.5.1.1.4.2.cmml"><mi 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end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT = { italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ caligraphic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∣ divide start_ARG 1 end_ARG start_ARG italic_A end_ARG ∑ start_POSTSUBSCRIPT italic_a = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT blackboard_1 ( italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT ) > divide start_ARG 1 end_ARG start_ARG 2 end_ARG } ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.p2.14">where each identified interval <math alttext="\mathcal{B}_{I,k}^{a}" class="ltx_Math" display="inline" id="S3.SS3.p2.13.m1.2"><semantics id="S3.SS3.p2.13.m1.2a"><msubsup id="S3.SS3.p2.13.m1.2.3" xref="S3.SS3.p2.13.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.13.m1.2.3.2.2" xref="S3.SS3.p2.13.m1.2.3.2.2.cmml">ℬ</mi><mrow id="S3.SS3.p2.13.m1.2.2.2.4" xref="S3.SS3.p2.13.m1.2.2.2.3.cmml"><mi id="S3.SS3.p2.13.m1.1.1.1.1" xref="S3.SS3.p2.13.m1.1.1.1.1.cmml">I</mi><mo id="S3.SS3.p2.13.m1.2.2.2.4.1" xref="S3.SS3.p2.13.m1.2.2.2.3.cmml">,</mo><mi id="S3.SS3.p2.13.m1.2.2.2.2" xref="S3.SS3.p2.13.m1.2.2.2.2.cmml">k</mi></mrow><mi id="S3.SS3.p2.13.m1.2.3.3" xref="S3.SS3.p2.13.m1.2.3.3.cmml">a</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.13.m1.2b"><apply id="S3.SS3.p2.13.m1.2.3.cmml" xref="S3.SS3.p2.13.m1.2.3"><csymbol cd="ambiguous" id="S3.SS3.p2.13.m1.2.3.1.cmml" xref="S3.SS3.p2.13.m1.2.3">superscript</csymbol><apply id="S3.SS3.p2.13.m1.2.3.2.cmml" xref="S3.SS3.p2.13.m1.2.3"><csymbol cd="ambiguous" id="S3.SS3.p2.13.m1.2.3.2.1.cmml" xref="S3.SS3.p2.13.m1.2.3">subscript</csymbol><ci id="S3.SS3.p2.13.m1.2.3.2.2.cmml" xref="S3.SS3.p2.13.m1.2.3.2.2">ℬ</ci><list id="S3.SS3.p2.13.m1.2.2.2.3.cmml" xref="S3.SS3.p2.13.m1.2.2.2.4"><ci id="S3.SS3.p2.13.m1.1.1.1.1.cmml" xref="S3.SS3.p2.13.m1.1.1.1.1">𝐼</ci><ci id="S3.SS3.p2.13.m1.2.2.2.2.cmml" xref="S3.SS3.p2.13.m1.2.2.2.2">𝑘</ci></list></apply><ci id="S3.SS3.p2.13.m1.2.3.3.cmml" xref="S3.SS3.p2.13.m1.2.3.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.13.m1.2c">\mathcal{B}_{I,k}^{a}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.13.m1.2d">caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT</annotation></semantics></math> should be estimated only at a confidence level <math alttext="1-\alpha/2" class="ltx_Math" display="inline" id="S3.SS3.p2.14.m2.1"><semantics id="S3.SS3.p2.14.m2.1a"><mrow id="S3.SS3.p2.14.m2.1.1" xref="S3.SS3.p2.14.m2.1.1.cmml"><mn id="S3.SS3.p2.14.m2.1.1.2" xref="S3.SS3.p2.14.m2.1.1.2.cmml">1</mn><mo id="S3.SS3.p2.14.m2.1.1.1" xref="S3.SS3.p2.14.m2.1.1.1.cmml">−</mo><mrow id="S3.SS3.p2.14.m2.1.1.3" xref="S3.SS3.p2.14.m2.1.1.3.cmml"><mi id="S3.SS3.p2.14.m2.1.1.3.2" xref="S3.SS3.p2.14.m2.1.1.3.2.cmml">α</mi><mo id="S3.SS3.p2.14.m2.1.1.3.1" xref="S3.SS3.p2.14.m2.1.1.3.1.cmml">/</mo><mn id="S3.SS3.p2.14.m2.1.1.3.3" xref="S3.SS3.p2.14.m2.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.14.m2.1b"><apply id="S3.SS3.p2.14.m2.1.1.cmml" xref="S3.SS3.p2.14.m2.1.1"><minus id="S3.SS3.p2.14.m2.1.1.1.cmml" xref="S3.SS3.p2.14.m2.1.1.1"></minus><cn id="S3.SS3.p2.14.m2.1.1.2.cmml" type="integer" xref="S3.SS3.p2.14.m2.1.1.2">1</cn><apply id="S3.SS3.p2.14.m2.1.1.3.cmml" xref="S3.SS3.p2.14.m2.1.1.3"><divide id="S3.SS3.p2.14.m2.1.1.3.1.cmml" xref="S3.SS3.p2.14.m2.1.1.3.1"></divide><ci id="S3.SS3.p2.14.m2.1.1.3.2.cmml" xref="S3.SS3.p2.14.m2.1.1.3.2">𝛼</ci><cn id="S3.SS3.p2.14.m2.1.1.3.3.cmml" type="integer" xref="S3.SS3.p2.14.m2.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.14.m2.1c">1-\alpha/2</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.14.m2.1d">1 - italic_α / 2</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS3.p3"> <p class="ltx_p" id="S3.SS3.p3.11">Remark that there is a trade-off to be made when using these combination strategies. In the case of combination by Expression (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.E11" title="In 3.3 Time-independent effects of covariates ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">11</span></a>), we have on the one hand that <math alttext="\hat{\mathcal{B}}_{I,k}" class="ltx_Math" display="inline" id="S3.SS3.p3.1.m1.2"><semantics id="S3.SS3.p3.1.m1.2a"><msub id="S3.SS3.p3.1.m1.2.3" xref="S3.SS3.p3.1.m1.2.3.cmml"><mover accent="true" id="S3.SS3.p3.1.m1.2.3.2" xref="S3.SS3.p3.1.m1.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p3.1.m1.2.3.2.2" xref="S3.SS3.p3.1.m1.2.3.2.2.cmml">ℬ</mi><mo id="S3.SS3.p3.1.m1.2.3.2.1" xref="S3.SS3.p3.1.m1.2.3.2.1.cmml">^</mo></mover><mrow id="S3.SS3.p3.1.m1.2.2.2.4" xref="S3.SS3.p3.1.m1.2.2.2.3.cmml"><mi id="S3.SS3.p3.1.m1.1.1.1.1" xref="S3.SS3.p3.1.m1.1.1.1.1.cmml">I</mi><mo id="S3.SS3.p3.1.m1.2.2.2.4.1" xref="S3.SS3.p3.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.SS3.p3.1.m1.2.2.2.2" xref="S3.SS3.p3.1.m1.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.1.m1.2b"><apply id="S3.SS3.p3.1.m1.2.3.cmml" xref="S3.SS3.p3.1.m1.2.3"><csymbol cd="ambiguous" id="S3.SS3.p3.1.m1.2.3.1.cmml" xref="S3.SS3.p3.1.m1.2.3">subscript</csymbol><apply id="S3.SS3.p3.1.m1.2.3.2.cmml" xref="S3.SS3.p3.1.m1.2.3.2"><ci id="S3.SS3.p3.1.m1.2.3.2.1.cmml" xref="S3.SS3.p3.1.m1.2.3.2.1">^</ci><ci id="S3.SS3.p3.1.m1.2.3.2.2.cmml" xref="S3.SS3.p3.1.m1.2.3.2.2">ℬ</ci></apply><list id="S3.SS3.p3.1.m1.2.2.2.3.cmml" xref="S3.SS3.p3.1.m1.2.2.2.4"><ci id="S3.SS3.p3.1.m1.1.1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1.1.1">𝐼</ci><ci id="S3.SS3.p3.1.m1.2.2.2.2.cmml" xref="S3.SS3.p3.1.m1.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.1.m1.2c">\hat{\mathcal{B}}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.1.m1.2d">over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> can only decrease in size by adding more intervals <math alttext="\hat{\mathcal{B}}_{I,k}^{a}" class="ltx_Math" display="inline" id="S3.SS3.p3.2.m2.2"><semantics id="S3.SS3.p3.2.m2.2a"><msubsup id="S3.SS3.p3.2.m2.2.3" xref="S3.SS3.p3.2.m2.2.3.cmml"><mover accent="true" id="S3.SS3.p3.2.m2.2.3.2.2" xref="S3.SS3.p3.2.m2.2.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p3.2.m2.2.3.2.2.2" xref="S3.SS3.p3.2.m2.2.3.2.2.2.cmml">ℬ</mi><mo id="S3.SS3.p3.2.m2.2.3.2.2.1" xref="S3.SS3.p3.2.m2.2.3.2.2.1.cmml">^</mo></mover><mrow id="S3.SS3.p3.2.m2.2.2.2.4" xref="S3.SS3.p3.2.m2.2.2.2.3.cmml"><mi id="S3.SS3.p3.2.m2.1.1.1.1" xref="S3.SS3.p3.2.m2.1.1.1.1.cmml">I</mi><mo id="S3.SS3.p3.2.m2.2.2.2.4.1" xref="S3.SS3.p3.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.SS3.p3.2.m2.2.2.2.2" xref="S3.SS3.p3.2.m2.2.2.2.2.cmml">k</mi></mrow><mi id="S3.SS3.p3.2.m2.2.3.3" xref="S3.SS3.p3.2.m2.2.3.3.cmml">a</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.2.m2.2b"><apply id="S3.SS3.p3.2.m2.2.3.cmml" xref="S3.SS3.p3.2.m2.2.3"><csymbol cd="ambiguous" id="S3.SS3.p3.2.m2.2.3.1.cmml" xref="S3.SS3.p3.2.m2.2.3">superscript</csymbol><apply id="S3.SS3.p3.2.m2.2.3.2.cmml" xref="S3.SS3.p3.2.m2.2.3"><csymbol cd="ambiguous" id="S3.SS3.p3.2.m2.2.3.2.1.cmml" xref="S3.SS3.p3.2.m2.2.3">subscript</csymbol><apply id="S3.SS3.p3.2.m2.2.3.2.2.cmml" xref="S3.SS3.p3.2.m2.2.3.2.2"><ci id="S3.SS3.p3.2.m2.2.3.2.2.1.cmml" xref="S3.SS3.p3.2.m2.2.3.2.2.1">^</ci><ci id="S3.SS3.p3.2.m2.2.3.2.2.2.cmml" xref="S3.SS3.p3.2.m2.2.3.2.2.2">ℬ</ci></apply><list id="S3.SS3.p3.2.m2.2.2.2.3.cmml" xref="S3.SS3.p3.2.m2.2.2.2.4"><ci id="S3.SS3.p3.2.m2.1.1.1.1.cmml" xref="S3.SS3.p3.2.m2.1.1.1.1">𝐼</ci><ci id="S3.SS3.p3.2.m2.2.2.2.2.cmml" xref="S3.SS3.p3.2.m2.2.2.2.2">𝑘</ci></list></apply><ci id="S3.SS3.p3.2.m2.2.3.3.cmml" xref="S3.SS3.p3.2.m2.2.3.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.2.m2.2c">\hat{\mathcal{B}}_{I,k}^{a}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.2.m2.2d">over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT</annotation></semantics></math> to the intersection, making it seem desirable to consider as many additional time points as possible. On the other hand, in doing so, the estimation of each interval <math alttext="\hat{\mathcal{B}}_{I,k}^{a}" class="ltx_Math" display="inline" id="S3.SS3.p3.3.m3.2"><semantics id="S3.SS3.p3.3.m3.2a"><msubsup id="S3.SS3.p3.3.m3.2.3" xref="S3.SS3.p3.3.m3.2.3.cmml"><mover accent="true" id="S3.SS3.p3.3.m3.2.3.2.2" xref="S3.SS3.p3.3.m3.2.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p3.3.m3.2.3.2.2.2" xref="S3.SS3.p3.3.m3.2.3.2.2.2.cmml">ℬ</mi><mo id="S3.SS3.p3.3.m3.2.3.2.2.1" xref="S3.SS3.p3.3.m3.2.3.2.2.1.cmml">^</mo></mover><mrow id="S3.SS3.p3.3.m3.2.2.2.4" xref="S3.SS3.p3.3.m3.2.2.2.3.cmml"><mi id="S3.SS3.p3.3.m3.1.1.1.1" xref="S3.SS3.p3.3.m3.1.1.1.1.cmml">I</mi><mo id="S3.SS3.p3.3.m3.2.2.2.4.1" xref="S3.SS3.p3.3.m3.2.2.2.3.cmml">,</mo><mi id="S3.SS3.p3.3.m3.2.2.2.2" xref="S3.SS3.p3.3.m3.2.2.2.2.cmml">k</mi></mrow><mi id="S3.SS3.p3.3.m3.2.3.3" xref="S3.SS3.p3.3.m3.2.3.3.cmml">a</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.3.m3.2b"><apply id="S3.SS3.p3.3.m3.2.3.cmml" xref="S3.SS3.p3.3.m3.2.3"><csymbol cd="ambiguous" id="S3.SS3.p3.3.m3.2.3.1.cmml" xref="S3.SS3.p3.3.m3.2.3">superscript</csymbol><apply id="S3.SS3.p3.3.m3.2.3.2.cmml" xref="S3.SS3.p3.3.m3.2.3"><csymbol cd="ambiguous" id="S3.SS3.p3.3.m3.2.3.2.1.cmml" xref="S3.SS3.p3.3.m3.2.3">subscript</csymbol><apply id="S3.SS3.p3.3.m3.2.3.2.2.cmml" xref="S3.SS3.p3.3.m3.2.3.2.2"><ci id="S3.SS3.p3.3.m3.2.3.2.2.1.cmml" xref="S3.SS3.p3.3.m3.2.3.2.2.1">^</ci><ci id="S3.SS3.p3.3.m3.2.3.2.2.2.cmml" xref="S3.SS3.p3.3.m3.2.3.2.2.2">ℬ</ci></apply><list id="S3.SS3.p3.3.m3.2.2.2.3.cmml" xref="S3.SS3.p3.3.m3.2.2.2.4"><ci id="S3.SS3.p3.3.m3.1.1.1.1.cmml" xref="S3.SS3.p3.3.m3.1.1.1.1">𝐼</ci><ci id="S3.SS3.p3.3.m3.2.2.2.2.cmml" xref="S3.SS3.p3.3.m3.2.2.2.2">𝑘</ci></list></apply><ci id="S3.SS3.p3.3.m3.2.3.3.cmml" xref="S3.SS3.p3.3.m3.2.3.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.3.m3.2c">\hat{\mathcal{B}}_{I,k}^{a}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.3.m3.2d">over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT</annotation></semantics></math> will have to be done at a larger confidence level, leading to wider intervals. Therefore, one could wonder about an appropriate value for <math alttext="A" class="ltx_Math" display="inline" id="S3.SS3.p3.4.m4.1"><semantics id="S3.SS3.p3.4.m4.1a"><mi id="S3.SS3.p3.4.m4.1.1" xref="S3.SS3.p3.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.4.m4.1b"><ci id="S3.SS3.p3.4.m4.1.1.cmml" xref="S3.SS3.p3.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.4.m4.1d">italic_A</annotation></semantics></math>, as well as an appropriate set of time points <math alttext="t_{1},\dots,t_{A}" class="ltx_Math" display="inline" id="S3.SS3.p3.5.m5.3"><semantics id="S3.SS3.p3.5.m5.3a"><mrow id="S3.SS3.p3.5.m5.3.3.2" xref="S3.SS3.p3.5.m5.3.3.3.cmml"><msub id="S3.SS3.p3.5.m5.2.2.1.1" xref="S3.SS3.p3.5.m5.2.2.1.1.cmml"><mi id="S3.SS3.p3.5.m5.2.2.1.1.2" xref="S3.SS3.p3.5.m5.2.2.1.1.2.cmml">t</mi><mn id="S3.SS3.p3.5.m5.2.2.1.1.3" xref="S3.SS3.p3.5.m5.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.SS3.p3.5.m5.3.3.2.3" xref="S3.SS3.p3.5.m5.3.3.3.cmml">,</mo><mi id="S3.SS3.p3.5.m5.1.1" mathvariant="normal" xref="S3.SS3.p3.5.m5.1.1.cmml">…</mi><mo id="S3.SS3.p3.5.m5.3.3.2.4" xref="S3.SS3.p3.5.m5.3.3.3.cmml">,</mo><msub id="S3.SS3.p3.5.m5.3.3.2.2" xref="S3.SS3.p3.5.m5.3.3.2.2.cmml"><mi id="S3.SS3.p3.5.m5.3.3.2.2.2" xref="S3.SS3.p3.5.m5.3.3.2.2.2.cmml">t</mi><mi id="S3.SS3.p3.5.m5.3.3.2.2.3" xref="S3.SS3.p3.5.m5.3.3.2.2.3.cmml">A</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.5.m5.3b"><list id="S3.SS3.p3.5.m5.3.3.3.cmml" xref="S3.SS3.p3.5.m5.3.3.2"><apply id="S3.SS3.p3.5.m5.2.2.1.1.cmml" xref="S3.SS3.p3.5.m5.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p3.5.m5.2.2.1.1.1.cmml" xref="S3.SS3.p3.5.m5.2.2.1.1">subscript</csymbol><ci id="S3.SS3.p3.5.m5.2.2.1.1.2.cmml" xref="S3.SS3.p3.5.m5.2.2.1.1.2">𝑡</ci><cn id="S3.SS3.p3.5.m5.2.2.1.1.3.cmml" type="integer" xref="S3.SS3.p3.5.m5.2.2.1.1.3">1</cn></apply><ci id="S3.SS3.p3.5.m5.1.1.cmml" xref="S3.SS3.p3.5.m5.1.1">…</ci><apply id="S3.SS3.p3.5.m5.3.3.2.2.cmml" xref="S3.SS3.p3.5.m5.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS3.p3.5.m5.3.3.2.2.1.cmml" xref="S3.SS3.p3.5.m5.3.3.2.2">subscript</csymbol><ci id="S3.SS3.p3.5.m5.3.3.2.2.2.cmml" xref="S3.SS3.p3.5.m5.3.3.2.2.2">𝑡</ci><ci id="S3.SS3.p3.5.m5.3.3.2.2.3.cmml" xref="S3.SS3.p3.5.m5.3.3.2.2.3">𝐴</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.5.m5.3c">t_{1},\dots,t_{A}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.5.m5.3d">italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_t start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math>. In view of keeping the required computations manageable, we recommend to let <math alttext="A\in\{3,4,5\}" class="ltx_Math" display="inline" id="S3.SS3.p3.6.m6.3"><semantics id="S3.SS3.p3.6.m6.3a"><mrow id="S3.SS3.p3.6.m6.3.4" xref="S3.SS3.p3.6.m6.3.4.cmml"><mi id="S3.SS3.p3.6.m6.3.4.2" xref="S3.SS3.p3.6.m6.3.4.2.cmml">A</mi><mo id="S3.SS3.p3.6.m6.3.4.1" xref="S3.SS3.p3.6.m6.3.4.1.cmml">∈</mo><mrow id="S3.SS3.p3.6.m6.3.4.3.2" xref="S3.SS3.p3.6.m6.3.4.3.1.cmml"><mo id="S3.SS3.p3.6.m6.3.4.3.2.1" stretchy="false" xref="S3.SS3.p3.6.m6.3.4.3.1.cmml">{</mo><mn id="S3.SS3.p3.6.m6.1.1" xref="S3.SS3.p3.6.m6.1.1.cmml">3</mn><mo id="S3.SS3.p3.6.m6.3.4.3.2.2" xref="S3.SS3.p3.6.m6.3.4.3.1.cmml">,</mo><mn id="S3.SS3.p3.6.m6.2.2" xref="S3.SS3.p3.6.m6.2.2.cmml">4</mn><mo id="S3.SS3.p3.6.m6.3.4.3.2.3" xref="S3.SS3.p3.6.m6.3.4.3.1.cmml">,</mo><mn id="S3.SS3.p3.6.m6.3.3" xref="S3.SS3.p3.6.m6.3.3.cmml">5</mn><mo id="S3.SS3.p3.6.m6.3.4.3.2.4" stretchy="false" xref="S3.SS3.p3.6.m6.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.6.m6.3b"><apply id="S3.SS3.p3.6.m6.3.4.cmml" xref="S3.SS3.p3.6.m6.3.4"><in id="S3.SS3.p3.6.m6.3.4.1.cmml" xref="S3.SS3.p3.6.m6.3.4.1"></in><ci id="S3.SS3.p3.6.m6.3.4.2.cmml" xref="S3.SS3.p3.6.m6.3.4.2">𝐴</ci><set id="S3.SS3.p3.6.m6.3.4.3.1.cmml" xref="S3.SS3.p3.6.m6.3.4.3.2"><cn id="S3.SS3.p3.6.m6.1.1.cmml" type="integer" xref="S3.SS3.p3.6.m6.1.1">3</cn><cn id="S3.SS3.p3.6.m6.2.2.cmml" type="integer" xref="S3.SS3.p3.6.m6.2.2">4</cn><cn id="S3.SS3.p3.6.m6.3.3.cmml" type="integer" xref="S3.SS3.p3.6.m6.3.3">5</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.6.m6.3c">A\in\{3,4,5\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.6.m6.3d">italic_A ∈ { 3 , 4 , 5 }</annotation></semantics></math>. The selected time points <math alttext="t_{1},\dots,t_{A}" class="ltx_Math" display="inline" id="S3.SS3.p3.7.m7.3"><semantics id="S3.SS3.p3.7.m7.3a"><mrow id="S3.SS3.p3.7.m7.3.3.2" xref="S3.SS3.p3.7.m7.3.3.3.cmml"><msub id="S3.SS3.p3.7.m7.2.2.1.1" xref="S3.SS3.p3.7.m7.2.2.1.1.cmml"><mi id="S3.SS3.p3.7.m7.2.2.1.1.2" xref="S3.SS3.p3.7.m7.2.2.1.1.2.cmml">t</mi><mn id="S3.SS3.p3.7.m7.2.2.1.1.3" xref="S3.SS3.p3.7.m7.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.SS3.p3.7.m7.3.3.2.3" xref="S3.SS3.p3.7.m7.3.3.3.cmml">,</mo><mi id="S3.SS3.p3.7.m7.1.1" mathvariant="normal" xref="S3.SS3.p3.7.m7.1.1.cmml">…</mi><mo id="S3.SS3.p3.7.m7.3.3.2.4" xref="S3.SS3.p3.7.m7.3.3.3.cmml">,</mo><msub id="S3.SS3.p3.7.m7.3.3.2.2" xref="S3.SS3.p3.7.m7.3.3.2.2.cmml"><mi id="S3.SS3.p3.7.m7.3.3.2.2.2" xref="S3.SS3.p3.7.m7.3.3.2.2.2.cmml">t</mi><mi id="S3.SS3.p3.7.m7.3.3.2.2.3" xref="S3.SS3.p3.7.m7.3.3.2.2.3.cmml">A</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.7.m7.3b"><list id="S3.SS3.p3.7.m7.3.3.3.cmml" xref="S3.SS3.p3.7.m7.3.3.2"><apply id="S3.SS3.p3.7.m7.2.2.1.1.cmml" xref="S3.SS3.p3.7.m7.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p3.7.m7.2.2.1.1.1.cmml" xref="S3.SS3.p3.7.m7.2.2.1.1">subscript</csymbol><ci id="S3.SS3.p3.7.m7.2.2.1.1.2.cmml" xref="S3.SS3.p3.7.m7.2.2.1.1.2">𝑡</ci><cn id="S3.SS3.p3.7.m7.2.2.1.1.3.cmml" type="integer" xref="S3.SS3.p3.7.m7.2.2.1.1.3">1</cn></apply><ci id="S3.SS3.p3.7.m7.1.1.cmml" xref="S3.SS3.p3.7.m7.1.1">…</ci><apply id="S3.SS3.p3.7.m7.3.3.2.2.cmml" xref="S3.SS3.p3.7.m7.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS3.p3.7.m7.3.3.2.2.1.cmml" xref="S3.SS3.p3.7.m7.3.3.2.2">subscript</csymbol><ci id="S3.SS3.p3.7.m7.3.3.2.2.2.cmml" xref="S3.SS3.p3.7.m7.3.3.2.2.2">𝑡</ci><ci id="S3.SS3.p3.7.m7.3.3.2.2.3.cmml" xref="S3.SS3.p3.7.m7.3.3.2.2.3">𝐴</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.7.m7.3c">t_{1},\dots,t_{A}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.7.m7.3d">italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_t start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> can be spread over the support of <math alttext="Y" class="ltx_Math" display="inline" id="S3.SS3.p3.8.m8.1"><semantics id="S3.SS3.p3.8.m8.1a"><mi id="S3.SS3.p3.8.m8.1.1" xref="S3.SS3.p3.8.m8.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.8.m8.1b"><ci id="S3.SS3.p3.8.m8.1.1.cmml" xref="S3.SS3.p3.8.m8.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.8.m8.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.8.m8.1d">italic_Y</annotation></semantics></math>, though we note that earlier time points are often more informative, since Peterson bounds widen as the considered time point <math alttext="t_{a}" class="ltx_Math" display="inline" id="S3.SS3.p3.9.m9.1"><semantics id="S3.SS3.p3.9.m9.1a"><msub id="S3.SS3.p3.9.m9.1.1" xref="S3.SS3.p3.9.m9.1.1.cmml"><mi id="S3.SS3.p3.9.m9.1.1.2" xref="S3.SS3.p3.9.m9.1.1.2.cmml">t</mi><mi id="S3.SS3.p3.9.m9.1.1.3" xref="S3.SS3.p3.9.m9.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.9.m9.1b"><apply id="S3.SS3.p3.9.m9.1.1.cmml" xref="S3.SS3.p3.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS3.p3.9.m9.1.1.1.cmml" xref="S3.SS3.p3.9.m9.1.1">subscript</csymbol><ci id="S3.SS3.p3.9.m9.1.1.2.cmml" xref="S3.SS3.p3.9.m9.1.1.2">𝑡</ci><ci id="S3.SS3.p3.9.m9.1.1.3.cmml" xref="S3.SS3.p3.9.m9.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.9.m9.1c">t_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.9.m9.1d">italic_t start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> increases, and hence the identified intervals <math alttext="\hat{\mathcal{B}}_{I,k}^{a}" class="ltx_Math" display="inline" id="S3.SS3.p3.10.m10.2"><semantics id="S3.SS3.p3.10.m10.2a"><msubsup id="S3.SS3.p3.10.m10.2.3" xref="S3.SS3.p3.10.m10.2.3.cmml"><mover accent="true" id="S3.SS3.p3.10.m10.2.3.2.2" xref="S3.SS3.p3.10.m10.2.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p3.10.m10.2.3.2.2.2" xref="S3.SS3.p3.10.m10.2.3.2.2.2.cmml">ℬ</mi><mo id="S3.SS3.p3.10.m10.2.3.2.2.1" xref="S3.SS3.p3.10.m10.2.3.2.2.1.cmml">^</mo></mover><mrow id="S3.SS3.p3.10.m10.2.2.2.4" xref="S3.SS3.p3.10.m10.2.2.2.3.cmml"><mi id="S3.SS3.p3.10.m10.1.1.1.1" xref="S3.SS3.p3.10.m10.1.1.1.1.cmml">I</mi><mo id="S3.SS3.p3.10.m10.2.2.2.4.1" xref="S3.SS3.p3.10.m10.2.2.2.3.cmml">,</mo><mi id="S3.SS3.p3.10.m10.2.2.2.2" xref="S3.SS3.p3.10.m10.2.2.2.2.cmml">k</mi></mrow><mi id="S3.SS3.p3.10.m10.2.3.3" xref="S3.SS3.p3.10.m10.2.3.3.cmml">a</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.10.m10.2b"><apply id="S3.SS3.p3.10.m10.2.3.cmml" xref="S3.SS3.p3.10.m10.2.3"><csymbol cd="ambiguous" id="S3.SS3.p3.10.m10.2.3.1.cmml" xref="S3.SS3.p3.10.m10.2.3">superscript</csymbol><apply id="S3.SS3.p3.10.m10.2.3.2.cmml" xref="S3.SS3.p3.10.m10.2.3"><csymbol cd="ambiguous" id="S3.SS3.p3.10.m10.2.3.2.1.cmml" xref="S3.SS3.p3.10.m10.2.3">subscript</csymbol><apply id="S3.SS3.p3.10.m10.2.3.2.2.cmml" xref="S3.SS3.p3.10.m10.2.3.2.2"><ci id="S3.SS3.p3.10.m10.2.3.2.2.1.cmml" xref="S3.SS3.p3.10.m10.2.3.2.2.1">^</ci><ci id="S3.SS3.p3.10.m10.2.3.2.2.2.cmml" xref="S3.SS3.p3.10.m10.2.3.2.2.2">ℬ</ci></apply><list id="S3.SS3.p3.10.m10.2.2.2.3.cmml" xref="S3.SS3.p3.10.m10.2.2.2.4"><ci id="S3.SS3.p3.10.m10.1.1.1.1.cmml" xref="S3.SS3.p3.10.m10.1.1.1.1">𝐼</ci><ci id="S3.SS3.p3.10.m10.2.2.2.2.cmml" xref="S3.SS3.p3.10.m10.2.2.2.2">𝑘</ci></list></apply><ci id="S3.SS3.p3.10.m10.2.3.3.cmml" xref="S3.SS3.p3.10.m10.2.3.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.10.m10.2c">\hat{\mathcal{B}}_{I,k}^{a}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.10.m10.2d">over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT</annotation></semantics></math> tend to do so as well. An analysis of both combination approaches as well as the effect of the selected time points <math alttext="t_{1},\dots,t_{A}" class="ltx_Math" display="inline" id="S3.SS3.p3.11.m11.3"><semantics id="S3.SS3.p3.11.m11.3a"><mrow id="S3.SS3.p3.11.m11.3.3.2" xref="S3.SS3.p3.11.m11.3.3.3.cmml"><msub id="S3.SS3.p3.11.m11.2.2.1.1" xref="S3.SS3.p3.11.m11.2.2.1.1.cmml"><mi id="S3.SS3.p3.11.m11.2.2.1.1.2" xref="S3.SS3.p3.11.m11.2.2.1.1.2.cmml">t</mi><mn id="S3.SS3.p3.11.m11.2.2.1.1.3" xref="S3.SS3.p3.11.m11.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.SS3.p3.11.m11.3.3.2.3" xref="S3.SS3.p3.11.m11.3.3.3.cmml">,</mo><mi id="S3.SS3.p3.11.m11.1.1" mathvariant="normal" xref="S3.SS3.p3.11.m11.1.1.cmml">…</mi><mo id="S3.SS3.p3.11.m11.3.3.2.4" xref="S3.SS3.p3.11.m11.3.3.3.cmml">,</mo><msub id="S3.SS3.p3.11.m11.3.3.2.2" xref="S3.SS3.p3.11.m11.3.3.2.2.cmml"><mi id="S3.SS3.p3.11.m11.3.3.2.2.2" xref="S3.SS3.p3.11.m11.3.3.2.2.2.cmml">t</mi><mi id="S3.SS3.p3.11.m11.3.3.2.2.3" xref="S3.SS3.p3.11.m11.3.3.2.2.3.cmml">A</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.11.m11.3b"><list id="S3.SS3.p3.11.m11.3.3.3.cmml" xref="S3.SS3.p3.11.m11.3.3.2"><apply id="S3.SS3.p3.11.m11.2.2.1.1.cmml" xref="S3.SS3.p3.11.m11.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p3.11.m11.2.2.1.1.1.cmml" xref="S3.SS3.p3.11.m11.2.2.1.1">subscript</csymbol><ci id="S3.SS3.p3.11.m11.2.2.1.1.2.cmml" xref="S3.SS3.p3.11.m11.2.2.1.1.2">𝑡</ci><cn id="S3.SS3.p3.11.m11.2.2.1.1.3.cmml" type="integer" xref="S3.SS3.p3.11.m11.2.2.1.1.3">1</cn></apply><ci id="S3.SS3.p3.11.m11.1.1.cmml" xref="S3.SS3.p3.11.m11.1.1">…</ci><apply id="S3.SS3.p3.11.m11.3.3.2.2.cmml" xref="S3.SS3.p3.11.m11.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS3.p3.11.m11.3.3.2.2.1.cmml" xref="S3.SS3.p3.11.m11.3.3.2.2">subscript</csymbol><ci id="S3.SS3.p3.11.m11.3.3.2.2.2.cmml" xref="S3.SS3.p3.11.m11.3.3.2.2.2">𝑡</ci><ci id="S3.SS3.p3.11.m11.3.3.2.2.3.cmml" xref="S3.SS3.p3.11.m11.3.3.2.2.3">𝐴</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.11.m11.3c">t_{1},\dots,t_{A}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.11.m11.3d">italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_t start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> is presented in Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Comparison_of_combination_methods</span>.</p> </div> <div class="ltx_para" id="S3.SS3.p4"> <p class="ltx_p" id="S3.SS3.p4.2">Lastly, we note that the approach as described above only assumes that <math alttext="\beta_{k}" class="ltx_Math" display="inline" id="S3.SS3.p4.1.m1.1"><semantics id="S3.SS3.p4.1.m1.1a"><msub id="S3.SS3.p4.1.m1.1.1" xref="S3.SS3.p4.1.m1.1.1.cmml"><mi id="S3.SS3.p4.1.m1.1.1.2" xref="S3.SS3.p4.1.m1.1.1.2.cmml">β</mi><mi id="S3.SS3.p4.1.m1.1.1.3" xref="S3.SS3.p4.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.1.m1.1b"><apply id="S3.SS3.p4.1.m1.1.1.cmml" xref="S3.SS3.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p4.1.m1.1.1.1.cmml" xref="S3.SS3.p4.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.p4.1.m1.1.1.2.cmml" xref="S3.SS3.p4.1.m1.1.1.2">𝛽</ci><ci id="S3.SS3.p4.1.m1.1.1.3.cmml" xref="S3.SS3.p4.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.1.m1.1c">\beta_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.1.m1.1d">italic_β start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is independent of time. Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Alternative_approach_to_time-independent_coefficients</span> describes a more direct approach that can take into account that all coefficients are time-independent. It works by augmenting the unconditional moment conditions in (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E6" title="In 2.3 Unconditional moment restrictions ‣ 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">6</span></a>) to immediately include all considered time points <math alttext="\{t_{1},\dots,t_{A}\}" class="ltx_Math" display="inline" id="S3.SS3.p4.2.m2.3"><semantics id="S3.SS3.p4.2.m2.3a"><mrow id="S3.SS3.p4.2.m2.3.3.2" xref="S3.SS3.p4.2.m2.3.3.3.cmml"><mo id="S3.SS3.p4.2.m2.3.3.2.3" stretchy="false" xref="S3.SS3.p4.2.m2.3.3.3.cmml">{</mo><msub id="S3.SS3.p4.2.m2.2.2.1.1" xref="S3.SS3.p4.2.m2.2.2.1.1.cmml"><mi id="S3.SS3.p4.2.m2.2.2.1.1.2" xref="S3.SS3.p4.2.m2.2.2.1.1.2.cmml">t</mi><mn id="S3.SS3.p4.2.m2.2.2.1.1.3" xref="S3.SS3.p4.2.m2.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.SS3.p4.2.m2.3.3.2.4" xref="S3.SS3.p4.2.m2.3.3.3.cmml">,</mo><mi id="S3.SS3.p4.2.m2.1.1" mathvariant="normal" xref="S3.SS3.p4.2.m2.1.1.cmml">…</mi><mo id="S3.SS3.p4.2.m2.3.3.2.5" xref="S3.SS3.p4.2.m2.3.3.3.cmml">,</mo><msub id="S3.SS3.p4.2.m2.3.3.2.2" xref="S3.SS3.p4.2.m2.3.3.2.2.cmml"><mi id="S3.SS3.p4.2.m2.3.3.2.2.2" xref="S3.SS3.p4.2.m2.3.3.2.2.2.cmml">t</mi><mi id="S3.SS3.p4.2.m2.3.3.2.2.3" xref="S3.SS3.p4.2.m2.3.3.2.2.3.cmml">A</mi></msub><mo id="S3.SS3.p4.2.m2.3.3.2.6" stretchy="false" xref="S3.SS3.p4.2.m2.3.3.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.2.m2.3b"><set id="S3.SS3.p4.2.m2.3.3.3.cmml" xref="S3.SS3.p4.2.m2.3.3.2"><apply id="S3.SS3.p4.2.m2.2.2.1.1.cmml" xref="S3.SS3.p4.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p4.2.m2.2.2.1.1.1.cmml" xref="S3.SS3.p4.2.m2.2.2.1.1">subscript</csymbol><ci id="S3.SS3.p4.2.m2.2.2.1.1.2.cmml" xref="S3.SS3.p4.2.m2.2.2.1.1.2">𝑡</ci><cn id="S3.SS3.p4.2.m2.2.2.1.1.3.cmml" type="integer" xref="S3.SS3.p4.2.m2.2.2.1.1.3">1</cn></apply><ci id="S3.SS3.p4.2.m2.1.1.cmml" xref="S3.SS3.p4.2.m2.1.1">…</ci><apply id="S3.SS3.p4.2.m2.3.3.2.2.cmml" xref="S3.SS3.p4.2.m2.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS3.p4.2.m2.3.3.2.2.1.cmml" xref="S3.SS3.p4.2.m2.3.3.2.2">subscript</csymbol><ci id="S3.SS3.p4.2.m2.3.3.2.2.2.cmml" xref="S3.SS3.p4.2.m2.3.3.2.2.2">𝑡</ci><ci id="S3.SS3.p4.2.m2.3.3.2.2.3.cmml" xref="S3.SS3.p4.2.m2.3.3.2.2.3">𝐴</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.2.m2.3c">\{t_{1},\dots,t_{A}\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.2.m2.3d">{ italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_t start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT }</annotation></semantics></math> and constructing a test for this augmented set of moment conditions. Such an approach, however, was found to be extremely computationally intensive, and it did not lead to worthwhile improvements.</p> </div> </section> <section class="ltx_subsection" id="S3.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.4 </span>Discussion</h3> <div class="ltx_para" id="S3.SS4.p1"> <p class="ltx_p" id="S3.SS4.p1.4">Below we elaborate on several more aspects of the model. Going forward, we will refer to <math alttext="\mathcal{B}_{I,k}" class="ltx_Math" display="inline" id="S3.SS4.p1.1.m1.2"><semantics id="S3.SS4.p1.1.m1.2a"><msub id="S3.SS4.p1.1.m1.2.3" xref="S3.SS4.p1.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p1.1.m1.2.3.2" xref="S3.SS4.p1.1.m1.2.3.2.cmml">ℬ</mi><mrow id="S3.SS4.p1.1.m1.2.2.2.4" xref="S3.SS4.p1.1.m1.2.2.2.3.cmml"><mi id="S3.SS4.p1.1.m1.1.1.1.1" xref="S3.SS4.p1.1.m1.1.1.1.1.cmml">I</mi><mo id="S3.SS4.p1.1.m1.2.2.2.4.1" xref="S3.SS4.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.SS4.p1.1.m1.2.2.2.2" xref="S3.SS4.p1.1.m1.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.1.m1.2b"><apply id="S3.SS4.p1.1.m1.2.3.cmml" xref="S3.SS4.p1.1.m1.2.3"><csymbol cd="ambiguous" id="S3.SS4.p1.1.m1.2.3.1.cmml" xref="S3.SS4.p1.1.m1.2.3">subscript</csymbol><ci id="S3.SS4.p1.1.m1.2.3.2.cmml" xref="S3.SS4.p1.1.m1.2.3.2">ℬ</ci><list id="S3.SS4.p1.1.m1.2.2.2.3.cmml" xref="S3.SS4.p1.1.m1.2.2.2.4"><ci id="S3.SS4.p1.1.m1.1.1.1.1.cmml" xref="S3.SS4.p1.1.m1.1.1.1.1">𝐼</ci><ci id="S3.SS4.p1.1.m1.2.2.2.2.cmml" xref="S3.SS4.p1.1.m1.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.1.m1.2c">\mathcal{B}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.1.m1.2d">caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and its estimator <math alttext="\hat{\mathcal{B}}_{I,k}" class="ltx_Math" display="inline" id="S3.SS4.p1.2.m2.2"><semantics id="S3.SS4.p1.2.m2.2a"><msub id="S3.SS4.p1.2.m2.2.3" xref="S3.SS4.p1.2.m2.2.3.cmml"><mover accent="true" id="S3.SS4.p1.2.m2.2.3.2" xref="S3.SS4.p1.2.m2.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p1.2.m2.2.3.2.2" xref="S3.SS4.p1.2.m2.2.3.2.2.cmml">ℬ</mi><mo id="S3.SS4.p1.2.m2.2.3.2.1" xref="S3.SS4.p1.2.m2.2.3.2.1.cmml">^</mo></mover><mrow id="S3.SS4.p1.2.m2.2.2.2.4" xref="S3.SS4.p1.2.m2.2.2.2.3.cmml"><mi id="S3.SS4.p1.2.m2.1.1.1.1" xref="S3.SS4.p1.2.m2.1.1.1.1.cmml">I</mi><mo id="S3.SS4.p1.2.m2.2.2.2.4.1" xref="S3.SS4.p1.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.SS4.p1.2.m2.2.2.2.2" xref="S3.SS4.p1.2.m2.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.2.m2.2b"><apply id="S3.SS4.p1.2.m2.2.3.cmml" xref="S3.SS4.p1.2.m2.2.3"><csymbol cd="ambiguous" id="S3.SS4.p1.2.m2.2.3.1.cmml" xref="S3.SS4.p1.2.m2.2.3">subscript</csymbol><apply id="S3.SS4.p1.2.m2.2.3.2.cmml" xref="S3.SS4.p1.2.m2.2.3.2"><ci id="S3.SS4.p1.2.m2.2.3.2.1.cmml" xref="S3.SS4.p1.2.m2.2.3.2.1">^</ci><ci id="S3.SS4.p1.2.m2.2.3.2.2.cmml" xref="S3.SS4.p1.2.m2.2.3.2.2">ℬ</ci></apply><list id="S3.SS4.p1.2.m2.2.2.2.3.cmml" xref="S3.SS4.p1.2.m2.2.2.2.4"><ci id="S3.SS4.p1.2.m2.1.1.1.1.cmml" xref="S3.SS4.p1.2.m2.1.1.1.1">𝐼</ci><ci id="S3.SS4.p1.2.m2.2.2.2.2.cmml" xref="S3.SS4.p1.2.m2.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.2.m2.2c">\hat{\mathcal{B}}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.2.m2.2d">over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> as the (estimated) <em class="ltx_emph ltx_font_italic" id="S3.SS4.p1.4.1">identified interval</em> throughout the rest of this paper, in order to distinguish it from the identified set <math alttext="\mathcal{B}_{I}" class="ltx_Math" display="inline" id="S3.SS4.p1.3.m3.1"><semantics id="S3.SS4.p1.3.m3.1a"><msub id="S3.SS4.p1.3.m3.1.1" xref="S3.SS4.p1.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p1.3.m3.1.1.2" xref="S3.SS4.p1.3.m3.1.1.2.cmml">ℬ</mi><mi id="S3.SS4.p1.3.m3.1.1.3" xref="S3.SS4.p1.3.m3.1.1.3.cmml">I</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.3.m3.1b"><apply id="S3.SS4.p1.3.m3.1.1.cmml" xref="S3.SS4.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS4.p1.3.m3.1.1.1.cmml" xref="S3.SS4.p1.3.m3.1.1">subscript</csymbol><ci id="S3.SS4.p1.3.m3.1.1.2.cmml" xref="S3.SS4.p1.3.m3.1.1.2">ℬ</ci><ci id="S3.SS4.p1.3.m3.1.1.3.cmml" xref="S3.SS4.p1.3.m3.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.3.m3.1c">\mathcal{B}_{I}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.3.m3.1d">caligraphic_B start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT</annotation></semantics></math> in the full parameter space. While this is a slight abuse of terminology – one can construct pathological examples where <math alttext="\mathcal{B}_{I,k}" class="ltx_Math" display="inline" id="S3.SS4.p1.4.m4.2"><semantics id="S3.SS4.p1.4.m4.2a"><msub id="S3.SS4.p1.4.m4.2.3" xref="S3.SS4.p1.4.m4.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p1.4.m4.2.3.2" xref="S3.SS4.p1.4.m4.2.3.2.cmml">ℬ</mi><mrow id="S3.SS4.p1.4.m4.2.2.2.4" xref="S3.SS4.p1.4.m4.2.2.2.3.cmml"><mi id="S3.SS4.p1.4.m4.1.1.1.1" xref="S3.SS4.p1.4.m4.1.1.1.1.cmml">I</mi><mo id="S3.SS4.p1.4.m4.2.2.2.4.1" xref="S3.SS4.p1.4.m4.2.2.2.3.cmml">,</mo><mi id="S3.SS4.p1.4.m4.2.2.2.2" xref="S3.SS4.p1.4.m4.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.4.m4.2b"><apply id="S3.SS4.p1.4.m4.2.3.cmml" xref="S3.SS4.p1.4.m4.2.3"><csymbol cd="ambiguous" id="S3.SS4.p1.4.m4.2.3.1.cmml" xref="S3.SS4.p1.4.m4.2.3">subscript</csymbol><ci id="S3.SS4.p1.4.m4.2.3.2.cmml" xref="S3.SS4.p1.4.m4.2.3.2">ℬ</ci><list id="S3.SS4.p1.4.m4.2.2.2.3.cmml" xref="S3.SS4.p1.4.m4.2.2.2.4"><ci id="S3.SS4.p1.4.m4.1.1.1.1.cmml" xref="S3.SS4.p1.4.m4.1.1.1.1">𝐼</ci><ci id="S3.SS4.p1.4.m4.2.2.2.2.cmml" xref="S3.SS4.p1.4.m4.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.4.m4.2c">\mathcal{B}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.4.m4.2d">caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is not an interval, cf. Example <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:example:_nonconvex_identified_interval</span> in Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Examples</span> – it will be valid in most practical cases. First, we discuss how the estimated identified interval should be interpreted. Next, we provide a specification test for the model. Lastly, we highlight several caveats. Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_future_research_directions</span> discusses possible avenues for future research.</p> </div> <div class="ltx_para" id="S3.SS4.p2"> <p class="ltx_p" id="S3.SS4.p2.10"><span class="ltx_text ltx_font_bold" id="S3.SS4.p2.10.1">Coverage.</span> In the partial identification literature there are two common coverage properties of the estimated identified interval (or, more generally, set). The first one states that <math alttext="\hat{\mathcal{B}}_{I,k}" class="ltx_Math" display="inline" id="S3.SS4.p2.1.m1.2"><semantics id="S3.SS4.p2.1.m1.2a"><msub id="S3.SS4.p2.1.m1.2.3" xref="S3.SS4.p2.1.m1.2.3.cmml"><mover accent="true" id="S3.SS4.p2.1.m1.2.3.2" xref="S3.SS4.p2.1.m1.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p2.1.m1.2.3.2.2" xref="S3.SS4.p2.1.m1.2.3.2.2.cmml">ℬ</mi><mo id="S3.SS4.p2.1.m1.2.3.2.1" xref="S3.SS4.p2.1.m1.2.3.2.1.cmml">^</mo></mover><mrow id="S3.SS4.p2.1.m1.2.2.2.4" xref="S3.SS4.p2.1.m1.2.2.2.3.cmml"><mi id="S3.SS4.p2.1.m1.1.1.1.1" xref="S3.SS4.p2.1.m1.1.1.1.1.cmml">I</mi><mo id="S3.SS4.p2.1.m1.2.2.2.4.1" xref="S3.SS4.p2.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.SS4.p2.1.m1.2.2.2.2" xref="S3.SS4.p2.1.m1.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.1.m1.2b"><apply id="S3.SS4.p2.1.m1.2.3.cmml" xref="S3.SS4.p2.1.m1.2.3"><csymbol cd="ambiguous" id="S3.SS4.p2.1.m1.2.3.1.cmml" xref="S3.SS4.p2.1.m1.2.3">subscript</csymbol><apply id="S3.SS4.p2.1.m1.2.3.2.cmml" xref="S3.SS4.p2.1.m1.2.3.2"><ci id="S3.SS4.p2.1.m1.2.3.2.1.cmml" xref="S3.SS4.p2.1.m1.2.3.2.1">^</ci><ci id="S3.SS4.p2.1.m1.2.3.2.2.cmml" xref="S3.SS4.p2.1.m1.2.3.2.2">ℬ</ci></apply><list id="S3.SS4.p2.1.m1.2.2.2.3.cmml" xref="S3.SS4.p2.1.m1.2.2.2.4"><ci id="S3.SS4.p2.1.m1.1.1.1.1.cmml" xref="S3.SS4.p2.1.m1.1.1.1.1">𝐼</ci><ci id="S3.SS4.p2.1.m1.2.2.2.2.cmml" xref="S3.SS4.p2.1.m1.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.1.m1.2c">\hat{\mathcal{B}}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.1.m1.2d">over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> covers each point in the true identified interval, <math alttext="\mathcal{B}_{I,k}" class="ltx_Math" display="inline" id="S3.SS4.p2.2.m2.2"><semantics id="S3.SS4.p2.2.m2.2a"><msub id="S3.SS4.p2.2.m2.2.3" xref="S3.SS4.p2.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p2.2.m2.2.3.2" xref="S3.SS4.p2.2.m2.2.3.2.cmml">ℬ</mi><mrow id="S3.SS4.p2.2.m2.2.2.2.4" xref="S3.SS4.p2.2.m2.2.2.2.3.cmml"><mi id="S3.SS4.p2.2.m2.1.1.1.1" xref="S3.SS4.p2.2.m2.1.1.1.1.cmml">I</mi><mo id="S3.SS4.p2.2.m2.2.2.2.4.1" xref="S3.SS4.p2.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.SS4.p2.2.m2.2.2.2.2" xref="S3.SS4.p2.2.m2.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.2.m2.2b"><apply id="S3.SS4.p2.2.m2.2.3.cmml" xref="S3.SS4.p2.2.m2.2.3"><csymbol cd="ambiguous" id="S3.SS4.p2.2.m2.2.3.1.cmml" xref="S3.SS4.p2.2.m2.2.3">subscript</csymbol><ci id="S3.SS4.p2.2.m2.2.3.2.cmml" xref="S3.SS4.p2.2.m2.2.3.2">ℬ</ci><list id="S3.SS4.p2.2.m2.2.2.2.3.cmml" xref="S3.SS4.p2.2.m2.2.2.2.4"><ci id="S3.SS4.p2.2.m2.1.1.1.1.cmml" xref="S3.SS4.p2.2.m2.1.1.1.1">𝐼</ci><ci id="S3.SS4.p2.2.m2.2.2.2.2.cmml" xref="S3.SS4.p2.2.m2.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.2.m2.2c">\mathcal{B}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.2.m2.2d">caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, with specified confidence level at least <math alttext="1-\alpha" class="ltx_Math" display="inline" id="S3.SS4.p2.3.m3.1"><semantics id="S3.SS4.p2.3.m3.1a"><mrow id="S3.SS4.p2.3.m3.1.1" xref="S3.SS4.p2.3.m3.1.1.cmml"><mn id="S3.SS4.p2.3.m3.1.1.2" xref="S3.SS4.p2.3.m3.1.1.2.cmml">1</mn><mo id="S3.SS4.p2.3.m3.1.1.1" xref="S3.SS4.p2.3.m3.1.1.1.cmml">−</mo><mi id="S3.SS4.p2.3.m3.1.1.3" xref="S3.SS4.p2.3.m3.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.3.m3.1b"><apply id="S3.SS4.p2.3.m3.1.1.cmml" xref="S3.SS4.p2.3.m3.1.1"><minus id="S3.SS4.p2.3.m3.1.1.1.cmml" xref="S3.SS4.p2.3.m3.1.1.1"></minus><cn id="S3.SS4.p2.3.m3.1.1.2.cmml" type="integer" xref="S3.SS4.p2.3.m3.1.1.2">1</cn><ci id="S3.SS4.p2.3.m3.1.1.3.cmml" xref="S3.SS4.p2.3.m3.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.3.m3.1c">1-\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.3.m3.1d">1 - italic_α</annotation></semantics></math>. The second one states that <math alttext="\hat{\mathcal{B}}_{I,k}" class="ltx_Math" display="inline" id="S3.SS4.p2.4.m4.2"><semantics id="S3.SS4.p2.4.m4.2a"><msub id="S3.SS4.p2.4.m4.2.3" xref="S3.SS4.p2.4.m4.2.3.cmml"><mover accent="true" id="S3.SS4.p2.4.m4.2.3.2" xref="S3.SS4.p2.4.m4.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p2.4.m4.2.3.2.2" xref="S3.SS4.p2.4.m4.2.3.2.2.cmml">ℬ</mi><mo id="S3.SS4.p2.4.m4.2.3.2.1" xref="S3.SS4.p2.4.m4.2.3.2.1.cmml">^</mo></mover><mrow id="S3.SS4.p2.4.m4.2.2.2.4" xref="S3.SS4.p2.4.m4.2.2.2.3.cmml"><mi id="S3.SS4.p2.4.m4.1.1.1.1" xref="S3.SS4.p2.4.m4.1.1.1.1.cmml">I</mi><mo id="S3.SS4.p2.4.m4.2.2.2.4.1" xref="S3.SS4.p2.4.m4.2.2.2.3.cmml">,</mo><mi id="S3.SS4.p2.4.m4.2.2.2.2" xref="S3.SS4.p2.4.m4.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.4.m4.2b"><apply id="S3.SS4.p2.4.m4.2.3.cmml" xref="S3.SS4.p2.4.m4.2.3"><csymbol cd="ambiguous" id="S3.SS4.p2.4.m4.2.3.1.cmml" xref="S3.SS4.p2.4.m4.2.3">subscript</csymbol><apply id="S3.SS4.p2.4.m4.2.3.2.cmml" xref="S3.SS4.p2.4.m4.2.3.2"><ci id="S3.SS4.p2.4.m4.2.3.2.1.cmml" xref="S3.SS4.p2.4.m4.2.3.2.1">^</ci><ci id="S3.SS4.p2.4.m4.2.3.2.2.cmml" xref="S3.SS4.p2.4.m4.2.3.2.2">ℬ</ci></apply><list id="S3.SS4.p2.4.m4.2.2.2.3.cmml" xref="S3.SS4.p2.4.m4.2.2.2.4"><ci id="S3.SS4.p2.4.m4.1.1.1.1.cmml" xref="S3.SS4.p2.4.m4.1.1.1.1">𝐼</ci><ci id="S3.SS4.p2.4.m4.2.2.2.2.cmml" xref="S3.SS4.p2.4.m4.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.4.m4.2c">\hat{\mathcal{B}}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.4.m4.2d">over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> covers <math alttext="\mathcal{B}_{I,k}" class="ltx_Math" display="inline" id="S3.SS4.p2.5.m5.2"><semantics id="S3.SS4.p2.5.m5.2a"><msub id="S3.SS4.p2.5.m5.2.3" xref="S3.SS4.p2.5.m5.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p2.5.m5.2.3.2" xref="S3.SS4.p2.5.m5.2.3.2.cmml">ℬ</mi><mrow id="S3.SS4.p2.5.m5.2.2.2.4" xref="S3.SS4.p2.5.m5.2.2.2.3.cmml"><mi id="S3.SS4.p2.5.m5.1.1.1.1" xref="S3.SS4.p2.5.m5.1.1.1.1.cmml">I</mi><mo id="S3.SS4.p2.5.m5.2.2.2.4.1" xref="S3.SS4.p2.5.m5.2.2.2.3.cmml">,</mo><mi id="S3.SS4.p2.5.m5.2.2.2.2" xref="S3.SS4.p2.5.m5.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.5.m5.2b"><apply id="S3.SS4.p2.5.m5.2.3.cmml" xref="S3.SS4.p2.5.m5.2.3"><csymbol cd="ambiguous" id="S3.SS4.p2.5.m5.2.3.1.cmml" xref="S3.SS4.p2.5.m5.2.3">subscript</csymbol><ci id="S3.SS4.p2.5.m5.2.3.2.cmml" xref="S3.SS4.p2.5.m5.2.3.2">ℬ</ci><list id="S3.SS4.p2.5.m5.2.2.2.3.cmml" xref="S3.SS4.p2.5.m5.2.2.2.4"><ci id="S3.SS4.p2.5.m5.1.1.1.1.cmml" xref="S3.SS4.p2.5.m5.1.1.1.1">𝐼</ci><ci id="S3.SS4.p2.5.m5.2.2.2.2.cmml" xref="S3.SS4.p2.5.m5.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.5.m5.2c">\mathcal{B}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.5.m5.2d">caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> entirely with specified confidence level at least <math alttext="1-\alpha" class="ltx_Math" display="inline" id="S3.SS4.p2.6.m6.1"><semantics id="S3.SS4.p2.6.m6.1a"><mrow id="S3.SS4.p2.6.m6.1.1" xref="S3.SS4.p2.6.m6.1.1.cmml"><mn id="S3.SS4.p2.6.m6.1.1.2" xref="S3.SS4.p2.6.m6.1.1.2.cmml">1</mn><mo id="S3.SS4.p2.6.m6.1.1.1" xref="S3.SS4.p2.6.m6.1.1.1.cmml">−</mo><mi id="S3.SS4.p2.6.m6.1.1.3" xref="S3.SS4.p2.6.m6.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.6.m6.1b"><apply id="S3.SS4.p2.6.m6.1.1.cmml" xref="S3.SS4.p2.6.m6.1.1"><minus id="S3.SS4.p2.6.m6.1.1.1.cmml" xref="S3.SS4.p2.6.m6.1.1.1"></minus><cn id="S3.SS4.p2.6.m6.1.1.2.cmml" type="integer" xref="S3.SS4.p2.6.m6.1.1.2">1</cn><ci id="S3.SS4.p2.6.m6.1.1.3.cmml" xref="S3.SS4.p2.6.m6.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.6.m6.1c">1-\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.6.m6.1d">1 - italic_α</annotation></semantics></math>. It can be seen that by nature of the test inversion approach, our estimator satisfies the first coverage property. Since the true value <math alttext="\beta_{\text{true},k}" class="ltx_Math" display="inline" id="S3.SS4.p2.7.m7.2"><semantics id="S3.SS4.p2.7.m7.2a"><msub id="S3.SS4.p2.7.m7.2.3" xref="S3.SS4.p2.7.m7.2.3.cmml"><mi id="S3.SS4.p2.7.m7.2.3.2" xref="S3.SS4.p2.7.m7.2.3.2.cmml">β</mi><mrow id="S3.SS4.p2.7.m7.2.2.2.4" xref="S3.SS4.p2.7.m7.2.2.2.3.cmml"><mtext id="S3.SS4.p2.7.m7.1.1.1.1" xref="S3.SS4.p2.7.m7.1.1.1.1a.cmml">true</mtext><mo id="S3.SS4.p2.7.m7.2.2.2.4.1" xref="S3.SS4.p2.7.m7.2.2.2.3.cmml">,</mo><mi id="S3.SS4.p2.7.m7.2.2.2.2" xref="S3.SS4.p2.7.m7.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.7.m7.2b"><apply id="S3.SS4.p2.7.m7.2.3.cmml" xref="S3.SS4.p2.7.m7.2.3"><csymbol cd="ambiguous" id="S3.SS4.p2.7.m7.2.3.1.cmml" xref="S3.SS4.p2.7.m7.2.3">subscript</csymbol><ci id="S3.SS4.p2.7.m7.2.3.2.cmml" xref="S3.SS4.p2.7.m7.2.3.2">𝛽</ci><list id="S3.SS4.p2.7.m7.2.2.2.3.cmml" xref="S3.SS4.p2.7.m7.2.2.2.4"><ci id="S3.SS4.p2.7.m7.1.1.1.1a.cmml" xref="S3.SS4.p2.7.m7.1.1.1.1"><mtext id="S3.SS4.p2.7.m7.1.1.1.1.cmml" mathsize="70%" xref="S3.SS4.p2.7.m7.1.1.1.1">true</mtext></ci><ci id="S3.SS4.p2.7.m7.2.2.2.2.cmml" xref="S3.SS4.p2.7.m7.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.7.m7.2c">\beta_{\text{true},k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.7.m7.2d">italic_β start_POSTSUBSCRIPT true , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is always an element of <math alttext="\mathcal{B}_{I,k}" class="ltx_Math" display="inline" id="S3.SS4.p2.8.m8.2"><semantics id="S3.SS4.p2.8.m8.2a"><msub id="S3.SS4.p2.8.m8.2.3" xref="S3.SS4.p2.8.m8.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p2.8.m8.2.3.2" xref="S3.SS4.p2.8.m8.2.3.2.cmml">ℬ</mi><mrow id="S3.SS4.p2.8.m8.2.2.2.4" xref="S3.SS4.p2.8.m8.2.2.2.3.cmml"><mi id="S3.SS4.p2.8.m8.1.1.1.1" xref="S3.SS4.p2.8.m8.1.1.1.1.cmml">I</mi><mo id="S3.SS4.p2.8.m8.2.2.2.4.1" xref="S3.SS4.p2.8.m8.2.2.2.3.cmml">,</mo><mi id="S3.SS4.p2.8.m8.2.2.2.2" xref="S3.SS4.p2.8.m8.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.8.m8.2b"><apply id="S3.SS4.p2.8.m8.2.3.cmml" xref="S3.SS4.p2.8.m8.2.3"><csymbol cd="ambiguous" id="S3.SS4.p2.8.m8.2.3.1.cmml" xref="S3.SS4.p2.8.m8.2.3">subscript</csymbol><ci id="S3.SS4.p2.8.m8.2.3.2.cmml" xref="S3.SS4.p2.8.m8.2.3.2">ℬ</ci><list id="S3.SS4.p2.8.m8.2.2.2.3.cmml" xref="S3.SS4.p2.8.m8.2.2.2.4"><ci id="S3.SS4.p2.8.m8.1.1.1.1.cmml" xref="S3.SS4.p2.8.m8.1.1.1.1">𝐼</ci><ci id="S3.SS4.p2.8.m8.2.2.2.2.cmml" xref="S3.SS4.p2.8.m8.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.8.m8.2c">\mathcal{B}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.8.m8.2d">caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, our estimated identified interval covers <math alttext="\beta_{\text{true},k}" class="ltx_Math" display="inline" id="S3.SS4.p2.9.m9.2"><semantics id="S3.SS4.p2.9.m9.2a"><msub id="S3.SS4.p2.9.m9.2.3" xref="S3.SS4.p2.9.m9.2.3.cmml"><mi id="S3.SS4.p2.9.m9.2.3.2" xref="S3.SS4.p2.9.m9.2.3.2.cmml">β</mi><mrow id="S3.SS4.p2.9.m9.2.2.2.4" xref="S3.SS4.p2.9.m9.2.2.2.3.cmml"><mtext id="S3.SS4.p2.9.m9.1.1.1.1" xref="S3.SS4.p2.9.m9.1.1.1.1a.cmml">true</mtext><mo id="S3.SS4.p2.9.m9.2.2.2.4.1" xref="S3.SS4.p2.9.m9.2.2.2.3.cmml">,</mo><mi id="S3.SS4.p2.9.m9.2.2.2.2" xref="S3.SS4.p2.9.m9.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.9.m9.2b"><apply id="S3.SS4.p2.9.m9.2.3.cmml" xref="S3.SS4.p2.9.m9.2.3"><csymbol cd="ambiguous" id="S3.SS4.p2.9.m9.2.3.1.cmml" xref="S3.SS4.p2.9.m9.2.3">subscript</csymbol><ci id="S3.SS4.p2.9.m9.2.3.2.cmml" xref="S3.SS4.p2.9.m9.2.3.2">𝛽</ci><list id="S3.SS4.p2.9.m9.2.2.2.3.cmml" xref="S3.SS4.p2.9.m9.2.2.2.4"><ci id="S3.SS4.p2.9.m9.1.1.1.1a.cmml" xref="S3.SS4.p2.9.m9.1.1.1.1"><mtext id="S3.SS4.p2.9.m9.1.1.1.1.cmml" mathsize="70%" xref="S3.SS4.p2.9.m9.1.1.1.1">true</mtext></ci><ci id="S3.SS4.p2.9.m9.2.2.2.2.cmml" xref="S3.SS4.p2.9.m9.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.9.m9.2c">\beta_{\text{true},k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.9.m9.2d">italic_β start_POSTSUBSCRIPT true , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> with specified probability at least <math alttext="1-\alpha" class="ltx_Math" display="inline" id="S3.SS4.p2.10.m10.1"><semantics id="S3.SS4.p2.10.m10.1a"><mrow id="S3.SS4.p2.10.m10.1.1" xref="S3.SS4.p2.10.m10.1.1.cmml"><mn id="S3.SS4.p2.10.m10.1.1.2" xref="S3.SS4.p2.10.m10.1.1.2.cmml">1</mn><mo id="S3.SS4.p2.10.m10.1.1.1" xref="S3.SS4.p2.10.m10.1.1.1.cmml">−</mo><mi id="S3.SS4.p2.10.m10.1.1.3" xref="S3.SS4.p2.10.m10.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.10.m10.1b"><apply id="S3.SS4.p2.10.m10.1.1.cmml" xref="S3.SS4.p2.10.m10.1.1"><minus id="S3.SS4.p2.10.m10.1.1.1.cmml" xref="S3.SS4.p2.10.m10.1.1.1"></minus><cn id="S3.SS4.p2.10.m10.1.1.2.cmml" type="integer" xref="S3.SS4.p2.10.m10.1.1.2">1</cn><ci id="S3.SS4.p2.10.m10.1.1.3.cmml" xref="S3.SS4.p2.10.m10.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.10.m10.1c">1-\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.10.m10.1d">1 - italic_α</annotation></semantics></math> and in this sense can be interpreted as a regular confidence interval. For a more comprehensive discussion on identified sets, we refer to <cite class="ltx_cite ltx_citemacro_cite">Canay and Shaikh, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib5" title="">2017</a>)</cite>.</p> </div> <div class="ltx_para" id="S3.SS4.p3"> <p class="ltx_p" id="S3.SS4.p3.2"><span class="ltx_text ltx_font_bold" id="S3.SS4.p3.2.1">Misspecification test.</span> Even though model (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E1" title="In 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">1</span></a>) with its underpinning assumptions <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I1.i1" title="item (A1) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A1)</span></a>-<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I3.i1" title="item (A9) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A9)</span></a> is flexible, it could still be misspecified. Handily, as is the case in many partial identification models, one can immediately derive a specification test from <math alttext="\hat{\mathcal{B}}_{I,k}" class="ltx_Math" display="inline" id="S3.SS4.p3.1.m1.2"><semantics id="S3.SS4.p3.1.m1.2a"><msub id="S3.SS4.p3.1.m1.2.3" xref="S3.SS4.p3.1.m1.2.3.cmml"><mover accent="true" id="S3.SS4.p3.1.m1.2.3.2" xref="S3.SS4.p3.1.m1.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p3.1.m1.2.3.2.2" xref="S3.SS4.p3.1.m1.2.3.2.2.cmml">ℬ</mi><mo id="S3.SS4.p3.1.m1.2.3.2.1" xref="S3.SS4.p3.1.m1.2.3.2.1.cmml">^</mo></mover><mrow id="S3.SS4.p3.1.m1.2.2.2.4" xref="S3.SS4.p3.1.m1.2.2.2.3.cmml"><mi id="S3.SS4.p3.1.m1.1.1.1.1" xref="S3.SS4.p3.1.m1.1.1.1.1.cmml">I</mi><mo id="S3.SS4.p3.1.m1.2.2.2.4.1" xref="S3.SS4.p3.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.SS4.p3.1.m1.2.2.2.2" xref="S3.SS4.p3.1.m1.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.1.m1.2b"><apply id="S3.SS4.p3.1.m1.2.3.cmml" xref="S3.SS4.p3.1.m1.2.3"><csymbol cd="ambiguous" id="S3.SS4.p3.1.m1.2.3.1.cmml" xref="S3.SS4.p3.1.m1.2.3">subscript</csymbol><apply id="S3.SS4.p3.1.m1.2.3.2.cmml" xref="S3.SS4.p3.1.m1.2.3.2"><ci id="S3.SS4.p3.1.m1.2.3.2.1.cmml" xref="S3.SS4.p3.1.m1.2.3.2.1">^</ci><ci id="S3.SS4.p3.1.m1.2.3.2.2.cmml" xref="S3.SS4.p3.1.m1.2.3.2.2">ℬ</ci></apply><list id="S3.SS4.p3.1.m1.2.2.2.3.cmml" xref="S3.SS4.p3.1.m1.2.2.2.4"><ci id="S3.SS4.p3.1.m1.1.1.1.1.cmml" xref="S3.SS4.p3.1.m1.1.1.1.1">𝐼</ci><ci id="S3.SS4.p3.1.m1.2.2.2.2.cmml" xref="S3.SS4.p3.1.m1.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.1.m1.2c">\hat{\mathcal{B}}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.1.m1.2d">over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. Indeed, the model can be rejected when <math alttext="\hat{\mathcal{B}}_{I,k}=\emptyset" class="ltx_Math" display="inline" id="S3.SS4.p3.2.m2.2"><semantics id="S3.SS4.p3.2.m2.2a"><mrow id="S3.SS4.p3.2.m2.2.3" xref="S3.SS4.p3.2.m2.2.3.cmml"><msub id="S3.SS4.p3.2.m2.2.3.2" xref="S3.SS4.p3.2.m2.2.3.2.cmml"><mover accent="true" id="S3.SS4.p3.2.m2.2.3.2.2" xref="S3.SS4.p3.2.m2.2.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p3.2.m2.2.3.2.2.2" xref="S3.SS4.p3.2.m2.2.3.2.2.2.cmml">ℬ</mi><mo id="S3.SS4.p3.2.m2.2.3.2.2.1" xref="S3.SS4.p3.2.m2.2.3.2.2.1.cmml">^</mo></mover><mrow id="S3.SS4.p3.2.m2.2.2.2.4" xref="S3.SS4.p3.2.m2.2.2.2.3.cmml"><mi id="S3.SS4.p3.2.m2.1.1.1.1" xref="S3.SS4.p3.2.m2.1.1.1.1.cmml">I</mi><mo id="S3.SS4.p3.2.m2.2.2.2.4.1" xref="S3.SS4.p3.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.SS4.p3.2.m2.2.2.2.2" xref="S3.SS4.p3.2.m2.2.2.2.2.cmml">k</mi></mrow></msub><mo id="S3.SS4.p3.2.m2.2.3.1" xref="S3.SS4.p3.2.m2.2.3.1.cmml">=</mo><mi id="S3.SS4.p3.2.m2.2.3.3" mathvariant="normal" xref="S3.SS4.p3.2.m2.2.3.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.2.m2.2b"><apply id="S3.SS4.p3.2.m2.2.3.cmml" xref="S3.SS4.p3.2.m2.2.3"><eq id="S3.SS4.p3.2.m2.2.3.1.cmml" xref="S3.SS4.p3.2.m2.2.3.1"></eq><apply id="S3.SS4.p3.2.m2.2.3.2.cmml" xref="S3.SS4.p3.2.m2.2.3.2"><csymbol cd="ambiguous" id="S3.SS4.p3.2.m2.2.3.2.1.cmml" xref="S3.SS4.p3.2.m2.2.3.2">subscript</csymbol><apply id="S3.SS4.p3.2.m2.2.3.2.2.cmml" xref="S3.SS4.p3.2.m2.2.3.2.2"><ci id="S3.SS4.p3.2.m2.2.3.2.2.1.cmml" xref="S3.SS4.p3.2.m2.2.3.2.2.1">^</ci><ci id="S3.SS4.p3.2.m2.2.3.2.2.2.cmml" xref="S3.SS4.p3.2.m2.2.3.2.2.2">ℬ</ci></apply><list id="S3.SS4.p3.2.m2.2.2.2.3.cmml" xref="S3.SS4.p3.2.m2.2.2.2.4"><ci id="S3.SS4.p3.2.m2.1.1.1.1.cmml" xref="S3.SS4.p3.2.m2.1.1.1.1">𝐼</ci><ci id="S3.SS4.p3.2.m2.2.2.2.2.cmml" xref="S3.SS4.p3.2.m2.2.2.2.2">𝑘</ci></list></apply><emptyset id="S3.SS4.p3.2.m2.2.3.3.cmml" xref="S3.SS4.p3.2.m2.2.3.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.2.m2.2c">\hat{\mathcal{B}}_{I,k}=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.2.m2.2d">over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT = ∅</annotation></semantics></math>. <cite class="ltx_cite ltx_citemacro_cite">Bugni et al., (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib4" title="">2015</a>)</cite> study several specification tests for partially identified models. They refer to this type of test as the “by-product” test (test BP). In their paper, they provide alternative specification tests which outperform test BP in certain situations. When power is of the essence, we therefore recommend to use such an alternative. Nevertheless, test BP is still a useful front line tool.</p> </div> <div class="ltx_para" id="S3.SS4.p4"> <p class="ltx_p" id="S3.SS4.p4.5"><span class="ltx_text ltx_font_bold" id="S3.SS4.p4.5.1">Pitfalls.</span> Lastly, we caution against two pitfalls of the method. A first pitfall is to include too many instrumental functions in the class <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S3.SS4.p4.1.m1.1"><semantics id="S3.SS4.p4.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p4.1.m1.1.1" xref="S3.SS4.p4.1.m1.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.1.m1.1b"><ci id="S3.SS4.p4.1.m1.1.1.cmml" xref="S3.SS4.p4.1.m1.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p4.1.m1.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.1.m1.1d">caligraphic_G</annotation></semantics></math>. At first, this may seem counter-intuitive. Indeed, including more instrumental functions will lead to more information being transferred when transforming the conditional to the unconditional moments. Hence, <math alttext="\mathcal{B}_{I,k}" class="ltx_Math" display="inline" id="S3.SS4.p4.2.m2.2"><semantics id="S3.SS4.p4.2.m2.2a"><msub id="S3.SS4.p4.2.m2.2.3" xref="S3.SS4.p4.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p4.2.m2.2.3.2" xref="S3.SS4.p4.2.m2.2.3.2.cmml">ℬ</mi><mrow id="S3.SS4.p4.2.m2.2.2.2.4" xref="S3.SS4.p4.2.m2.2.2.2.3.cmml"><mi id="S3.SS4.p4.2.m2.1.1.1.1" xref="S3.SS4.p4.2.m2.1.1.1.1.cmml">I</mi><mo id="S3.SS4.p4.2.m2.2.2.2.4.1" xref="S3.SS4.p4.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.SS4.p4.2.m2.2.2.2.2" xref="S3.SS4.p4.2.m2.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.2.m2.2b"><apply id="S3.SS4.p4.2.m2.2.3.cmml" xref="S3.SS4.p4.2.m2.2.3"><csymbol cd="ambiguous" id="S3.SS4.p4.2.m2.2.3.1.cmml" xref="S3.SS4.p4.2.m2.2.3">subscript</csymbol><ci id="S3.SS4.p4.2.m2.2.3.2.cmml" xref="S3.SS4.p4.2.m2.2.3.2">ℬ</ci><list id="S3.SS4.p4.2.m2.2.2.2.3.cmml" xref="S3.SS4.p4.2.m2.2.2.2.4"><ci id="S3.SS4.p4.2.m2.1.1.1.1.cmml" xref="S3.SS4.p4.2.m2.1.1.1.1">𝐼</ci><ci id="S3.SS4.p4.2.m2.2.2.2.2.cmml" xref="S3.SS4.p4.2.m2.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p4.2.m2.2c">\mathcal{B}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.2.m2.2d">caligraphic_B start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> can only be made smaller in this way and it seems reasonable to assume that this effect will also be visible in the estimator <math alttext="\hat{\mathcal{B}}_{I,k}" class="ltx_Math" display="inline" id="S3.SS4.p4.3.m3.2"><semantics id="S3.SS4.p4.3.m3.2a"><msub id="S3.SS4.p4.3.m3.2.3" xref="S3.SS4.p4.3.m3.2.3.cmml"><mover accent="true" id="S3.SS4.p4.3.m3.2.3.2" xref="S3.SS4.p4.3.m3.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p4.3.m3.2.3.2.2" xref="S3.SS4.p4.3.m3.2.3.2.2.cmml">ℬ</mi><mo id="S3.SS4.p4.3.m3.2.3.2.1" xref="S3.SS4.p4.3.m3.2.3.2.1.cmml">^</mo></mover><mrow id="S3.SS4.p4.3.m3.2.2.2.4" xref="S3.SS4.p4.3.m3.2.2.2.3.cmml"><mi id="S3.SS4.p4.3.m3.1.1.1.1" xref="S3.SS4.p4.3.m3.1.1.1.1.cmml">I</mi><mo id="S3.SS4.p4.3.m3.2.2.2.4.1" xref="S3.SS4.p4.3.m3.2.2.2.3.cmml">,</mo><mi id="S3.SS4.p4.3.m3.2.2.2.2" xref="S3.SS4.p4.3.m3.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.3.m3.2b"><apply id="S3.SS4.p4.3.m3.2.3.cmml" xref="S3.SS4.p4.3.m3.2.3"><csymbol cd="ambiguous" id="S3.SS4.p4.3.m3.2.3.1.cmml" xref="S3.SS4.p4.3.m3.2.3">subscript</csymbol><apply id="S3.SS4.p4.3.m3.2.3.2.cmml" xref="S3.SS4.p4.3.m3.2.3.2"><ci id="S3.SS4.p4.3.m3.2.3.2.1.cmml" xref="S3.SS4.p4.3.m3.2.3.2.1">^</ci><ci id="S3.SS4.p4.3.m3.2.3.2.2.cmml" xref="S3.SS4.p4.3.m3.2.3.2.2">ℬ</ci></apply><list id="S3.SS4.p4.3.m3.2.2.2.3.cmml" xref="S3.SS4.p4.3.m3.2.2.2.4"><ci id="S3.SS4.p4.3.m3.1.1.1.1.cmml" xref="S3.SS4.p4.3.m3.1.1.1.1">𝐼</ci><ci id="S3.SS4.p4.3.m3.2.2.2.2.cmml" xref="S3.SS4.p4.3.m3.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p4.3.m3.2c">\hat{\mathcal{B}}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.3.m3.2d">over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. While this is true up to some point, there is a balance to be struck: including more instrumental functions will result in more unconditional moments to test, increasing the variance of the test statistic, which in turn has the effect of making <math alttext="\hat{\mathcal{B}}_{I,k}" class="ltx_Math" display="inline" id="S3.SS4.p4.4.m4.2"><semantics id="S3.SS4.p4.4.m4.2a"><msub id="S3.SS4.p4.4.m4.2.3" xref="S3.SS4.p4.4.m4.2.3.cmml"><mover accent="true" id="S3.SS4.p4.4.m4.2.3.2" xref="S3.SS4.p4.4.m4.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p4.4.m4.2.3.2.2" xref="S3.SS4.p4.4.m4.2.3.2.2.cmml">ℬ</mi><mo id="S3.SS4.p4.4.m4.2.3.2.1" xref="S3.SS4.p4.4.m4.2.3.2.1.cmml">^</mo></mover><mrow id="S3.SS4.p4.4.m4.2.2.2.4" xref="S3.SS4.p4.4.m4.2.2.2.3.cmml"><mi id="S3.SS4.p4.4.m4.1.1.1.1" xref="S3.SS4.p4.4.m4.1.1.1.1.cmml">I</mi><mo id="S3.SS4.p4.4.m4.2.2.2.4.1" xref="S3.SS4.p4.4.m4.2.2.2.3.cmml">,</mo><mi id="S3.SS4.p4.4.m4.2.2.2.2" xref="S3.SS4.p4.4.m4.2.2.2.2.cmml">k</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.4.m4.2b"><apply id="S3.SS4.p4.4.m4.2.3.cmml" xref="S3.SS4.p4.4.m4.2.3"><csymbol cd="ambiguous" id="S3.SS4.p4.4.m4.2.3.1.cmml" xref="S3.SS4.p4.4.m4.2.3">subscript</csymbol><apply id="S3.SS4.p4.4.m4.2.3.2.cmml" xref="S3.SS4.p4.4.m4.2.3.2"><ci id="S3.SS4.p4.4.m4.2.3.2.1.cmml" xref="S3.SS4.p4.4.m4.2.3.2.1">^</ci><ci id="S3.SS4.p4.4.m4.2.3.2.2.cmml" xref="S3.SS4.p4.4.m4.2.3.2.2">ℬ</ci></apply><list id="S3.SS4.p4.4.m4.2.2.2.3.cmml" xref="S3.SS4.p4.4.m4.2.2.2.4"><ci id="S3.SS4.p4.4.m4.1.1.1.1.cmml" xref="S3.SS4.p4.4.m4.1.1.1.1">𝐼</ci><ci id="S3.SS4.p4.4.m4.2.2.2.2.cmml" xref="S3.SS4.p4.4.m4.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p4.4.m4.2c">\hat{\mathcal{B}}_{I,k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.4.m4.2d">over^ start_ARG caligraphic_B end_ARG start_POSTSUBSCRIPT italic_I , italic_k end_POSTSUBSCRIPT</annotation></semantics></math> larger. It is therefore recommended to try out several choices for <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S3.SS4.p4.5.m5.1"><semantics id="S3.SS4.p4.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p4.5.m5.1.1" xref="S3.SS4.p4.5.m5.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.5.m5.1b"><ci id="S3.SS4.p4.5.m5.1.1.cmml" xref="S3.SS4.p4.5.m5.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p4.5.m5.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.5.m5.1d">caligraphic_G</annotation></semantics></math> when applying this method. Secondly, when many covariates are included in the analysis, we recommend that they be standardized so as to keep the size of their linear combinations manageable, since they are the argument to an exponential function.</p> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Simulation study</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.29">To assess the performance of the proposed methodology, a simulation study is carried out. 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The parameter vector is specified as <math alttext="\beta_{\text{true}}(t)=(\log(t),1,-1)" class="ltx_Math" display="inline" id="S4.p1.3.m3.6"><semantics id="S4.p1.3.m3.6a"><mrow id="S4.p1.3.m3.6.6" xref="S4.p1.3.m3.6.6.cmml"><mrow id="S4.p1.3.m3.6.6.4" xref="S4.p1.3.m3.6.6.4.cmml"><msub id="S4.p1.3.m3.6.6.4.2" xref="S4.p1.3.m3.6.6.4.2.cmml"><mi id="S4.p1.3.m3.6.6.4.2.2" xref="S4.p1.3.m3.6.6.4.2.2.cmml">β</mi><mtext id="S4.p1.3.m3.6.6.4.2.3" xref="S4.p1.3.m3.6.6.4.2.3a.cmml">true</mtext></msub><mo id="S4.p1.3.m3.6.6.4.1" xref="S4.p1.3.m3.6.6.4.1.cmml">⁢</mo><mrow id="S4.p1.3.m3.6.6.4.3.2" xref="S4.p1.3.m3.6.6.4.cmml"><mo id="S4.p1.3.m3.6.6.4.3.2.1" stretchy="false" xref="S4.p1.3.m3.6.6.4.cmml">(</mo><mi id="S4.p1.3.m3.1.1" xref="S4.p1.3.m3.1.1.cmml">t</mi><mo id="S4.p1.3.m3.6.6.4.3.2.2" stretchy="false" xref="S4.p1.3.m3.6.6.4.cmml">)</mo></mrow></mrow><mo id="S4.p1.3.m3.6.6.3" xref="S4.p1.3.m3.6.6.3.cmml">=</mo><mrow id="S4.p1.3.m3.6.6.2.2" xref="S4.p1.3.m3.6.6.2.3.cmml"><mo id="S4.p1.3.m3.6.6.2.2.3" stretchy="false" xref="S4.p1.3.m3.6.6.2.3.cmml">(</mo><mrow id="S4.p1.3.m3.5.5.1.1.1.2" xref="S4.p1.3.m3.5.5.1.1.1.1.cmml"><mi id="S4.p1.3.m3.2.2" xref="S4.p1.3.m3.2.2.cmml">log</mi><mo id="S4.p1.3.m3.5.5.1.1.1.2a" xref="S4.p1.3.m3.5.5.1.1.1.1.cmml">⁡</mo><mrow id="S4.p1.3.m3.5.5.1.1.1.2.1" xref="S4.p1.3.m3.5.5.1.1.1.1.cmml"><mo id="S4.p1.3.m3.5.5.1.1.1.2.1.1" stretchy="false" xref="S4.p1.3.m3.5.5.1.1.1.1.cmml">(</mo><mi id="S4.p1.3.m3.3.3" xref="S4.p1.3.m3.3.3.cmml">t</mi><mo id="S4.p1.3.m3.5.5.1.1.1.2.1.2" stretchy="false" xref="S4.p1.3.m3.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.p1.3.m3.6.6.2.2.4" xref="S4.p1.3.m3.6.6.2.3.cmml">,</mo><mn id="S4.p1.3.m3.4.4" xref="S4.p1.3.m3.4.4.cmml">1</mn><mo id="S4.p1.3.m3.6.6.2.2.5" xref="S4.p1.3.m3.6.6.2.3.cmml">,</mo><mrow id="S4.p1.3.m3.6.6.2.2.2" xref="S4.p1.3.m3.6.6.2.2.2.cmml"><mo id="S4.p1.3.m3.6.6.2.2.2a" xref="S4.p1.3.m3.6.6.2.2.2.cmml">−</mo><mn id="S4.p1.3.m3.6.6.2.2.2.2" xref="S4.p1.3.m3.6.6.2.2.2.2.cmml">1</mn></mrow><mo id="S4.p1.3.m3.6.6.2.2.6" stretchy="false" xref="S4.p1.3.m3.6.6.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.3.m3.6b"><apply id="S4.p1.3.m3.6.6.cmml" xref="S4.p1.3.m3.6.6"><eq id="S4.p1.3.m3.6.6.3.cmml" xref="S4.p1.3.m3.6.6.3"></eq><apply id="S4.p1.3.m3.6.6.4.cmml" xref="S4.p1.3.m3.6.6.4"><times id="S4.p1.3.m3.6.6.4.1.cmml" xref="S4.p1.3.m3.6.6.4.1"></times><apply id="S4.p1.3.m3.6.6.4.2.cmml" xref="S4.p1.3.m3.6.6.4.2"><csymbol cd="ambiguous" id="S4.p1.3.m3.6.6.4.2.1.cmml" xref="S4.p1.3.m3.6.6.4.2">subscript</csymbol><ci id="S4.p1.3.m3.6.6.4.2.2.cmml" xref="S4.p1.3.m3.6.6.4.2.2">𝛽</ci><ci id="S4.p1.3.m3.6.6.4.2.3a.cmml" xref="S4.p1.3.m3.6.6.4.2.3"><mtext id="S4.p1.3.m3.6.6.4.2.3.cmml" mathsize="70%" xref="S4.p1.3.m3.6.6.4.2.3">true</mtext></ci></apply><ci id="S4.p1.3.m3.1.1.cmml" xref="S4.p1.3.m3.1.1">𝑡</ci></apply><vector id="S4.p1.3.m3.6.6.2.3.cmml" xref="S4.p1.3.m3.6.6.2.2"><apply id="S4.p1.3.m3.5.5.1.1.1.1.cmml" xref="S4.p1.3.m3.5.5.1.1.1.2"><log id="S4.p1.3.m3.2.2.cmml" xref="S4.p1.3.m3.2.2"></log><ci id="S4.p1.3.m3.3.3.cmml" xref="S4.p1.3.m3.3.3">𝑡</ci></apply><cn id="S4.p1.3.m3.4.4.cmml" type="integer" xref="S4.p1.3.m3.4.4">1</cn><apply id="S4.p1.3.m3.6.6.2.2.2.cmml" xref="S4.p1.3.m3.6.6.2.2.2"><minus id="S4.p1.3.m3.6.6.2.2.2.1.cmml" xref="S4.p1.3.m3.6.6.2.2.2"></minus><cn id="S4.p1.3.m3.6.6.2.2.2.2.cmml" type="integer" xref="S4.p1.3.m3.6.6.2.2.2.2">1</cn></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.3.m3.6c">\beta_{\text{true}}(t)=(\log(t),1,-1)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.3.m3.6d">italic_β start_POSTSUBSCRIPT true end_POSTSUBSCRIPT ( italic_t ) = ( roman_log ( italic_t ) , 1 , - 1 )</annotation></semantics></math>. 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italic_i 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_i 2 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> according to their respective distributions, we simulate latent times from the conditional joint distribution <math alttext="F_{T,C|X}(t,c|x)=\mathcal{C}_{\theta}(F_{T|X}(t|x),F_{C}(c))" class="ltx_Math" display="inline" id="S4.p1.5.m5.7"><semantics id="S4.p1.5.m5.7a"><mrow id="S4.p1.5.m5.7.7" xref="S4.p1.5.m5.7.7.cmml"><mrow id="S4.p1.5.m5.5.5.1" xref="S4.p1.5.m5.5.5.1.cmml"><msub id="S4.p1.5.m5.5.5.1.3" xref="S4.p1.5.m5.5.5.1.3.cmml"><mi id="S4.p1.5.m5.5.5.1.3.2" xref="S4.p1.5.m5.5.5.1.3.2.cmml">F</mi><mrow id="S4.p1.5.m5.2.2.2.2" xref="S4.p1.5.m5.2.2.2.3.cmml"><mi id="S4.p1.5.m5.1.1.1.1" xref="S4.p1.5.m5.1.1.1.1.cmml">T</mi><mo id="S4.p1.5.m5.2.2.2.2.2" xref="S4.p1.5.m5.2.2.2.3.cmml">,</mo><mrow id="S4.p1.5.m5.2.2.2.2.1" xref="S4.p1.5.m5.2.2.2.2.1.cmml"><mi 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id="S4.p1.5.m5.7d">italic_F start_POSTSUBSCRIPT italic_T , italic_C | italic_X end_POSTSUBSCRIPT ( italic_t , italic_c | italic_x ) = caligraphic_C start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_F start_POSTSUBSCRIPT italic_T | italic_X end_POSTSUBSCRIPT ( italic_t | italic_x ) , italic_F start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_c ) )</annotation></semantics></math>. In this formulation, <math alttext="F_{T|X}(\cdot|x)" class="ltx_math_unparsed" display="inline" id="S4.p1.6.m6.1"><semantics id="S4.p1.6.m6.1a"><mrow id="S4.p1.6.m6.1b"><msub id="S4.p1.6.m6.1.1"><mi id="S4.p1.6.m6.1.1.2">F</mi><mrow id="S4.p1.6.m6.1.1.3"><mi id="S4.p1.6.m6.1.1.3.2">T</mi><mo fence="false" id="S4.p1.6.m6.1.1.3.1">|</mo><mi id="S4.p1.6.m6.1.1.3.3">X</mi></mrow></msub><mrow id="S4.p1.6.m6.1.2"><mo id="S4.p1.6.m6.1.2.1" stretchy="false">(</mo><mo id="S4.p1.6.m6.1.2.2" lspace="0em" rspace="0em">⋅</mo><mo fence="false" id="S4.p1.6.m6.1.2.3" rspace="0.167em" stretchy="false">|</mo><mi id="S4.p1.6.m6.1.2.4">x</mi><mo id="S4.p1.6.m6.1.2.5" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S4.p1.6.m6.1c">F_{T|X}(\cdot|x)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.6.m6.1d">italic_F start_POSTSUBSCRIPT italic_T | italic_X end_POSTSUBSCRIPT ( ⋅ | italic_x )</annotation></semantics></math> takes the form as in model (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E1" title="In 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">1</span></a>), for which we will consider <math alttext="\Lambda(\cdot)=1-\exp(-\exp(\cdot))" class="ltx_Math" display="inline" id="S4.p1.7.m7.5"><semantics id="S4.p1.7.m7.5a"><mrow id="S4.p1.7.m7.5.5" xref="S4.p1.7.m7.5.5.cmml"><mrow id="S4.p1.7.m7.5.5.3" xref="S4.p1.7.m7.5.5.3.cmml"><mi id="S4.p1.7.m7.5.5.3.2" mathvariant="normal" xref="S4.p1.7.m7.5.5.3.2.cmml">Λ</mi><mo id="S4.p1.7.m7.5.5.3.1" xref="S4.p1.7.m7.5.5.3.1.cmml">⁢</mo><mrow id="S4.p1.7.m7.5.5.3.3.2" xref="S4.p1.7.m7.5.5.3.cmml"><mo id="S4.p1.7.m7.5.5.3.3.2.1" stretchy="false" xref="S4.p1.7.m7.5.5.3.cmml">(</mo><mo id="S4.p1.7.m7.1.1" lspace="0em" rspace="0em" xref="S4.p1.7.m7.1.1.cmml">⋅</mo><mo id="S4.p1.7.m7.5.5.3.3.2.2" stretchy="false" xref="S4.p1.7.m7.5.5.3.cmml">)</mo></mrow></mrow><mo id="S4.p1.7.m7.5.5.2" xref="S4.p1.7.m7.5.5.2.cmml">=</mo><mrow id="S4.p1.7.m7.5.5.1" xref="S4.p1.7.m7.5.5.1.cmml"><mn id="S4.p1.7.m7.5.5.1.3" xref="S4.p1.7.m7.5.5.1.3.cmml">1</mn><mo id="S4.p1.7.m7.5.5.1.2" xref="S4.p1.7.m7.5.5.1.2.cmml">−</mo><mrow id="S4.p1.7.m7.5.5.1.1.1" xref="S4.p1.7.m7.5.5.1.1.2.cmml"><mi id="S4.p1.7.m7.4.4" xref="S4.p1.7.m7.4.4.cmml">exp</mi><mo id="S4.p1.7.m7.5.5.1.1.1a" xref="S4.p1.7.m7.5.5.1.1.2.cmml">⁡</mo><mrow id="S4.p1.7.m7.5.5.1.1.1.1" xref="S4.p1.7.m7.5.5.1.1.2.cmml"><mo id="S4.p1.7.m7.5.5.1.1.1.1.2" stretchy="false" xref="S4.p1.7.m7.5.5.1.1.2.cmml">(</mo><mrow id="S4.p1.7.m7.5.5.1.1.1.1.1" xref="S4.p1.7.m7.5.5.1.1.1.1.1.cmml"><mo id="S4.p1.7.m7.5.5.1.1.1.1.1a" rspace="0.167em" xref="S4.p1.7.m7.5.5.1.1.1.1.1.cmml">−</mo><mrow id="S4.p1.7.m7.5.5.1.1.1.1.1.2.2" xref="S4.p1.7.m7.5.5.1.1.1.1.1.2.1.cmml"><mi id="S4.p1.7.m7.2.2" xref="S4.p1.7.m7.2.2.cmml">exp</mi><mo id="S4.p1.7.m7.5.5.1.1.1.1.1.2.2a" xref="S4.p1.7.m7.5.5.1.1.1.1.1.2.1.cmml">⁡</mo><mrow id="S4.p1.7.m7.5.5.1.1.1.1.1.2.2.1" xref="S4.p1.7.m7.5.5.1.1.1.1.1.2.1.cmml"><mo id="S4.p1.7.m7.5.5.1.1.1.1.1.2.2.1.1" stretchy="false" xref="S4.p1.7.m7.5.5.1.1.1.1.1.2.1.cmml">(</mo><mo id="S4.p1.7.m7.3.3" lspace="0em" rspace="0em" xref="S4.p1.7.m7.3.3.cmml">⋅</mo><mo id="S4.p1.7.m7.5.5.1.1.1.1.1.2.2.1.2" stretchy="false" xref="S4.p1.7.m7.5.5.1.1.1.1.1.2.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.p1.7.m7.5.5.1.1.1.1.3" stretchy="false" xref="S4.p1.7.m7.5.5.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.7.m7.5b"><apply id="S4.p1.7.m7.5.5.cmml" xref="S4.p1.7.m7.5.5"><eq id="S4.p1.7.m7.5.5.2.cmml" xref="S4.p1.7.m7.5.5.2"></eq><apply id="S4.p1.7.m7.5.5.3.cmml" xref="S4.p1.7.m7.5.5.3"><times id="S4.p1.7.m7.5.5.3.1.cmml" xref="S4.p1.7.m7.5.5.3.1"></times><ci id="S4.p1.7.m7.5.5.3.2.cmml" xref="S4.p1.7.m7.5.5.3.2">Λ</ci><ci id="S4.p1.7.m7.1.1.cmml" xref="S4.p1.7.m7.1.1">⋅</ci></apply><apply id="S4.p1.7.m7.5.5.1.cmml" xref="S4.p1.7.m7.5.5.1"><minus id="S4.p1.7.m7.5.5.1.2.cmml" xref="S4.p1.7.m7.5.5.1.2"></minus><cn id="S4.p1.7.m7.5.5.1.3.cmml" type="integer" xref="S4.p1.7.m7.5.5.1.3">1</cn><apply id="S4.p1.7.m7.5.5.1.1.2.cmml" xref="S4.p1.7.m7.5.5.1.1.1"><exp id="S4.p1.7.m7.4.4.cmml" xref="S4.p1.7.m7.4.4"></exp><apply id="S4.p1.7.m7.5.5.1.1.1.1.1.cmml" xref="S4.p1.7.m7.5.5.1.1.1.1.1"><minus id="S4.p1.7.m7.5.5.1.1.1.1.1.1.cmml" xref="S4.p1.7.m7.5.5.1.1.1.1.1"></minus><apply id="S4.p1.7.m7.5.5.1.1.1.1.1.2.1.cmml" xref="S4.p1.7.m7.5.5.1.1.1.1.1.2.2"><exp id="S4.p1.7.m7.2.2.cmml" xref="S4.p1.7.m7.2.2"></exp><ci id="S4.p1.7.m7.3.3.cmml" xref="S4.p1.7.m7.3.3">⋅</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.7.m7.5c">\Lambda(\cdot)=1-\exp(-\exp(\cdot))</annotation><annotation encoding="application/x-llamapun" id="S4.p1.7.m7.5d">roman_Λ ( ⋅ ) = 1 - roman_exp ( - roman_exp ( ⋅ ) )</annotation></semantics></math>, referred to as the Cox link function. The censoring distribution <math alttext="F_{C}" class="ltx_Math" display="inline" id="S4.p1.8.m8.1"><semantics id="S4.p1.8.m8.1a"><msub id="S4.p1.8.m8.1.1" xref="S4.p1.8.m8.1.1.cmml"><mi id="S4.p1.8.m8.1.1.2" xref="S4.p1.8.m8.1.1.2.cmml">F</mi><mi id="S4.p1.8.m8.1.1.3" xref="S4.p1.8.m8.1.1.3.cmml">C</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p1.8.m8.1b"><apply id="S4.p1.8.m8.1.1.cmml" xref="S4.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.p1.8.m8.1.1.1.cmml" xref="S4.p1.8.m8.1.1">subscript</csymbol><ci id="S4.p1.8.m8.1.1.2.cmml" xref="S4.p1.8.m8.1.1.2">𝐹</ci><ci id="S4.p1.8.m8.1.1.3.cmml" xref="S4.p1.8.m8.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.8.m8.1c">F_{C}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.8.m8.1d">italic_F start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> is taken to be an exponential distribution with parameter <math alttext="\lambda\in(0,2)" class="ltx_Math" display="inline" id="S4.p1.9.m9.2"><semantics id="S4.p1.9.m9.2a"><mrow id="S4.p1.9.m9.2.3" xref="S4.p1.9.m9.2.3.cmml"><mi id="S4.p1.9.m9.2.3.2" xref="S4.p1.9.m9.2.3.2.cmml">λ</mi><mo id="S4.p1.9.m9.2.3.1" xref="S4.p1.9.m9.2.3.1.cmml">∈</mo><mrow id="S4.p1.9.m9.2.3.3.2" xref="S4.p1.9.m9.2.3.3.1.cmml"><mo id="S4.p1.9.m9.2.3.3.2.1" stretchy="false" xref="S4.p1.9.m9.2.3.3.1.cmml">(</mo><mn id="S4.p1.9.m9.1.1" xref="S4.p1.9.m9.1.1.cmml">0</mn><mo id="S4.p1.9.m9.2.3.3.2.2" xref="S4.p1.9.m9.2.3.3.1.cmml">,</mo><mn id="S4.p1.9.m9.2.2" xref="S4.p1.9.m9.2.2.cmml">2</mn><mo id="S4.p1.9.m9.2.3.3.2.3" stretchy="false" xref="S4.p1.9.m9.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.9.m9.2b"><apply id="S4.p1.9.m9.2.3.cmml" xref="S4.p1.9.m9.2.3"><in id="S4.p1.9.m9.2.3.1.cmml" xref="S4.p1.9.m9.2.3.1"></in><ci id="S4.p1.9.m9.2.3.2.cmml" xref="S4.p1.9.m9.2.3.2">𝜆</ci><interval closure="open" id="S4.p1.9.m9.2.3.3.1.cmml" xref="S4.p1.9.m9.2.3.3.2"><cn id="S4.p1.9.m9.1.1.cmml" type="integer" xref="S4.p1.9.m9.1.1">0</cn><cn id="S4.p1.9.m9.2.2.cmml" type="integer" xref="S4.p1.9.m9.2.2">2</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.9.m9.2c">\lambda\in(0,2)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.9.m9.2d">italic_λ ∈ ( 0 , 2 )</annotation></semantics></math>, which is adapted in each design in order to control the percentage of censored observations. The copula <math alttext="\mathcal{C}_{\theta}" class="ltx_Math" display="inline" id="S4.p1.10.m10.1"><semantics id="S4.p1.10.m10.1a"><msub id="S4.p1.10.m10.1.1" xref="S4.p1.10.m10.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.p1.10.m10.1.1.2" xref="S4.p1.10.m10.1.1.2.cmml">𝒞</mi><mi id="S4.p1.10.m10.1.1.3" xref="S4.p1.10.m10.1.1.3.cmml">θ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p1.10.m10.1b"><apply id="S4.p1.10.m10.1.1.cmml" xref="S4.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S4.p1.10.m10.1.1.1.cmml" xref="S4.p1.10.m10.1.1">subscript</csymbol><ci id="S4.p1.10.m10.1.1.2.cmml" xref="S4.p1.10.m10.1.1.2">𝒞</ci><ci id="S4.p1.10.m10.1.1.3.cmml" xref="S4.p1.10.m10.1.1.3">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.10.m10.1c">\mathcal{C}_{\theta}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.10.m10.1d">caligraphic_C start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT</annotation></semantics></math> is taken to be a Frank copula where we let <math alttext="\theta" class="ltx_Math" display="inline" id="S4.p1.11.m11.1"><semantics id="S4.p1.11.m11.1a"><mi id="S4.p1.11.m11.1.1" xref="S4.p1.11.m11.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.p1.11.m11.1b"><ci id="S4.p1.11.m11.1.1.cmml" xref="S4.p1.11.m11.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.11.m11.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.p1.11.m11.1d">italic_θ</annotation></semantics></math> equal <math alttext="6" class="ltx_Math" display="inline" id="S4.p1.12.m12.1"><semantics id="S4.p1.12.m12.1a"><mn id="S4.p1.12.m12.1.1" xref="S4.p1.12.m12.1.1.cmml">6</mn><annotation-xml encoding="MathML-Content" id="S4.p1.12.m12.1b"><cn id="S4.p1.12.m12.1.1.cmml" type="integer" xref="S4.p1.12.m12.1.1">6</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.12.m12.1c">6</annotation><annotation encoding="application/x-llamapun" id="S4.p1.12.m12.1d">6</annotation></semantics></math>, <math alttext="0" class="ltx_Math" display="inline" id="S4.p1.13.m13.1"><semantics id="S4.p1.13.m13.1a"><mn id="S4.p1.13.m13.1.1" xref="S4.p1.13.m13.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.p1.13.m13.1b"><cn id="S4.p1.13.m13.1.1.cmml" type="integer" xref="S4.p1.13.m13.1.1">0</cn></annotation-xml></semantics></math> or <math alttext="-6" class="ltx_Math" display="inline" id="S4.p1.14.m14.1"><semantics id="S4.p1.14.m14.1a"><mrow id="S4.p1.14.m14.1.1" xref="S4.p1.14.m14.1.1.cmml"><mo id="S4.p1.14.m14.1.1a" xref="S4.p1.14.m14.1.1.cmml">−</mo><mn id="S4.p1.14.m14.1.1.2" xref="S4.p1.14.m14.1.1.2.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.14.m14.1b"><apply id="S4.p1.14.m14.1.1.cmml" xref="S4.p1.14.m14.1.1"><minus id="S4.p1.14.m14.1.1.1.cmml" xref="S4.p1.14.m14.1.1"></minus><cn id="S4.p1.14.m14.1.1.2.cmml" type="integer" xref="S4.p1.14.m14.1.1.2">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.14.m14.1c">-6</annotation><annotation encoding="application/x-llamapun" id="S4.p1.14.m14.1d">- 6</annotation></semantics></math>, leading to Kendall’s tau equal to <math alttext="0.51" class="ltx_Math" display="inline" id="S4.p1.15.m15.1"><semantics id="S4.p1.15.m15.1a"><mn id="S4.p1.15.m15.1.1" xref="S4.p1.15.m15.1.1.cmml">0.51</mn><annotation-xml encoding="MathML-Content" id="S4.p1.15.m15.1b"><cn id="S4.p1.15.m15.1.1.cmml" type="float" xref="S4.p1.15.m15.1.1">0.51</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.15.m15.1c">0.51</annotation><annotation encoding="application/x-llamapun" id="S4.p1.15.m15.1d">0.51</annotation></semantics></math>, <math alttext="0" class="ltx_Math" display="inline" id="S4.p1.16.m16.1"><semantics id="S4.p1.16.m16.1a"><mn id="S4.p1.16.m16.1.1" xref="S4.p1.16.m16.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.p1.16.m16.1b"><cn id="S4.p1.16.m16.1.1.cmml" type="integer" xref="S4.p1.16.m16.1.1">0</cn></annotation-xml></semantics></math> or <math alttext="-0.51" class="ltx_Math" display="inline" id="S4.p1.17.m17.1"><semantics id="S4.p1.17.m17.1a"><mrow id="S4.p1.17.m17.1.1" xref="S4.p1.17.m17.1.1.cmml"><mo id="S4.p1.17.m17.1.1a" xref="S4.p1.17.m17.1.1.cmml">−</mo><mn id="S4.p1.17.m17.1.1.2" xref="S4.p1.17.m17.1.1.2.cmml">0.51</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.17.m17.1b"><apply id="S4.p1.17.m17.1.1.cmml" xref="S4.p1.17.m17.1.1"><minus id="S4.p1.17.m17.1.1.1.cmml" xref="S4.p1.17.m17.1.1"></minus><cn id="S4.p1.17.m17.1.1.2.cmml" type="float" xref="S4.p1.17.m17.1.1.2">0.51</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.17.m17.1c">-0.51</annotation><annotation encoding="application/x-llamapun" id="S4.p1.17.m17.1d">- 0.51</annotation></semantics></math>, respectively. Latent times are then simulated by first generating pairs <math alttext="(u_{i1},u_{i2})_{i=1}^{n}" class="ltx_Math" display="inline" id="S4.p1.18.m18.2"><semantics id="S4.p1.18.m18.2a"><msubsup id="S4.p1.18.m18.2.2" xref="S4.p1.18.m18.2.2.cmml"><mrow id="S4.p1.18.m18.2.2.2.2.2" xref="S4.p1.18.m18.2.2.2.2.3.cmml"><mo id="S4.p1.18.m18.2.2.2.2.2.3" stretchy="false" xref="S4.p1.18.m18.2.2.2.2.3.cmml">(</mo><msub id="S4.p1.18.m18.1.1.1.1.1.1" xref="S4.p1.18.m18.1.1.1.1.1.1.cmml"><mi id="S4.p1.18.m18.1.1.1.1.1.1.2" xref="S4.p1.18.m18.1.1.1.1.1.1.2.cmml">u</mi><mrow id="S4.p1.18.m18.1.1.1.1.1.1.3" xref="S4.p1.18.m18.1.1.1.1.1.1.3.cmml"><mi id="S4.p1.18.m18.1.1.1.1.1.1.3.2" xref="S4.p1.18.m18.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="S4.p1.18.m18.1.1.1.1.1.1.3.1" xref="S4.p1.18.m18.1.1.1.1.1.1.3.1.cmml">⁢</mo><mn id="S4.p1.18.m18.1.1.1.1.1.1.3.3" xref="S4.p1.18.m18.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S4.p1.18.m18.2.2.2.2.2.4" xref="S4.p1.18.m18.2.2.2.2.3.cmml">,</mo><msub id="S4.p1.18.m18.2.2.2.2.2.2" xref="S4.p1.18.m18.2.2.2.2.2.2.cmml"><mi id="S4.p1.18.m18.2.2.2.2.2.2.2" xref="S4.p1.18.m18.2.2.2.2.2.2.2.cmml">u</mi><mrow id="S4.p1.18.m18.2.2.2.2.2.2.3" xref="S4.p1.18.m18.2.2.2.2.2.2.3.cmml"><mi id="S4.p1.18.m18.2.2.2.2.2.2.3.2" xref="S4.p1.18.m18.2.2.2.2.2.2.3.2.cmml">i</mi><mo id="S4.p1.18.m18.2.2.2.2.2.2.3.1" xref="S4.p1.18.m18.2.2.2.2.2.2.3.1.cmml">⁢</mo><mn id="S4.p1.18.m18.2.2.2.2.2.2.3.3" xref="S4.p1.18.m18.2.2.2.2.2.2.3.3.cmml">2</mn></mrow></msub><mo id="S4.p1.18.m18.2.2.2.2.2.5" stretchy="false" xref="S4.p1.18.m18.2.2.2.2.3.cmml">)</mo></mrow><mrow id="S4.p1.18.m18.2.2.2.4" xref="S4.p1.18.m18.2.2.2.4.cmml"><mi id="S4.p1.18.m18.2.2.2.4.2" xref="S4.p1.18.m18.2.2.2.4.2.cmml">i</mi><mo id="S4.p1.18.m18.2.2.2.4.1" xref="S4.p1.18.m18.2.2.2.4.1.cmml">=</mo><mn id="S4.p1.18.m18.2.2.2.4.3" xref="S4.p1.18.m18.2.2.2.4.3.cmml">1</mn></mrow><mi id="S4.p1.18.m18.2.2.4" xref="S4.p1.18.m18.2.2.4.cmml">n</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.p1.18.m18.2b"><apply id="S4.p1.18.m18.2.2.cmml" xref="S4.p1.18.m18.2.2"><csymbol cd="ambiguous" id="S4.p1.18.m18.2.2.3.cmml" xref="S4.p1.18.m18.2.2">superscript</csymbol><apply id="S4.p1.18.m18.2.2.2.cmml" xref="S4.p1.18.m18.2.2"><csymbol cd="ambiguous" id="S4.p1.18.m18.2.2.2.3.cmml" xref="S4.p1.18.m18.2.2">subscript</csymbol><interval closure="open" id="S4.p1.18.m18.2.2.2.2.3.cmml" xref="S4.p1.18.m18.2.2.2.2.2"><apply id="S4.p1.18.m18.1.1.1.1.1.1.cmml" xref="S4.p1.18.m18.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.p1.18.m18.1.1.1.1.1.1.1.cmml" xref="S4.p1.18.m18.1.1.1.1.1.1">subscript</csymbol><ci id="S4.p1.18.m18.1.1.1.1.1.1.2.cmml" xref="S4.p1.18.m18.1.1.1.1.1.1.2">𝑢</ci><apply id="S4.p1.18.m18.1.1.1.1.1.1.3.cmml" xref="S4.p1.18.m18.1.1.1.1.1.1.3"><times id="S4.p1.18.m18.1.1.1.1.1.1.3.1.cmml" xref="S4.p1.18.m18.1.1.1.1.1.1.3.1"></times><ci id="S4.p1.18.m18.1.1.1.1.1.1.3.2.cmml" xref="S4.p1.18.m18.1.1.1.1.1.1.3.2">𝑖</ci><cn id="S4.p1.18.m18.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.p1.18.m18.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="S4.p1.18.m18.2.2.2.2.2.2.cmml" xref="S4.p1.18.m18.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.p1.18.m18.2.2.2.2.2.2.1.cmml" xref="S4.p1.18.m18.2.2.2.2.2.2">subscript</csymbol><ci id="S4.p1.18.m18.2.2.2.2.2.2.2.cmml" xref="S4.p1.18.m18.2.2.2.2.2.2.2">𝑢</ci><apply id="S4.p1.18.m18.2.2.2.2.2.2.3.cmml" xref="S4.p1.18.m18.2.2.2.2.2.2.3"><times id="S4.p1.18.m18.2.2.2.2.2.2.3.1.cmml" xref="S4.p1.18.m18.2.2.2.2.2.2.3.1"></times><ci id="S4.p1.18.m18.2.2.2.2.2.2.3.2.cmml" xref="S4.p1.18.m18.2.2.2.2.2.2.3.2">𝑖</ci><cn id="S4.p1.18.m18.2.2.2.2.2.2.3.3.cmml" type="integer" xref="S4.p1.18.m18.2.2.2.2.2.2.3.3">2</cn></apply></apply></interval><apply id="S4.p1.18.m18.2.2.2.4.cmml" xref="S4.p1.18.m18.2.2.2.4"><eq id="S4.p1.18.m18.2.2.2.4.1.cmml" xref="S4.p1.18.m18.2.2.2.4.1"></eq><ci id="S4.p1.18.m18.2.2.2.4.2.cmml" xref="S4.p1.18.m18.2.2.2.4.2">𝑖</ci><cn id="S4.p1.18.m18.2.2.2.4.3.cmml" type="integer" xref="S4.p1.18.m18.2.2.2.4.3">1</cn></apply></apply><ci id="S4.p1.18.m18.2.2.4.cmml" xref="S4.p1.18.m18.2.2.4">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.18.m18.2c">(u_{i1},u_{i2})_{i=1}^{n}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.18.m18.2d">( italic_u start_POSTSUBSCRIPT italic_i 1 end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT italic_i 2 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> of possibly dependent uniform variables <math alttext="(U_{1},U_{2})" class="ltx_Math" display="inline" id="S4.p1.19.m19.2"><semantics id="S4.p1.19.m19.2a"><mrow id="S4.p1.19.m19.2.2.2" xref="S4.p1.19.m19.2.2.3.cmml"><mo id="S4.p1.19.m19.2.2.2.3" stretchy="false" xref="S4.p1.19.m19.2.2.3.cmml">(</mo><msub id="S4.p1.19.m19.1.1.1.1" xref="S4.p1.19.m19.1.1.1.1.cmml"><mi id="S4.p1.19.m19.1.1.1.1.2" xref="S4.p1.19.m19.1.1.1.1.2.cmml">U</mi><mn id="S4.p1.19.m19.1.1.1.1.3" xref="S4.p1.19.m19.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.p1.19.m19.2.2.2.4" xref="S4.p1.19.m19.2.2.3.cmml">,</mo><msub id="S4.p1.19.m19.2.2.2.2" xref="S4.p1.19.m19.2.2.2.2.cmml"><mi id="S4.p1.19.m19.2.2.2.2.2" xref="S4.p1.19.m19.2.2.2.2.2.cmml">U</mi><mn id="S4.p1.19.m19.2.2.2.2.3" xref="S4.p1.19.m19.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.p1.19.m19.2.2.2.5" stretchy="false" xref="S4.p1.19.m19.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.19.m19.2b"><interval closure="open" id="S4.p1.19.m19.2.2.3.cmml" xref="S4.p1.19.m19.2.2.2"><apply id="S4.p1.19.m19.1.1.1.1.cmml" xref="S4.p1.19.m19.1.1.1.1"><csymbol cd="ambiguous" id="S4.p1.19.m19.1.1.1.1.1.cmml" xref="S4.p1.19.m19.1.1.1.1">subscript</csymbol><ci id="S4.p1.19.m19.1.1.1.1.2.cmml" xref="S4.p1.19.m19.1.1.1.1.2">𝑈</ci><cn id="S4.p1.19.m19.1.1.1.1.3.cmml" type="integer" xref="S4.p1.19.m19.1.1.1.1.3">1</cn></apply><apply id="S4.p1.19.m19.2.2.2.2.cmml" xref="S4.p1.19.m19.2.2.2.2"><csymbol cd="ambiguous" id="S4.p1.19.m19.2.2.2.2.1.cmml" xref="S4.p1.19.m19.2.2.2.2">subscript</csymbol><ci id="S4.p1.19.m19.2.2.2.2.2.cmml" xref="S4.p1.19.m19.2.2.2.2.2">𝑈</ci><cn id="S4.p1.19.m19.2.2.2.2.3.cmml" type="integer" xref="S4.p1.19.m19.2.2.2.2.3">2</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.19.m19.2c">(U_{1},U_{2})</annotation><annotation encoding="application/x-llamapun" id="S4.p1.19.m19.2d">( italic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> according to the copula <math alttext="\mathcal{C}_{\theta}" class="ltx_Math" display="inline" id="S4.p1.20.m20.1"><semantics id="S4.p1.20.m20.1a"><msub id="S4.p1.20.m20.1.1" xref="S4.p1.20.m20.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.p1.20.m20.1.1.2" xref="S4.p1.20.m20.1.1.2.cmml">𝒞</mi><mi id="S4.p1.20.m20.1.1.3" xref="S4.p1.20.m20.1.1.3.cmml">θ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p1.20.m20.1b"><apply id="S4.p1.20.m20.1.1.cmml" xref="S4.p1.20.m20.1.1"><csymbol cd="ambiguous" id="S4.p1.20.m20.1.1.1.cmml" xref="S4.p1.20.m20.1.1">subscript</csymbol><ci id="S4.p1.20.m20.1.1.2.cmml" xref="S4.p1.20.m20.1.1.2">𝒞</ci><ci id="S4.p1.20.m20.1.1.3.cmml" xref="S4.p1.20.m20.1.1.3">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.20.m20.1c">\mathcal{C}_{\theta}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.20.m20.1d">caligraphic_C start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT</annotation></semantics></math> and then applying the probability integral transform to obtain realizations <math alttext="(t_{i},c_{i})_{i=1}^{n}" class="ltx_Math" display="inline" id="S4.p1.21.m21.2"><semantics id="S4.p1.21.m21.2a"><msubsup id="S4.p1.21.m21.2.2" xref="S4.p1.21.m21.2.2.cmml"><mrow id="S4.p1.21.m21.2.2.2.2.2" xref="S4.p1.21.m21.2.2.2.2.3.cmml"><mo id="S4.p1.21.m21.2.2.2.2.2.3" stretchy="false" xref="S4.p1.21.m21.2.2.2.2.3.cmml">(</mo><msub id="S4.p1.21.m21.1.1.1.1.1.1" xref="S4.p1.21.m21.1.1.1.1.1.1.cmml"><mi id="S4.p1.21.m21.1.1.1.1.1.1.2" xref="S4.p1.21.m21.1.1.1.1.1.1.2.cmml">t</mi><mi id="S4.p1.21.m21.1.1.1.1.1.1.3" xref="S4.p1.21.m21.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.p1.21.m21.2.2.2.2.2.4" xref="S4.p1.21.m21.2.2.2.2.3.cmml">,</mo><msub id="S4.p1.21.m21.2.2.2.2.2.2" xref="S4.p1.21.m21.2.2.2.2.2.2.cmml"><mi id="S4.p1.21.m21.2.2.2.2.2.2.2" xref="S4.p1.21.m21.2.2.2.2.2.2.2.cmml">c</mi><mi id="S4.p1.21.m21.2.2.2.2.2.2.3" xref="S4.p1.21.m21.2.2.2.2.2.2.3.cmml">i</mi></msub><mo id="S4.p1.21.m21.2.2.2.2.2.5" stretchy="false" xref="S4.p1.21.m21.2.2.2.2.3.cmml">)</mo></mrow><mrow id="S4.p1.21.m21.2.2.2.4" xref="S4.p1.21.m21.2.2.2.4.cmml"><mi id="S4.p1.21.m21.2.2.2.4.2" xref="S4.p1.21.m21.2.2.2.4.2.cmml">i</mi><mo id="S4.p1.21.m21.2.2.2.4.1" xref="S4.p1.21.m21.2.2.2.4.1.cmml">=</mo><mn 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xref="S4.p1.23.m23.1.1.1.1.1.1.2.2">𝑢</ci><apply id="S4.p1.23.m23.1.1.1.1.1.1.2.3.cmml" xref="S4.p1.23.m23.1.1.1.1.1.1.2.3"><times id="S4.p1.23.m23.1.1.1.1.1.1.2.3.1.cmml" xref="S4.p1.23.m23.1.1.1.1.1.1.2.3.1"></times><ci id="S4.p1.23.m23.1.1.1.1.1.1.2.3.2.cmml" xref="S4.p1.23.m23.1.1.1.1.1.1.2.3.2">𝑖</ci><cn id="S4.p1.23.m23.1.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S4.p1.23.m23.1.1.1.1.1.1.2.3.3">1</cn></apply></apply><apply id="S4.p1.23.m23.1.1.1.1.1.1.3.cmml" xref="S4.p1.23.m23.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.p1.23.m23.1.1.1.1.1.1.3.1.cmml" xref="S4.p1.23.m23.1.1.1.1.1.1.3">subscript</csymbol><ci id="S4.p1.23.m23.1.1.1.1.1.1.3.2.cmml" xref="S4.p1.23.m23.1.1.1.1.1.1.3.2">𝑥</ci><ci id="S4.p1.23.m23.1.1.1.1.1.1.3.3.cmml" xref="S4.p1.23.m23.1.1.1.1.1.1.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.23.m23.1c">t_{i}=F^{-1}_{T|X}(u_{i1}|x_{i})</annotation><annotation encoding="application/x-llamapun" id="S4.p1.23.m23.1d">italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_T | italic_X end_POSTSUBSCRIPT ( italic_u start_POSTSUBSCRIPT italic_i 1 end_POSTSUBSCRIPT | italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="c_{i}=F^{-1}_{C}(u_{i2})" class="ltx_Math" display="inline" id="S4.p1.24.m24.1"><semantics id="S4.p1.24.m24.1a"><mrow id="S4.p1.24.m24.1.1" xref="S4.p1.24.m24.1.1.cmml"><msub id="S4.p1.24.m24.1.1.3" xref="S4.p1.24.m24.1.1.3.cmml"><mi id="S4.p1.24.m24.1.1.3.2" xref="S4.p1.24.m24.1.1.3.2.cmml">c</mi><mi id="S4.p1.24.m24.1.1.3.3" xref="S4.p1.24.m24.1.1.3.3.cmml">i</mi></msub><mo id="S4.p1.24.m24.1.1.2" xref="S4.p1.24.m24.1.1.2.cmml">=</mo><mrow id="S4.p1.24.m24.1.1.1" xref="S4.p1.24.m24.1.1.1.cmml"><msubsup id="S4.p1.24.m24.1.1.1.3" xref="S4.p1.24.m24.1.1.1.3.cmml"><mi id="S4.p1.24.m24.1.1.1.3.2.2" xref="S4.p1.24.m24.1.1.1.3.2.2.cmml">F</mi><mi id="S4.p1.24.m24.1.1.1.3.3" xref="S4.p1.24.m24.1.1.1.3.3.cmml">C</mi><mrow id="S4.p1.24.m24.1.1.1.3.2.3" xref="S4.p1.24.m24.1.1.1.3.2.3.cmml"><mo id="S4.p1.24.m24.1.1.1.3.2.3a" xref="S4.p1.24.m24.1.1.1.3.2.3.cmml">−</mo><mn id="S4.p1.24.m24.1.1.1.3.2.3.2" xref="S4.p1.24.m24.1.1.1.3.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S4.p1.24.m24.1.1.1.2" xref="S4.p1.24.m24.1.1.1.2.cmml">⁢</mo><mrow id="S4.p1.24.m24.1.1.1.1.1" xref="S4.p1.24.m24.1.1.1.1.1.1.cmml"><mo id="S4.p1.24.m24.1.1.1.1.1.2" stretchy="false" xref="S4.p1.24.m24.1.1.1.1.1.1.cmml">(</mo><msub id="S4.p1.24.m24.1.1.1.1.1.1" xref="S4.p1.24.m24.1.1.1.1.1.1.cmml"><mi id="S4.p1.24.m24.1.1.1.1.1.1.2" xref="S4.p1.24.m24.1.1.1.1.1.1.2.cmml">u</mi><mrow id="S4.p1.24.m24.1.1.1.1.1.1.3" xref="S4.p1.24.m24.1.1.1.1.1.1.3.cmml"><mi id="S4.p1.24.m24.1.1.1.1.1.1.3.2" xref="S4.p1.24.m24.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="S4.p1.24.m24.1.1.1.1.1.1.3.1" xref="S4.p1.24.m24.1.1.1.1.1.1.3.1.cmml">⁢</mo><mn id="S4.p1.24.m24.1.1.1.1.1.1.3.3" xref="S4.p1.24.m24.1.1.1.1.1.1.3.3.cmml">2</mn></mrow></msub><mo id="S4.p1.24.m24.1.1.1.1.1.3" stretchy="false" xref="S4.p1.24.m24.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.24.m24.1b"><apply id="S4.p1.24.m24.1.1.cmml" xref="S4.p1.24.m24.1.1"><eq id="S4.p1.24.m24.1.1.2.cmml" xref="S4.p1.24.m24.1.1.2"></eq><apply id="S4.p1.24.m24.1.1.3.cmml" xref="S4.p1.24.m24.1.1.3"><csymbol cd="ambiguous" id="S4.p1.24.m24.1.1.3.1.cmml" xref="S4.p1.24.m24.1.1.3">subscript</csymbol><ci id="S4.p1.24.m24.1.1.3.2.cmml" xref="S4.p1.24.m24.1.1.3.2">𝑐</ci><ci id="S4.p1.24.m24.1.1.3.3.cmml" xref="S4.p1.24.m24.1.1.3.3">𝑖</ci></apply><apply id="S4.p1.24.m24.1.1.1.cmml" xref="S4.p1.24.m24.1.1.1"><times id="S4.p1.24.m24.1.1.1.2.cmml" xref="S4.p1.24.m24.1.1.1.2"></times><apply id="S4.p1.24.m24.1.1.1.3.cmml" xref="S4.p1.24.m24.1.1.1.3"><csymbol cd="ambiguous" id="S4.p1.24.m24.1.1.1.3.1.cmml" xref="S4.p1.24.m24.1.1.1.3">subscript</csymbol><apply id="S4.p1.24.m24.1.1.1.3.2.cmml" xref="S4.p1.24.m24.1.1.1.3"><csymbol cd="ambiguous" id="S4.p1.24.m24.1.1.1.3.2.1.cmml" xref="S4.p1.24.m24.1.1.1.3">superscript</csymbol><ci id="S4.p1.24.m24.1.1.1.3.2.2.cmml" xref="S4.p1.24.m24.1.1.1.3.2.2">𝐹</ci><apply id="S4.p1.24.m24.1.1.1.3.2.3.cmml" xref="S4.p1.24.m24.1.1.1.3.2.3"><minus id="S4.p1.24.m24.1.1.1.3.2.3.1.cmml" xref="S4.p1.24.m24.1.1.1.3.2.3"></minus><cn id="S4.p1.24.m24.1.1.1.3.2.3.2.cmml" type="integer" xref="S4.p1.24.m24.1.1.1.3.2.3.2">1</cn></apply></apply><ci id="S4.p1.24.m24.1.1.1.3.3.cmml" xref="S4.p1.24.m24.1.1.1.3.3">𝐶</ci></apply><apply id="S4.p1.24.m24.1.1.1.1.1.1.cmml" xref="S4.p1.24.m24.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.p1.24.m24.1.1.1.1.1.1.1.cmml" xref="S4.p1.24.m24.1.1.1.1.1">subscript</csymbol><ci id="S4.p1.24.m24.1.1.1.1.1.1.2.cmml" xref="S4.p1.24.m24.1.1.1.1.1.1.2">𝑢</ci><apply id="S4.p1.24.m24.1.1.1.1.1.1.3.cmml" xref="S4.p1.24.m24.1.1.1.1.1.1.3"><times id="S4.p1.24.m24.1.1.1.1.1.1.3.1.cmml" xref="S4.p1.24.m24.1.1.1.1.1.1.3.1"></times><ci id="S4.p1.24.m24.1.1.1.1.1.1.3.2.cmml" xref="S4.p1.24.m24.1.1.1.1.1.1.3.2">𝑖</ci><cn id="S4.p1.24.m24.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.p1.24.m24.1.1.1.1.1.1.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.24.m24.1c">c_{i}=F^{-1}_{C}(u_{i2})</annotation><annotation encoding="application/x-llamapun" id="S4.p1.24.m24.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_u start_POSTSUBSCRIPT italic_i 2 end_POSTSUBSCRIPT )</annotation></semantics></math>. As usual, we construct <math alttext="Y=\min(T,C)" class="ltx_Math" display="inline" id="S4.p1.25.m25.3"><semantics id="S4.p1.25.m25.3a"><mrow id="S4.p1.25.m25.3.4" xref="S4.p1.25.m25.3.4.cmml"><mi id="S4.p1.25.m25.3.4.2" xref="S4.p1.25.m25.3.4.2.cmml">Y</mi><mo id="S4.p1.25.m25.3.4.1" xref="S4.p1.25.m25.3.4.1.cmml">=</mo><mrow id="S4.p1.25.m25.3.4.3.2" xref="S4.p1.25.m25.3.4.3.1.cmml"><mi id="S4.p1.25.m25.1.1" xref="S4.p1.25.m25.1.1.cmml">min</mi><mo id="S4.p1.25.m25.3.4.3.2a" xref="S4.p1.25.m25.3.4.3.1.cmml">⁡</mo><mrow id="S4.p1.25.m25.3.4.3.2.1" xref="S4.p1.25.m25.3.4.3.1.cmml"><mo id="S4.p1.25.m25.3.4.3.2.1.1" stretchy="false" xref="S4.p1.25.m25.3.4.3.1.cmml">(</mo><mi id="S4.p1.25.m25.2.2" xref="S4.p1.25.m25.2.2.cmml">T</mi><mo id="S4.p1.25.m25.3.4.3.2.1.2" xref="S4.p1.25.m25.3.4.3.1.cmml">,</mo><mi id="S4.p1.25.m25.3.3" xref="S4.p1.25.m25.3.3.cmml">C</mi><mo id="S4.p1.25.m25.3.4.3.2.1.3" stretchy="false" xref="S4.p1.25.m25.3.4.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.25.m25.3b"><apply id="S4.p1.25.m25.3.4.cmml" xref="S4.p1.25.m25.3.4"><eq id="S4.p1.25.m25.3.4.1.cmml" xref="S4.p1.25.m25.3.4.1"></eq><ci id="S4.p1.25.m25.3.4.2.cmml" xref="S4.p1.25.m25.3.4.2">𝑌</ci><apply id="S4.p1.25.m25.3.4.3.1.cmml" xref="S4.p1.25.m25.3.4.3.2"><min id="S4.p1.25.m25.1.1.cmml" xref="S4.p1.25.m25.1.1"></min><ci id="S4.p1.25.m25.2.2.cmml" xref="S4.p1.25.m25.2.2">𝑇</ci><ci id="S4.p1.25.m25.3.3.cmml" xref="S4.p1.25.m25.3.3">𝐶</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.25.m25.3c">Y=\min(T,C)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.25.m25.3d">italic_Y = roman_min ( italic_T , italic_C )</annotation></semantics></math> and <math alttext="\Delta=\mathbbm{1}(Y=T)" class="ltx_Math" display="inline" id="S4.p1.26.m26.1"><semantics id="S4.p1.26.m26.1a"><mrow id="S4.p1.26.m26.1.1" xref="S4.p1.26.m26.1.1.cmml"><mi id="S4.p1.26.m26.1.1.3" mathvariant="normal" xref="S4.p1.26.m26.1.1.3.cmml">Δ</mi><mo id="S4.p1.26.m26.1.1.2" xref="S4.p1.26.m26.1.1.2.cmml">=</mo><mrow id="S4.p1.26.m26.1.1.1" xref="S4.p1.26.m26.1.1.1.cmml"><mn id="S4.p1.26.m26.1.1.1.3" xref="S4.p1.26.m26.1.1.1.3.cmml">𝟙</mn><mo id="S4.p1.26.m26.1.1.1.2" xref="S4.p1.26.m26.1.1.1.2.cmml">⁢</mo><mrow id="S4.p1.26.m26.1.1.1.1.1" xref="S4.p1.26.m26.1.1.1.1.1.1.cmml"><mo id="S4.p1.26.m26.1.1.1.1.1.2" stretchy="false" xref="S4.p1.26.m26.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.p1.26.m26.1.1.1.1.1.1" xref="S4.p1.26.m26.1.1.1.1.1.1.cmml"><mi id="S4.p1.26.m26.1.1.1.1.1.1.2" xref="S4.p1.26.m26.1.1.1.1.1.1.2.cmml">Y</mi><mo id="S4.p1.26.m26.1.1.1.1.1.1.1" xref="S4.p1.26.m26.1.1.1.1.1.1.1.cmml">=</mo><mi id="S4.p1.26.m26.1.1.1.1.1.1.3" xref="S4.p1.26.m26.1.1.1.1.1.1.3.cmml">T</mi></mrow><mo id="S4.p1.26.m26.1.1.1.1.1.3" stretchy="false" xref="S4.p1.26.m26.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.26.m26.1b"><apply id="S4.p1.26.m26.1.1.cmml" xref="S4.p1.26.m26.1.1"><eq id="S4.p1.26.m26.1.1.2.cmml" xref="S4.p1.26.m26.1.1.2"></eq><ci id="S4.p1.26.m26.1.1.3.cmml" xref="S4.p1.26.m26.1.1.3">Δ</ci><apply id="S4.p1.26.m26.1.1.1.cmml" xref="S4.p1.26.m26.1.1.1"><times id="S4.p1.26.m26.1.1.1.2.cmml" xref="S4.p1.26.m26.1.1.1.2"></times><cn id="S4.p1.26.m26.1.1.1.3.cmml" type="integer" xref="S4.p1.26.m26.1.1.1.3">1</cn><apply id="S4.p1.26.m26.1.1.1.1.1.1.cmml" xref="S4.p1.26.m26.1.1.1.1.1"><eq id="S4.p1.26.m26.1.1.1.1.1.1.1.cmml" xref="S4.p1.26.m26.1.1.1.1.1.1.1"></eq><ci id="S4.p1.26.m26.1.1.1.1.1.1.2.cmml" xref="S4.p1.26.m26.1.1.1.1.1.1.2">𝑌</ci><ci id="S4.p1.26.m26.1.1.1.1.1.1.3.cmml" xref="S4.p1.26.m26.1.1.1.1.1.1.3">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.26.m26.1c">\Delta=\mathbbm{1}(Y=T)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.26.m26.1d">roman_Δ = blackboard_1 ( italic_Y = italic_T )</annotation></semantics></math>. Inference will be done for <math alttext="\beta_{1}" class="ltx_Math" display="inline" id="S4.p1.27.m27.1"><semantics id="S4.p1.27.m27.1a"><msub id="S4.p1.27.m27.1.1" xref="S4.p1.27.m27.1.1.cmml"><mi id="S4.p1.27.m27.1.1.2" xref="S4.p1.27.m27.1.1.2.cmml">β</mi><mn id="S4.p1.27.m27.1.1.3" xref="S4.p1.27.m27.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.p1.27.m27.1b"><apply id="S4.p1.27.m27.1.1.cmml" xref="S4.p1.27.m27.1.1"><csymbol cd="ambiguous" id="S4.p1.27.m27.1.1.1.cmml" xref="S4.p1.27.m27.1.1">subscript</csymbol><ci id="S4.p1.27.m27.1.1.2.cmml" xref="S4.p1.27.m27.1.1.2">𝛽</ci><cn id="S4.p1.27.m27.1.1.3.cmml" type="integer" xref="S4.p1.27.m27.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.27.m27.1c">\beta_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.27.m27.1d">italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, at the selected time point of interest <math alttext="t=1" class="ltx_Math" display="inline" id="S4.p1.28.m28.1"><semantics id="S4.p1.28.m28.1a"><mrow id="S4.p1.28.m28.1.1" xref="S4.p1.28.m28.1.1.cmml"><mi id="S4.p1.28.m28.1.1.2" xref="S4.p1.28.m28.1.1.2.cmml">t</mi><mo id="S4.p1.28.m28.1.1.1" xref="S4.p1.28.m28.1.1.1.cmml">=</mo><mn id="S4.p1.28.m28.1.1.3" xref="S4.p1.28.m28.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.28.m28.1b"><apply id="S4.p1.28.m28.1.1.cmml" xref="S4.p1.28.m28.1.1"><eq id="S4.p1.28.m28.1.1.1.cmml" xref="S4.p1.28.m28.1.1.1"></eq><ci id="S4.p1.28.m28.1.1.2.cmml" xref="S4.p1.28.m28.1.1.2">𝑡</ci><cn id="S4.p1.28.m28.1.1.3.cmml" type="integer" xref="S4.p1.28.m28.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.28.m28.1c">t=1</annotation><annotation encoding="application/x-llamapun" id="S4.p1.28.m28.1d">italic_t = 1</annotation></semantics></math>. Simulation analyses using <math alttext="\Lambda(\cdot)=1-(1+\exp(\cdot))^{-1}" class="ltx_Math" display="inline" id="S4.p1.29.m29.4"><semantics id="S4.p1.29.m29.4a"><mrow id="S4.p1.29.m29.4.4" xref="S4.p1.29.m29.4.4.cmml"><mrow id="S4.p1.29.m29.4.4.3" xref="S4.p1.29.m29.4.4.3.cmml"><mi id="S4.p1.29.m29.4.4.3.2" mathvariant="normal" xref="S4.p1.29.m29.4.4.3.2.cmml">Λ</mi><mo id="S4.p1.29.m29.4.4.3.1" xref="S4.p1.29.m29.4.4.3.1.cmml">⁢</mo><mrow id="S4.p1.29.m29.4.4.3.3.2" xref="S4.p1.29.m29.4.4.3.cmml"><mo id="S4.p1.29.m29.4.4.3.3.2.1" stretchy="false" xref="S4.p1.29.m29.4.4.3.cmml">(</mo><mo id="S4.p1.29.m29.1.1" lspace="0em" rspace="0em" xref="S4.p1.29.m29.1.1.cmml">⋅</mo><mo id="S4.p1.29.m29.4.4.3.3.2.2" stretchy="false" xref="S4.p1.29.m29.4.4.3.cmml">)</mo></mrow></mrow><mo id="S4.p1.29.m29.4.4.2" xref="S4.p1.29.m29.4.4.2.cmml">=</mo><mrow id="S4.p1.29.m29.4.4.1" xref="S4.p1.29.m29.4.4.1.cmml"><mn id="S4.p1.29.m29.4.4.1.3" xref="S4.p1.29.m29.4.4.1.3.cmml">1</mn><mo id="S4.p1.29.m29.4.4.1.2" xref="S4.p1.29.m29.4.4.1.2.cmml">−</mo><msup id="S4.p1.29.m29.4.4.1.1" xref="S4.p1.29.m29.4.4.1.1.cmml"><mrow id="S4.p1.29.m29.4.4.1.1.1.1" xref="S4.p1.29.m29.4.4.1.1.1.1.1.cmml"><mo id="S4.p1.29.m29.4.4.1.1.1.1.2" stretchy="false" xref="S4.p1.29.m29.4.4.1.1.1.1.1.cmml">(</mo><mrow id="S4.p1.29.m29.4.4.1.1.1.1.1" xref="S4.p1.29.m29.4.4.1.1.1.1.1.cmml"><mn id="S4.p1.29.m29.4.4.1.1.1.1.1.2" xref="S4.p1.29.m29.4.4.1.1.1.1.1.2.cmml">1</mn><mo id="S4.p1.29.m29.4.4.1.1.1.1.1.1" xref="S4.p1.29.m29.4.4.1.1.1.1.1.1.cmml">+</mo><mrow id="S4.p1.29.m29.4.4.1.1.1.1.1.3.2" xref="S4.p1.29.m29.4.4.1.1.1.1.1.3.1.cmml"><mi id="S4.p1.29.m29.2.2" xref="S4.p1.29.m29.2.2.cmml">exp</mi><mo id="S4.p1.29.m29.4.4.1.1.1.1.1.3.2a" xref="S4.p1.29.m29.4.4.1.1.1.1.1.3.1.cmml">⁡</mo><mrow id="S4.p1.29.m29.4.4.1.1.1.1.1.3.2.1" xref="S4.p1.29.m29.4.4.1.1.1.1.1.3.1.cmml"><mo id="S4.p1.29.m29.4.4.1.1.1.1.1.3.2.1.1" stretchy="false" xref="S4.p1.29.m29.4.4.1.1.1.1.1.3.1.cmml">(</mo><mo id="S4.p1.29.m29.3.3" lspace="0em" rspace="0em" xref="S4.p1.29.m29.3.3.cmml">⋅</mo><mo id="S4.p1.29.m29.4.4.1.1.1.1.1.3.2.1.2" stretchy="false" xref="S4.p1.29.m29.4.4.1.1.1.1.1.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.p1.29.m29.4.4.1.1.1.1.3" stretchy="false" xref="S4.p1.29.m29.4.4.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S4.p1.29.m29.4.4.1.1.3" xref="S4.p1.29.m29.4.4.1.1.3.cmml"><mo id="S4.p1.29.m29.4.4.1.1.3a" xref="S4.p1.29.m29.4.4.1.1.3.cmml">−</mo><mn id="S4.p1.29.m29.4.4.1.1.3.2" xref="S4.p1.29.m29.4.4.1.1.3.2.cmml">1</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.29.m29.4b"><apply id="S4.p1.29.m29.4.4.cmml" xref="S4.p1.29.m29.4.4"><eq id="S4.p1.29.m29.4.4.2.cmml" xref="S4.p1.29.m29.4.4.2"></eq><apply id="S4.p1.29.m29.4.4.3.cmml" xref="S4.p1.29.m29.4.4.3"><times id="S4.p1.29.m29.4.4.3.1.cmml" xref="S4.p1.29.m29.4.4.3.1"></times><ci id="S4.p1.29.m29.4.4.3.2.cmml" xref="S4.p1.29.m29.4.4.3.2">Λ</ci><ci id="S4.p1.29.m29.1.1.cmml" xref="S4.p1.29.m29.1.1">⋅</ci></apply><apply id="S4.p1.29.m29.4.4.1.cmml" xref="S4.p1.29.m29.4.4.1"><minus id="S4.p1.29.m29.4.4.1.2.cmml" xref="S4.p1.29.m29.4.4.1.2"></minus><cn id="S4.p1.29.m29.4.4.1.3.cmml" type="integer" xref="S4.p1.29.m29.4.4.1.3">1</cn><apply id="S4.p1.29.m29.4.4.1.1.cmml" xref="S4.p1.29.m29.4.4.1.1"><csymbol cd="ambiguous" id="S4.p1.29.m29.4.4.1.1.2.cmml" xref="S4.p1.29.m29.4.4.1.1">superscript</csymbol><apply id="S4.p1.29.m29.4.4.1.1.1.1.1.cmml" xref="S4.p1.29.m29.4.4.1.1.1.1"><plus id="S4.p1.29.m29.4.4.1.1.1.1.1.1.cmml" xref="S4.p1.29.m29.4.4.1.1.1.1.1.1"></plus><cn id="S4.p1.29.m29.4.4.1.1.1.1.1.2.cmml" type="integer" xref="S4.p1.29.m29.4.4.1.1.1.1.1.2">1</cn><apply id="S4.p1.29.m29.4.4.1.1.1.1.1.3.1.cmml" xref="S4.p1.29.m29.4.4.1.1.1.1.1.3.2"><exp id="S4.p1.29.m29.2.2.cmml" xref="S4.p1.29.m29.2.2"></exp><ci id="S4.p1.29.m29.3.3.cmml" xref="S4.p1.29.m29.3.3">⋅</ci></apply></apply><apply id="S4.p1.29.m29.4.4.1.1.3.cmml" xref="S4.p1.29.m29.4.4.1.1.3"><minus id="S4.p1.29.m29.4.4.1.1.3.1.cmml" xref="S4.p1.29.m29.4.4.1.1.3"></minus><cn id="S4.p1.29.m29.4.4.1.1.3.2.cmml" type="integer" xref="S4.p1.29.m29.4.4.1.1.3.2">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.29.m29.4c">\Lambda(\cdot)=1-(1+\exp(\cdot))^{-1}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.29.m29.4d">roman_Λ ( ⋅ ) = 1 - ( 1 + roman_exp ( ⋅ ) ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, referred to as the AFT link function, are presented in Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Simulations</span>, though in general, they lead to very similar conclusions.</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.5">The parameter space is defined as <math alttext="\mathcal{B}=[-10,10]^{3}" class="ltx_Math" display="inline" id="S4.p2.1.m1.2"><semantics id="S4.p2.1.m1.2a"><mrow id="S4.p2.1.m1.2.2" xref="S4.p2.1.m1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.p2.1.m1.2.2.3" xref="S4.p2.1.m1.2.2.3.cmml">ℬ</mi><mo id="S4.p2.1.m1.2.2.2" xref="S4.p2.1.m1.2.2.2.cmml">=</mo><msup id="S4.p2.1.m1.2.2.1" xref="S4.p2.1.m1.2.2.1.cmml"><mrow id="S4.p2.1.m1.2.2.1.1.1" xref="S4.p2.1.m1.2.2.1.1.2.cmml"><mo id="S4.p2.1.m1.2.2.1.1.1.2" stretchy="false" xref="S4.p2.1.m1.2.2.1.1.2.cmml">[</mo><mrow id="S4.p2.1.m1.2.2.1.1.1.1" xref="S4.p2.1.m1.2.2.1.1.1.1.cmml"><mo id="S4.p2.1.m1.2.2.1.1.1.1a" xref="S4.p2.1.m1.2.2.1.1.1.1.cmml">−</mo><mn id="S4.p2.1.m1.2.2.1.1.1.1.2" xref="S4.p2.1.m1.2.2.1.1.1.1.2.cmml">10</mn></mrow><mo id="S4.p2.1.m1.2.2.1.1.1.3" xref="S4.p2.1.m1.2.2.1.1.2.cmml">,</mo><mn id="S4.p2.1.m1.1.1" xref="S4.p2.1.m1.1.1.cmml">10</mn><mo id="S4.p2.1.m1.2.2.1.1.1.4" stretchy="false" xref="S4.p2.1.m1.2.2.1.1.2.cmml">]</mo></mrow><mn id="S4.p2.1.m1.2.2.1.3" xref="S4.p2.1.m1.2.2.1.3.cmml">3</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.1.m1.2b"><apply id="S4.p2.1.m1.2.2.cmml" xref="S4.p2.1.m1.2.2"><eq id="S4.p2.1.m1.2.2.2.cmml" xref="S4.p2.1.m1.2.2.2"></eq><ci id="S4.p2.1.m1.2.2.3.cmml" xref="S4.p2.1.m1.2.2.3">ℬ</ci><apply id="S4.p2.1.m1.2.2.1.cmml" xref="S4.p2.1.m1.2.2.1"><csymbol cd="ambiguous" id="S4.p2.1.m1.2.2.1.2.cmml" xref="S4.p2.1.m1.2.2.1">superscript</csymbol><interval closure="closed" id="S4.p2.1.m1.2.2.1.1.2.cmml" xref="S4.p2.1.m1.2.2.1.1.1"><apply id="S4.p2.1.m1.2.2.1.1.1.1.cmml" xref="S4.p2.1.m1.2.2.1.1.1.1"><minus id="S4.p2.1.m1.2.2.1.1.1.1.1.cmml" xref="S4.p2.1.m1.2.2.1.1.1.1"></minus><cn id="S4.p2.1.m1.2.2.1.1.1.1.2.cmml" type="integer" xref="S4.p2.1.m1.2.2.1.1.1.1.2">10</cn></apply><cn id="S4.p2.1.m1.1.1.cmml" type="integer" xref="S4.p2.1.m1.1.1">10</cn></interval><cn id="S4.p2.1.m1.2.2.1.3.cmml" type="integer" xref="S4.p2.1.m1.2.2.1.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.1.m1.2c">\mathcal{B}=[-10,10]^{3}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.1.m1.2d">caligraphic_B = [ - 10 , 10 ] start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math>. The class of instrumental functions used is constructed by taking cubic B-splines for the continuous covariate and indicators for the categorical (binary) covariate, and combining them via Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.E10" title="In 3.2 Instrumental functions ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">10</span></a>). Furthermore, the test by <cite class="ltx_cite ltx_citemacro_cite">Bei, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib2" title="">2024</a>)</cite> requires the specification of several hyperparameters. Where possible, we will use the values proposed in their paper. One notable hyperparameter that cannot be specified in this way relates to Assumption <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I1.i7" title="item (A7) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A7)</span></a> and defines the lower bound of the empirical variances. We set its value to <math alttext="10^{-6}" class="ltx_Math" display="inline" id="S4.p2.2.m2.1"><semantics id="S4.p2.2.m2.1a"><msup id="S4.p2.2.m2.1.1" xref="S4.p2.2.m2.1.1.cmml"><mn id="S4.p2.2.m2.1.1.2" xref="S4.p2.2.m2.1.1.2.cmml">10</mn><mrow id="S4.p2.2.m2.1.1.3" xref="S4.p2.2.m2.1.1.3.cmml"><mo id="S4.p2.2.m2.1.1.3a" xref="S4.p2.2.m2.1.1.3.cmml">−</mo><mn id="S4.p2.2.m2.1.1.3.2" xref="S4.p2.2.m2.1.1.3.2.cmml">6</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.p2.2.m2.1b"><apply id="S4.p2.2.m2.1.1.cmml" xref="S4.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.p2.2.m2.1.1.1.cmml" xref="S4.p2.2.m2.1.1">superscript</csymbol><cn id="S4.p2.2.m2.1.1.2.cmml" type="integer" xref="S4.p2.2.m2.1.1.2">10</cn><apply id="S4.p2.2.m2.1.1.3.cmml" xref="S4.p2.2.m2.1.1.3"><minus id="S4.p2.2.m2.1.1.3.1.cmml" xref="S4.p2.2.m2.1.1.3"></minus><cn id="S4.p2.2.m2.1.1.3.2.cmml" type="integer" xref="S4.p2.2.m2.1.1.3.2">6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.2.m2.1c">10^{-6}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m2.1d">10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT</annotation></semantics></math>. Preliminary simulations showed that smaller values could lead to overrejection. Other notable design choices are the number of bootstrap samples used to compute the critical values, <math alttext="B=600" class="ltx_Math" display="inline" id="S4.p2.3.m3.1"><semantics id="S4.p2.3.m3.1a"><mrow id="S4.p2.3.m3.1.1" xref="S4.p2.3.m3.1.1.cmml"><mi id="S4.p2.3.m3.1.1.2" xref="S4.p2.3.m3.1.1.2.cmml">B</mi><mo id="S4.p2.3.m3.1.1.1" xref="S4.p2.3.m3.1.1.1.cmml">=</mo><mn id="S4.p2.3.m3.1.1.3" xref="S4.p2.3.m3.1.1.3.cmml">600</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.3.m3.1b"><apply id="S4.p2.3.m3.1.1.cmml" xref="S4.p2.3.m3.1.1"><eq id="S4.p2.3.m3.1.1.1.cmml" xref="S4.p2.3.m3.1.1.1"></eq><ci id="S4.p2.3.m3.1.1.2.cmml" xref="S4.p2.3.m3.1.1.2">𝐵</ci><cn id="S4.p2.3.m3.1.1.3.cmml" type="integer" xref="S4.p2.3.m3.1.1.3">600</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.3.m3.1c">B=600</annotation><annotation encoding="application/x-llamapun" id="S4.p2.3.m3.1d">italic_B = 600</annotation></semantics></math>, and the nominal level of the test, <math alttext="\alpha=0.05" class="ltx_Math" display="inline" id="S4.p2.4.m4.1"><semantics id="S4.p2.4.m4.1a"><mrow id="S4.p2.4.m4.1.1" xref="S4.p2.4.m4.1.1.cmml"><mi id="S4.p2.4.m4.1.1.2" xref="S4.p2.4.m4.1.1.2.cmml">α</mi><mo id="S4.p2.4.m4.1.1.1" xref="S4.p2.4.m4.1.1.1.cmml">=</mo><mn id="S4.p2.4.m4.1.1.3" xref="S4.p2.4.m4.1.1.3.cmml">0.05</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.4.m4.1b"><apply id="S4.p2.4.m4.1.1.cmml" xref="S4.p2.4.m4.1.1"><eq id="S4.p2.4.m4.1.1.1.cmml" xref="S4.p2.4.m4.1.1.1"></eq><ci id="S4.p2.4.m4.1.1.2.cmml" xref="S4.p2.4.m4.1.1.2">𝛼</ci><cn id="S4.p2.4.m4.1.1.3.cmml" type="float" xref="S4.p2.4.m4.1.1.3">0.05</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.4.m4.1c">\alpha=0.05</annotation><annotation encoding="application/x-llamapun" id="S4.p2.4.m4.1d">italic_α = 0.05</annotation></semantics></math>. Finally, each simulation design is run with <math alttext="500" class="ltx_Math" display="inline" id="S4.p2.5.m5.1"><semantics id="S4.p2.5.m5.1a"><mn id="S4.p2.5.m5.1.1" xref="S4.p2.5.m5.1.1.cmml">500</mn><annotation-xml encoding="MathML-Content" id="S4.p2.5.m5.1b"><cn id="S4.p2.5.m5.1.1.cmml" type="integer" xref="S4.p2.5.m5.1.1">500</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.5.m5.1c">500</annotation><annotation encoding="application/x-llamapun" id="S4.p2.5.m5.1d">500</annotation></semantics></math> repetitions.</p> </div> <div class="ltx_para" id="S4.p3"> <p class="ltx_p" id="S4.p3.4">The results of the simulation can be found in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S4.T1" title="Table 1 ‣ 4 Simulation study ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">1</span></a>. For each considered design, this table lists (<em class="ltx_emph ltx_font_italic" id="S4.p3.4.1">Bounds</em>) the average lower and upper bound of the estimated identified intervals, (<em class="ltx_emph ltx_font_italic" id="S4.p3.4.2">Var</em>) the variance of the width of the bounds, (<em class="ltx_emph ltx_font_italic" id="S4.p3.4.3">Sig</em>) the percentage of repetitions in which zero was not contained in the bounds and (<em class="ltx_emph ltx_font_italic" id="S4.p3.4.4">Cov</em>) the percentage of repetitions in which the true value, <math alttext="\beta_{\text{true},1}=1" class="ltx_Math" display="inline" id="S4.p3.1.m1.2"><semantics id="S4.p3.1.m1.2a"><mrow id="S4.p3.1.m1.2.3" xref="S4.p3.1.m1.2.3.cmml"><msub id="S4.p3.1.m1.2.3.2" xref="S4.p3.1.m1.2.3.2.cmml"><mi id="S4.p3.1.m1.2.3.2.2" xref="S4.p3.1.m1.2.3.2.2.cmml">β</mi><mrow id="S4.p3.1.m1.2.2.2.4" xref="S4.p3.1.m1.2.2.2.3.cmml"><mtext id="S4.p3.1.m1.1.1.1.1" xref="S4.p3.1.m1.1.1.1.1a.cmml">true</mtext><mo id="S4.p3.1.m1.2.2.2.4.1" xref="S4.p3.1.m1.2.2.2.3.cmml">,</mo><mn id="S4.p3.1.m1.2.2.2.2" xref="S4.p3.1.m1.2.2.2.2.cmml">1</mn></mrow></msub><mo id="S4.p3.1.m1.2.3.1" xref="S4.p3.1.m1.2.3.1.cmml">=</mo><mn id="S4.p3.1.m1.2.3.3" xref="S4.p3.1.m1.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.1.m1.2b"><apply id="S4.p3.1.m1.2.3.cmml" xref="S4.p3.1.m1.2.3"><eq id="S4.p3.1.m1.2.3.1.cmml" xref="S4.p3.1.m1.2.3.1"></eq><apply id="S4.p3.1.m1.2.3.2.cmml" xref="S4.p3.1.m1.2.3.2"><csymbol cd="ambiguous" id="S4.p3.1.m1.2.3.2.1.cmml" xref="S4.p3.1.m1.2.3.2">subscript</csymbol><ci id="S4.p3.1.m1.2.3.2.2.cmml" xref="S4.p3.1.m1.2.3.2.2">𝛽</ci><list id="S4.p3.1.m1.2.2.2.3.cmml" xref="S4.p3.1.m1.2.2.2.4"><ci id="S4.p3.1.m1.1.1.1.1a.cmml" xref="S4.p3.1.m1.1.1.1.1"><mtext id="S4.p3.1.m1.1.1.1.1.cmml" mathsize="70%" xref="S4.p3.1.m1.1.1.1.1">true</mtext></ci><cn id="S4.p3.1.m1.2.2.2.2.cmml" type="integer" xref="S4.p3.1.m1.2.2.2.2">1</cn></list></apply><cn id="S4.p3.1.m1.2.3.3.cmml" type="integer" xref="S4.p3.1.m1.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.1.m1.2c">\beta_{\text{true},1}=1</annotation><annotation encoding="application/x-llamapun" id="S4.p3.1.m1.2d">italic_β start_POSTSUBSCRIPT true , 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math>, was contained in the computed bounds. Remark that all quantities are necessarily computed only based on the repetitions that did not conclude model misspecification (i.e. empty estimated identified interval). The model was incorrectly determined to be misspecified in no more than <math alttext="5" class="ltx_Math" display="inline" id="S4.p3.2.m2.1"><semantics id="S4.p3.2.m2.1a"><mn id="S4.p3.2.m2.1.1" xref="S4.p3.2.m2.1.1.cmml">5</mn><annotation-xml encoding="MathML-Content" id="S4.p3.2.m2.1b"><cn id="S4.p3.2.m2.1.1.cmml" type="integer" xref="S4.p3.2.m2.1.1">5</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.2.m2.1c">5</annotation><annotation encoding="application/x-llamapun" id="S4.p3.2.m2.1d">5</annotation></semantics></math> out of the <math alttext="500" class="ltx_Math" display="inline" id="S4.p3.3.m3.1"><semantics id="S4.p3.3.m3.1a"><mn id="S4.p3.3.m3.1.1" xref="S4.p3.3.m3.1.1.cmml">500</mn><annotation-xml encoding="MathML-Content" id="S4.p3.3.m3.1b"><cn id="S4.p3.3.m3.1.1.cmml" type="integer" xref="S4.p3.3.m3.1.1">500</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.3.m3.1c">500</annotation><annotation encoding="application/x-llamapun" id="S4.p3.3.m3.1d">500</annotation></semantics></math> repetitions, throughout all considered designs. The quantity <em class="ltx_emph ltx_font_italic" id="S4.p3.4.5">Var</em> is interesting as it provides information on the stability of the method. <em class="ltx_emph ltx_font_italic" id="S4.p3.4.6">Sig</em> quantifies the proportion of repetitions for which the derived bounds are informative, in the sense that the covariate effect is concluded to be different from zero (i.e. significant). Lastly, <em class="ltx_emph ltx_font_italic" id="S4.p3.4.7">Cov</em> is equal to the proportion of repetitions in which the true value was contained within the estimated interval and, as such, provides information on the type-I error of the method. <em class="ltx_emph ltx_font_italic" id="S4.p3.4.8">Cov</em> is equal to one in almost all designs. This is not indicative of the method being conservative, since the nominal level <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.p3.4.m4.1"><semantics id="S4.p3.4.m4.1a"><mi id="S4.p3.4.m4.1.1" xref="S4.p3.4.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.p3.4.m4.1b"><ci id="S4.p3.4.m4.1.1.cmml" xref="S4.p3.4.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.4.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.p3.4.m4.1d">italic_α</annotation></semantics></math> holds in particular at the bounds of the identified interval, while the actual type-I error of the test on interior points of the interval will be much smaller.</p> </div> <div class="ltx_para" id="S4.p4"> <p class="ltx_p" id="S4.p4.11">From Table <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S4.T1" title="Table 1 ‣ 4 Simulation study ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">1</span></a> it can be seen that the bounds narrow and the method becomes more stable as the sample size increases. Interestingly, it can be seen that for the simulation designs considered here, an increase in the number of instrumental functions <math alttext="(N_{IF})" class="ltx_Math" display="inline" id="S4.p4.1.m1.1"><semantics id="S4.p4.1.m1.1a"><mrow id="S4.p4.1.m1.1.1.1" xref="S4.p4.1.m1.1.1.1.1.cmml"><mo id="S4.p4.1.m1.1.1.1.2" stretchy="false" xref="S4.p4.1.m1.1.1.1.1.cmml">(</mo><msub id="S4.p4.1.m1.1.1.1.1" xref="S4.p4.1.m1.1.1.1.1.cmml"><mi id="S4.p4.1.m1.1.1.1.1.2" xref="S4.p4.1.m1.1.1.1.1.2.cmml">N</mi><mrow id="S4.p4.1.m1.1.1.1.1.3" xref="S4.p4.1.m1.1.1.1.1.3.cmml"><mi id="S4.p4.1.m1.1.1.1.1.3.2" xref="S4.p4.1.m1.1.1.1.1.3.2.cmml">I</mi><mo id="S4.p4.1.m1.1.1.1.1.3.1" xref="S4.p4.1.m1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.p4.1.m1.1.1.1.1.3.3" xref="S4.p4.1.m1.1.1.1.1.3.3.cmml">F</mi></mrow></msub><mo id="S4.p4.1.m1.1.1.1.3" stretchy="false" xref="S4.p4.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.1.m1.1b"><apply id="S4.p4.1.m1.1.1.1.1.cmml" xref="S4.p4.1.m1.1.1.1"><csymbol cd="ambiguous" id="S4.p4.1.m1.1.1.1.1.1.cmml" xref="S4.p4.1.m1.1.1.1">subscript</csymbol><ci id="S4.p4.1.m1.1.1.1.1.2.cmml" xref="S4.p4.1.m1.1.1.1.1.2">𝑁</ci><apply id="S4.p4.1.m1.1.1.1.1.3.cmml" xref="S4.p4.1.m1.1.1.1.1.3"><times id="S4.p4.1.m1.1.1.1.1.3.1.cmml" xref="S4.p4.1.m1.1.1.1.1.3.1"></times><ci id="S4.p4.1.m1.1.1.1.1.3.2.cmml" xref="S4.p4.1.m1.1.1.1.1.3.2">𝐼</ci><ci id="S4.p4.1.m1.1.1.1.1.3.3.cmml" xref="S4.p4.1.m1.1.1.1.1.3.3">𝐹</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.1.m1.1c">(N_{IF})</annotation><annotation encoding="application/x-llamapun" id="S4.p4.1.m1.1d">( italic_N start_POSTSUBSCRIPT italic_I italic_F end_POSTSUBSCRIPT )</annotation></semantics></math> used – specifically, using <math alttext="8" class="ltx_Math" display="inline" id="S4.p4.2.m2.1"><semantics id="S4.p4.2.m2.1a"><mn id="S4.p4.2.m2.1.1" xref="S4.p4.2.m2.1.1.cmml">8</mn><annotation-xml encoding="MathML-Content" id="S4.p4.2.m2.1b"><cn id="S4.p4.2.m2.1.1.cmml" type="integer" xref="S4.p4.2.m2.1.1">8</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.2.m2.1c">8</annotation><annotation encoding="application/x-llamapun" id="S4.p4.2.m2.1d">8</annotation></semantics></math> splines for <math alttext="X_{1}" class="ltx_Math" display="inline" id="S4.p4.3.m3.1"><semantics id="S4.p4.3.m3.1a"><msub id="S4.p4.3.m3.1.1" xref="S4.p4.3.m3.1.1.cmml"><mi id="S4.p4.3.m3.1.1.2" xref="S4.p4.3.m3.1.1.2.cmml">X</mi><mn id="S4.p4.3.m3.1.1.3" xref="S4.p4.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.p4.3.m3.1b"><apply id="S4.p4.3.m3.1.1.cmml" xref="S4.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S4.p4.3.m3.1.1.1.cmml" xref="S4.p4.3.m3.1.1">subscript</csymbol><ci id="S4.p4.3.m3.1.1.2.cmml" xref="S4.p4.3.m3.1.1.2">𝑋</ci><cn id="S4.p4.3.m3.1.1.3.cmml" type="integer" xref="S4.p4.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.3.m3.1c">X_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.p4.3.m3.1d">italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> instead of <math alttext="6" class="ltx_Math" display="inline" id="S4.p4.4.m4.1"><semantics id="S4.p4.4.m4.1a"><mn id="S4.p4.4.m4.1.1" xref="S4.p4.4.m4.1.1.cmml">6</mn><annotation-xml encoding="MathML-Content" id="S4.p4.4.m4.1b"><cn id="S4.p4.4.m4.1.1.cmml" type="integer" xref="S4.p4.4.m4.1.1">6</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.4.m4.1c">6</annotation><annotation encoding="application/x-llamapun" id="S4.p4.4.m4.1d">6</annotation></semantics></math> – does not substantially narrow the bounds. This indicates that using a moderate amount of instrumental functions will already transfer most of the information from the conditional to the unconditional moments. It can also be seen that positive dependence between <math alttext="T" class="ltx_Math" display="inline" id="S4.p4.5.m5.1"><semantics id="S4.p4.5.m5.1a"><mi id="S4.p4.5.m5.1.1" xref="S4.p4.5.m5.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.p4.5.m5.1b"><ci id="S4.p4.5.m5.1.1.cmml" xref="S4.p4.5.m5.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.5.m5.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.p4.5.m5.1d">italic_T</annotation></semantics></math> and <math alttext="C" class="ltx_Math" display="inline" id="S4.p4.6.m6.1"><semantics id="S4.p4.6.m6.1a"><mi id="S4.p4.6.m6.1.1" xref="S4.p4.6.m6.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S4.p4.6.m6.1b"><ci id="S4.p4.6.m6.1.1.cmml" xref="S4.p4.6.m6.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.6.m6.1c">C</annotation><annotation encoding="application/x-llamapun" id="S4.p4.6.m6.1d">italic_C</annotation></semantics></math> leads to the narrowest bounds, while designs with negative dependence are substantially more difficult. This could be explained by remarking that it is easier to observe the full support of both <math alttext="T" class="ltx_Math" display="inline" id="S4.p4.7.m7.1"><semantics id="S4.p4.7.m7.1a"><mi id="S4.p4.7.m7.1.1" xref="S4.p4.7.m7.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.p4.7.m7.1b"><ci id="S4.p4.7.m7.1.1.cmml" xref="S4.p4.7.m7.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.7.m7.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.p4.7.m7.1d">italic_T</annotation></semantics></math> and <math alttext="C" class="ltx_Math" display="inline" id="S4.p4.8.m8.1"><semantics id="S4.p4.8.m8.1a"><mi id="S4.p4.8.m8.1.1" xref="S4.p4.8.m8.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S4.p4.8.m8.1b"><ci id="S4.p4.8.m8.1.1.cmml" xref="S4.p4.8.m8.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.8.m8.1c">C</annotation><annotation encoding="application/x-llamapun" id="S4.p4.8.m8.1d">italic_C</annotation></semantics></math> when they are positively dependent, while it will be difficult to observe the upper parts of their support in the negative dependence case. In this sense, data generated under positive dependence is more informative, which is reflected in the bounds. In the same line, bounds under heavy censoring <math alttext="(65\%)" class="ltx_Math" display="inline" id="S4.p4.9.m9.1"><semantics id="S4.p4.9.m9.1a"><mrow id="S4.p4.9.m9.1.1.1" xref="S4.p4.9.m9.1.1.1.1.cmml"><mo id="S4.p4.9.m9.1.1.1.2" stretchy="false" xref="S4.p4.9.m9.1.1.1.1.cmml">(</mo><mrow id="S4.p4.9.m9.1.1.1.1" xref="S4.p4.9.m9.1.1.1.1.cmml"><mn id="S4.p4.9.m9.1.1.1.1.2" xref="S4.p4.9.m9.1.1.1.1.2.cmml">65</mn><mo id="S4.p4.9.m9.1.1.1.1.1" xref="S4.p4.9.m9.1.1.1.1.1.cmml">%</mo></mrow><mo id="S4.p4.9.m9.1.1.1.3" stretchy="false" xref="S4.p4.9.m9.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.9.m9.1b"><apply id="S4.p4.9.m9.1.1.1.1.cmml" xref="S4.p4.9.m9.1.1.1"><csymbol cd="latexml" id="S4.p4.9.m9.1.1.1.1.1.cmml" xref="S4.p4.9.m9.1.1.1.1.1">percent</csymbol><cn id="S4.p4.9.m9.1.1.1.1.2.cmml" type="integer" xref="S4.p4.9.m9.1.1.1.1.2">65</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.9.m9.1c">(65\%)</annotation><annotation encoding="application/x-llamapun" id="S4.p4.9.m9.1d">( 65 % )</annotation></semantics></math> are considerably wider than bounds under light censoring <math alttext="(30\%)" class="ltx_Math" display="inline" id="S4.p4.10.m10.1"><semantics id="S4.p4.10.m10.1a"><mrow id="S4.p4.10.m10.1.1.1" xref="S4.p4.10.m10.1.1.1.1.cmml"><mo id="S4.p4.10.m10.1.1.1.2" stretchy="false" xref="S4.p4.10.m10.1.1.1.1.cmml">(</mo><mrow id="S4.p4.10.m10.1.1.1.1" xref="S4.p4.10.m10.1.1.1.1.cmml"><mn id="S4.p4.10.m10.1.1.1.1.2" xref="S4.p4.10.m10.1.1.1.1.2.cmml">30</mn><mo id="S4.p4.10.m10.1.1.1.1.1" xref="S4.p4.10.m10.1.1.1.1.1.cmml">%</mo></mrow><mo id="S4.p4.10.m10.1.1.1.3" stretchy="false" xref="S4.p4.10.m10.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.10.m10.1b"><apply id="S4.p4.10.m10.1.1.1.1.cmml" xref="S4.p4.10.m10.1.1.1"><csymbol cd="latexml" id="S4.p4.10.m10.1.1.1.1.1.cmml" xref="S4.p4.10.m10.1.1.1.1.1">percent</csymbol><cn id="S4.p4.10.m10.1.1.1.1.2.cmml" type="integer" xref="S4.p4.10.m10.1.1.1.1.2">30</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.10.m10.1c">(30\%)</annotation><annotation encoding="application/x-llamapun" id="S4.p4.10.m10.1d">( 30 % )</annotation></semantics></math>. However, Table <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S4.T1" title="Table 1 ‣ 4 Simulation study ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">1</span></a> shows that even in cases with heavy censoring, bounds can still be informative. Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_True_bounds</span> provides an analysis of the true identified interval of <math alttext="\beta_{\text{true},1}" class="ltx_Math" display="inline" id="S4.p4.11.m11.2"><semantics id="S4.p4.11.m11.2a"><msub id="S4.p4.11.m11.2.3" xref="S4.p4.11.m11.2.3.cmml"><mi id="S4.p4.11.m11.2.3.2" xref="S4.p4.11.m11.2.3.2.cmml">β</mi><mrow id="S4.p4.11.m11.2.2.2.4" xref="S4.p4.11.m11.2.2.2.3.cmml"><mtext id="S4.p4.11.m11.1.1.1.1" xref="S4.p4.11.m11.1.1.1.1a.cmml">true</mtext><mo id="S4.p4.11.m11.2.2.2.4.1" xref="S4.p4.11.m11.2.2.2.3.cmml">,</mo><mn id="S4.p4.11.m11.2.2.2.2" xref="S4.p4.11.m11.2.2.2.2.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.p4.11.m11.2b"><apply id="S4.p4.11.m11.2.3.cmml" xref="S4.p4.11.m11.2.3"><csymbol cd="ambiguous" id="S4.p4.11.m11.2.3.1.cmml" xref="S4.p4.11.m11.2.3">subscript</csymbol><ci id="S4.p4.11.m11.2.3.2.cmml" xref="S4.p4.11.m11.2.3.2">𝛽</ci><list id="S4.p4.11.m11.2.2.2.3.cmml" xref="S4.p4.11.m11.2.2.2.4"><ci id="S4.p4.11.m11.1.1.1.1a.cmml" xref="S4.p4.11.m11.1.1.1.1"><mtext id="S4.p4.11.m11.1.1.1.1.cmml" mathsize="70%" xref="S4.p4.11.m11.1.1.1.1">true</mtext></ci><cn id="S4.p4.11.m11.2.2.2.2.cmml" type="integer" xref="S4.p4.11.m11.2.2.2.2">1</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.11.m11.2c">\beta_{\text{true},1}</annotation><annotation encoding="application/x-llamapun" id="S4.p4.11.m11.2d">italic_β start_POSTSUBSCRIPT true , 1 end_POSTSUBSCRIPT</annotation></semantics></math> for each design. We find that the true bounds are typically not substantially smaller than the estimated ones.</p> </div> <figure class="ltx_table" id="S4.T1"> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S4.T1.9"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T1.2.2"> <th class="ltx_td ltx_th ltx_th_row ltx_border_tt" id="S4.T1.2.2.3"></th> <th class="ltx_td ltx_th ltx_th_column ltx_th_row ltx_border_tt" id="S4.T1.2.2.4"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="4" id="S4.T1.1.1.1"><math alttext="N_{IF}=12" class="ltx_Math" display="inline" id="S4.T1.1.1.1.m1.1"><semantics id="S4.T1.1.1.1.m1.1a"><mrow id="S4.T1.1.1.1.m1.1.1" xref="S4.T1.1.1.1.m1.1.1.cmml"><msub id="S4.T1.1.1.1.m1.1.1.2" xref="S4.T1.1.1.1.m1.1.1.2.cmml"><mi id="S4.T1.1.1.1.m1.1.1.2.2" xref="S4.T1.1.1.1.m1.1.1.2.2.cmml">N</mi><mrow id="S4.T1.1.1.1.m1.1.1.2.3" xref="S4.T1.1.1.1.m1.1.1.2.3.cmml"><mi id="S4.T1.1.1.1.m1.1.1.2.3.2" xref="S4.T1.1.1.1.m1.1.1.2.3.2.cmml">I</mi><mo id="S4.T1.1.1.1.m1.1.1.2.3.1" xref="S4.T1.1.1.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S4.T1.1.1.1.m1.1.1.2.3.3" xref="S4.T1.1.1.1.m1.1.1.2.3.3.cmml">F</mi></mrow></msub><mo id="S4.T1.1.1.1.m1.1.1.1" xref="S4.T1.1.1.1.m1.1.1.1.cmml">=</mo><mn id="S4.T1.1.1.1.m1.1.1.3" xref="S4.T1.1.1.1.m1.1.1.3.cmml">12</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.1.1.1.m1.1b"><apply id="S4.T1.1.1.1.m1.1.1.cmml" xref="S4.T1.1.1.1.m1.1.1"><eq id="S4.T1.1.1.1.m1.1.1.1.cmml" xref="S4.T1.1.1.1.m1.1.1.1"></eq><apply id="S4.T1.1.1.1.m1.1.1.2.cmml" xref="S4.T1.1.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T1.1.1.1.m1.1.1.2.1.cmml" xref="S4.T1.1.1.1.m1.1.1.2">subscript</csymbol><ci id="S4.T1.1.1.1.m1.1.1.2.2.cmml" xref="S4.T1.1.1.1.m1.1.1.2.2">𝑁</ci><apply id="S4.T1.1.1.1.m1.1.1.2.3.cmml" xref="S4.T1.1.1.1.m1.1.1.2.3"><times id="S4.T1.1.1.1.m1.1.1.2.3.1.cmml" xref="S4.T1.1.1.1.m1.1.1.2.3.1"></times><ci id="S4.T1.1.1.1.m1.1.1.2.3.2.cmml" xref="S4.T1.1.1.1.m1.1.1.2.3.2">𝐼</ci><ci id="S4.T1.1.1.1.m1.1.1.2.3.3.cmml" xref="S4.T1.1.1.1.m1.1.1.2.3.3">𝐹</ci></apply></apply><cn id="S4.T1.1.1.1.m1.1.1.3.cmml" type="integer" xref="S4.T1.1.1.1.m1.1.1.3">12</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.1.1.1.m1.1c">N_{IF}=12</annotation><annotation encoding="application/x-llamapun" id="S4.T1.1.1.1.m1.1d">italic_N start_POSTSUBSCRIPT italic_I italic_F end_POSTSUBSCRIPT = 12</annotation></semantics></math></th> <th class="ltx_td ltx_th ltx_th_column ltx_border_tt" id="S4.T1.2.2.5"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="4" id="S4.T1.2.2.2"><math alttext="N_{IF}=16" class="ltx_Math" display="inline" id="S4.T1.2.2.2.m1.1"><semantics id="S4.T1.2.2.2.m1.1a"><mrow id="S4.T1.2.2.2.m1.1.1" xref="S4.T1.2.2.2.m1.1.1.cmml"><msub id="S4.T1.2.2.2.m1.1.1.2" xref="S4.T1.2.2.2.m1.1.1.2.cmml"><mi id="S4.T1.2.2.2.m1.1.1.2.2" xref="S4.T1.2.2.2.m1.1.1.2.2.cmml">N</mi><mrow id="S4.T1.2.2.2.m1.1.1.2.3" xref="S4.T1.2.2.2.m1.1.1.2.3.cmml"><mi id="S4.T1.2.2.2.m1.1.1.2.3.2" xref="S4.T1.2.2.2.m1.1.1.2.3.2.cmml">I</mi><mo id="S4.T1.2.2.2.m1.1.1.2.3.1" xref="S4.T1.2.2.2.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S4.T1.2.2.2.m1.1.1.2.3.3" xref="S4.T1.2.2.2.m1.1.1.2.3.3.cmml">F</mi></mrow></msub><mo id="S4.T1.2.2.2.m1.1.1.1" xref="S4.T1.2.2.2.m1.1.1.1.cmml">=</mo><mn id="S4.T1.2.2.2.m1.1.1.3" xref="S4.T1.2.2.2.m1.1.1.3.cmml">16</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.2.2.2.m1.1b"><apply id="S4.T1.2.2.2.m1.1.1.cmml" xref="S4.T1.2.2.2.m1.1.1"><eq id="S4.T1.2.2.2.m1.1.1.1.cmml" xref="S4.T1.2.2.2.m1.1.1.1"></eq><apply id="S4.T1.2.2.2.m1.1.1.2.cmml" xref="S4.T1.2.2.2.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T1.2.2.2.m1.1.1.2.1.cmml" xref="S4.T1.2.2.2.m1.1.1.2">subscript</csymbol><ci id="S4.T1.2.2.2.m1.1.1.2.2.cmml" xref="S4.T1.2.2.2.m1.1.1.2.2">𝑁</ci><apply id="S4.T1.2.2.2.m1.1.1.2.3.cmml" xref="S4.T1.2.2.2.m1.1.1.2.3"><times id="S4.T1.2.2.2.m1.1.1.2.3.1.cmml" xref="S4.T1.2.2.2.m1.1.1.2.3.1"></times><ci id="S4.T1.2.2.2.m1.1.1.2.3.2.cmml" xref="S4.T1.2.2.2.m1.1.1.2.3.2">𝐼</ci><ci id="S4.T1.2.2.2.m1.1.1.2.3.3.cmml" xref="S4.T1.2.2.2.m1.1.1.2.3.3">𝐹</ci></apply></apply><cn id="S4.T1.2.2.2.m1.1.1.3.cmml" type="integer" xref="S4.T1.2.2.2.m1.1.1.3">16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.2.2.2.m1.1c">N_{IF}=16</annotation><annotation encoding="application/x-llamapun" id="S4.T1.2.2.2.m1.1d">italic_N start_POSTSUBSCRIPT italic_I italic_F end_POSTSUBSCRIPT = 16</annotation></semantics></math></th> </tr> <tr class="ltx_tr" id="S4.T1.3.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row" id="S4.T1.3.3.2">Setting</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row" id="S4.T1.3.3.1"><math alttext="n" class="ltx_Math" display="inline" id="S4.T1.3.3.1.m1.1"><semantics id="S4.T1.3.3.1.m1.1a"><mi id="S4.T1.3.3.1.m1.1.1" xref="S4.T1.3.3.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.T1.3.3.1.m1.1b"><ci id="S4.T1.3.3.1.m1.1.1.cmml" xref="S4.T1.3.3.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.3.3.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.T1.3.3.1.m1.1d">italic_n</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T1.3.3.3">Bounds</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T1.3.3.4">Var</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T1.3.3.5">Sig</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T1.3.3.6">Cov</th> <th class="ltx_td ltx_th ltx_th_column" id="S4.T1.3.3.7"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T1.3.3.8">Bounds</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T1.3.3.9">Var</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T1.3.3.10">Sig</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S4.T1.3.3.11">Cov</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T1.4.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S4.T1.4.4.1" rowspan="3"><span class="ltx_text" id="S4.T1.4.4.1.1"> <span class="ltx_inline-block ltx_align_center" id="S4.T1.4.4.1.1.1"> <span class="ltx_p" id="S4.T1.4.4.1.1.1.2">Indep.</span> <span class="ltx_p" id="S4.T1.4.4.1.1.1.1"><math alttext="\sim 30\%" class="ltx_Math" display="inline" id="S4.T1.4.4.1.1.1.1.m1.1"><semantics id="S4.T1.4.4.1.1.1.1.m1.1a"><mrow id="S4.T1.4.4.1.1.1.1.m1.1.1" xref="S4.T1.4.4.1.1.1.1.m1.1.1.cmml"><mi id="S4.T1.4.4.1.1.1.1.m1.1.1.2" xref="S4.T1.4.4.1.1.1.1.m1.1.1.2.cmml"></mi><mo id="S4.T1.4.4.1.1.1.1.m1.1.1.1" xref="S4.T1.4.4.1.1.1.1.m1.1.1.1.cmml">∼</mo><mrow id="S4.T1.4.4.1.1.1.1.m1.1.1.3" xref="S4.T1.4.4.1.1.1.1.m1.1.1.3.cmml"><mn id="S4.T1.4.4.1.1.1.1.m1.1.1.3.2" xref="S4.T1.4.4.1.1.1.1.m1.1.1.3.2.cmml">30</mn><mo id="S4.T1.4.4.1.1.1.1.m1.1.1.3.1" xref="S4.T1.4.4.1.1.1.1.m1.1.1.3.1.cmml">%</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.4.4.1.1.1.1.m1.1b"><apply id="S4.T1.4.4.1.1.1.1.m1.1.1.cmml" xref="S4.T1.4.4.1.1.1.1.m1.1.1"><csymbol cd="latexml" id="S4.T1.4.4.1.1.1.1.m1.1.1.1.cmml" xref="S4.T1.4.4.1.1.1.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.T1.4.4.1.1.1.1.m1.1.1.2.cmml" xref="S4.T1.4.4.1.1.1.1.m1.1.1.2">absent</csymbol><apply id="S4.T1.4.4.1.1.1.1.m1.1.1.3.cmml" xref="S4.T1.4.4.1.1.1.1.m1.1.1.3"><csymbol cd="latexml" id="S4.T1.4.4.1.1.1.1.m1.1.1.3.1.cmml" xref="S4.T1.4.4.1.1.1.1.m1.1.1.3.1">percent</csymbol><cn id="S4.T1.4.4.1.1.1.1.m1.1.1.3.2.cmml" type="integer" xref="S4.T1.4.4.1.1.1.1.m1.1.1.3.2">30</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.4.4.1.1.1.1.m1.1c">\sim 30\%</annotation><annotation encoding="application/x-llamapun" id="S4.T1.4.4.1.1.1.1.m1.1d">∼ 30 %</annotation></semantics></math> cens.</span> </span></span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S4.T1.4.4.2">500</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.3">[0.47, 1.56]</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.4">0.03</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.5">1.00</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.6">1.00</td> <td class="ltx_td ltx_border_t" id="S4.T1.4.4.7"></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.8">[0.50, 1.55]</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.9">0.03</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.10">1.00</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.11">1.00</td> </tr> <tr class="ltx_tr" id="S4.T1.9.10.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T1.9.10.1.1">1000</th> <td class="ltx_td ltx_align_center" id="S4.T1.9.10.1.2">[0.49, 1.39]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.10.1.3">0.01</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.10.1.4">1.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.10.1.5">1.00</td> <td class="ltx_td" id="S4.T1.9.10.1.6"></td> <td class="ltx_td ltx_align_center" id="S4.T1.9.10.1.7">[0.52, 1.37]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.10.1.8">0.01</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.10.1.9">1.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.10.1.10">1.00</td> </tr> <tr class="ltx_tr" id="S4.T1.9.11.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T1.9.11.2.1">2000</th> <td class="ltx_td ltx_align_center" id="S4.T1.9.11.2.2">[0.51, 1.29]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.11.2.3">0.01</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.11.2.4">1.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.11.2.5">1.00</td> <td class="ltx_td" id="S4.T1.9.11.2.6"></td> <td class="ltx_td ltx_align_center" id="S4.T1.9.11.2.7">[0.53, 1.27]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.11.2.8">0.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.11.2.9">1.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.11.2.10">1.00</td> </tr> <tr class="ltx_tr" id="S4.T1.5.5"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S4.T1.5.5.1" rowspan="3"><span class="ltx_text" id="S4.T1.5.5.1.1"> <span class="ltx_inline-block ltx_align_center" id="S4.T1.5.5.1.1.1"> <span class="ltx_p" id="S4.T1.5.5.1.1.1.2">Pos. dep.</span> <span class="ltx_p" id="S4.T1.5.5.1.1.1.1"><math alttext="\sim 30\%" class="ltx_Math" display="inline" id="S4.T1.5.5.1.1.1.1.m1.1"><semantics id="S4.T1.5.5.1.1.1.1.m1.1a"><mrow id="S4.T1.5.5.1.1.1.1.m1.1.1" xref="S4.T1.5.5.1.1.1.1.m1.1.1.cmml"><mi id="S4.T1.5.5.1.1.1.1.m1.1.1.2" xref="S4.T1.5.5.1.1.1.1.m1.1.1.2.cmml"></mi><mo id="S4.T1.5.5.1.1.1.1.m1.1.1.1" xref="S4.T1.5.5.1.1.1.1.m1.1.1.1.cmml">∼</mo><mrow id="S4.T1.5.5.1.1.1.1.m1.1.1.3" xref="S4.T1.5.5.1.1.1.1.m1.1.1.3.cmml"><mn id="S4.T1.5.5.1.1.1.1.m1.1.1.3.2" xref="S4.T1.5.5.1.1.1.1.m1.1.1.3.2.cmml">30</mn><mo id="S4.T1.5.5.1.1.1.1.m1.1.1.3.1" xref="S4.T1.5.5.1.1.1.1.m1.1.1.3.1.cmml">%</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.5.5.1.1.1.1.m1.1b"><apply id="S4.T1.5.5.1.1.1.1.m1.1.1.cmml" xref="S4.T1.5.5.1.1.1.1.m1.1.1"><csymbol cd="latexml" id="S4.T1.5.5.1.1.1.1.m1.1.1.1.cmml" xref="S4.T1.5.5.1.1.1.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.T1.5.5.1.1.1.1.m1.1.1.2.cmml" xref="S4.T1.5.5.1.1.1.1.m1.1.1.2">absent</csymbol><apply id="S4.T1.5.5.1.1.1.1.m1.1.1.3.cmml" xref="S4.T1.5.5.1.1.1.1.m1.1.1.3"><csymbol cd="latexml" id="S4.T1.5.5.1.1.1.1.m1.1.1.3.1.cmml" xref="S4.T1.5.5.1.1.1.1.m1.1.1.3.1">percent</csymbol><cn id="S4.T1.5.5.1.1.1.1.m1.1.1.3.2.cmml" type="integer" xref="S4.T1.5.5.1.1.1.1.m1.1.1.3.2">30</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.5.5.1.1.1.1.m1.1c">\sim 30\%</annotation><annotation encoding="application/x-llamapun" id="S4.T1.5.5.1.1.1.1.m1.1d">∼ 30 %</annotation></semantics></math> cens.</span> </span></span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S4.T1.5.5.2">500</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.5.5.3">[0.60, 1.69]</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.5.5.4">0.03</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.5.5.5">1.00</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.5.5.6">1.00</td> <td class="ltx_td ltx_border_t" id="S4.T1.5.5.7"></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.5.5.8">[0.66, 1.68]</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.5.5.9">0.04</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.5.5.10">1.00</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.5.5.11">0.99</td> </tr> <tr class="ltx_tr" id="S4.T1.9.12.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T1.9.12.3.1">1000</th> <td class="ltx_td ltx_align_center" id="S4.T1.9.12.3.2">[0.63, 1.50]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.12.3.3">0.01</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.12.3.4">1.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.12.3.5">1.00</td> <td class="ltx_td" id="S4.T1.9.12.3.6"></td> <td class="ltx_td ltx_align_center" id="S4.T1.9.12.3.7">[0.68, 1.50]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.12.3.8">0.01</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.12.3.9">1.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.12.3.10">0.99</td> </tr> <tr class="ltx_tr" id="S4.T1.9.13.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T1.9.13.4.1">2000</th> <td class="ltx_td ltx_align_center" id="S4.T1.9.13.4.2">[0.66, 1.40]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.13.4.3">0.01</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.13.4.4">1.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.13.4.5">1.00</td> <td class="ltx_td" id="S4.T1.9.13.4.6"></td> <td class="ltx_td ltx_align_center" id="S4.T1.9.13.4.7">[0.70, 1.38]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.13.4.8">0.01</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.13.4.9">1.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.13.4.10">1.00</td> </tr> <tr class="ltx_tr" id="S4.T1.6.6"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S4.T1.6.6.1" rowspan="3"><span class="ltx_text" id="S4.T1.6.6.1.1"> <span class="ltx_inline-block ltx_align_center" id="S4.T1.6.6.1.1.1"> <span class="ltx_p" id="S4.T1.6.6.1.1.1.2">Neg. dep.</span> <span class="ltx_p" id="S4.T1.6.6.1.1.1.1"><math alttext="\sim 30\%" class="ltx_Math" display="inline" id="S4.T1.6.6.1.1.1.1.m1.1"><semantics id="S4.T1.6.6.1.1.1.1.m1.1a"><mrow id="S4.T1.6.6.1.1.1.1.m1.1.1" xref="S4.T1.6.6.1.1.1.1.m1.1.1.cmml"><mi id="S4.T1.6.6.1.1.1.1.m1.1.1.2" xref="S4.T1.6.6.1.1.1.1.m1.1.1.2.cmml"></mi><mo id="S4.T1.6.6.1.1.1.1.m1.1.1.1" xref="S4.T1.6.6.1.1.1.1.m1.1.1.1.cmml">∼</mo><mrow id="S4.T1.6.6.1.1.1.1.m1.1.1.3" xref="S4.T1.6.6.1.1.1.1.m1.1.1.3.cmml"><mn id="S4.T1.6.6.1.1.1.1.m1.1.1.3.2" xref="S4.T1.6.6.1.1.1.1.m1.1.1.3.2.cmml">30</mn><mo id="S4.T1.6.6.1.1.1.1.m1.1.1.3.1" xref="S4.T1.6.6.1.1.1.1.m1.1.1.3.1.cmml">%</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.6.6.1.1.1.1.m1.1b"><apply id="S4.T1.6.6.1.1.1.1.m1.1.1.cmml" xref="S4.T1.6.6.1.1.1.1.m1.1.1"><csymbol cd="latexml" id="S4.T1.6.6.1.1.1.1.m1.1.1.1.cmml" xref="S4.T1.6.6.1.1.1.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.T1.6.6.1.1.1.1.m1.1.1.2.cmml" xref="S4.T1.6.6.1.1.1.1.m1.1.1.2">absent</csymbol><apply id="S4.T1.6.6.1.1.1.1.m1.1.1.3.cmml" xref="S4.T1.6.6.1.1.1.1.m1.1.1.3"><csymbol cd="latexml" id="S4.T1.6.6.1.1.1.1.m1.1.1.3.1.cmml" xref="S4.T1.6.6.1.1.1.1.m1.1.1.3.1">percent</csymbol><cn id="S4.T1.6.6.1.1.1.1.m1.1.1.3.2.cmml" type="integer" xref="S4.T1.6.6.1.1.1.1.m1.1.1.3.2">30</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.6.6.1.1.1.1.m1.1c">\sim 30\%</annotation><annotation encoding="application/x-llamapun" id="S4.T1.6.6.1.1.1.1.m1.1d">∼ 30 %</annotation></semantics></math> cens.</span> </span></span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S4.T1.6.6.2">500</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.6.6.3">[0.46, 1.64]</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.6.6.4">0.04</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.6.6.5">1.00</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.6.6.6">1.00</td> <td class="ltx_td ltx_border_t" id="S4.T1.6.6.7"></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.6.6.8">[0.49, 1.63]</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.6.6.9">0.04</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.6.6.10">1.00</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.6.6.11">1.00</td> </tr> <tr class="ltx_tr" id="S4.T1.9.14.5"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T1.9.14.5.1">1000</th> <td class="ltx_td ltx_align_center" id="S4.T1.9.14.5.2">[0.48, 1.46]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.14.5.3">0.01</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.14.5.4">1.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.14.5.5">1.00</td> <td class="ltx_td" id="S4.T1.9.14.5.6"></td> <td class="ltx_td ltx_align_center" id="S4.T1.9.14.5.7">[0.50, 1.44]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.14.5.8">0.01</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.14.5.9">1.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.14.5.10">1.00</td> </tr> <tr class="ltx_tr" id="S4.T1.9.15.6"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T1.9.15.6.1">2000</th> <td class="ltx_td ltx_align_center" id="S4.T1.9.15.6.2">[0.50, 1.35]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.15.6.3">0.01</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.15.6.4">1.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.15.6.5">1.00</td> <td class="ltx_td" id="S4.T1.9.15.6.6"></td> <td class="ltx_td ltx_align_center" id="S4.T1.9.15.6.7">[0.51, 1.33]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.15.6.8">0.01</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.15.6.9">1.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.15.6.10">1.00</td> </tr> <tr class="ltx_tr" id="S4.T1.7.7"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S4.T1.7.7.1" rowspan="3"><span class="ltx_text" id="S4.T1.7.7.1.1"> <span class="ltx_inline-block ltx_align_center" id="S4.T1.7.7.1.1.1"> <span class="ltx_p" id="S4.T1.7.7.1.1.1.2">Indep.</span> <span class="ltx_p" id="S4.T1.7.7.1.1.1.1"><math alttext="\sim 65\%" class="ltx_Math" display="inline" id="S4.T1.7.7.1.1.1.1.m1.1"><semantics id="S4.T1.7.7.1.1.1.1.m1.1a"><mrow id="S4.T1.7.7.1.1.1.1.m1.1.1" xref="S4.T1.7.7.1.1.1.1.m1.1.1.cmml"><mi id="S4.T1.7.7.1.1.1.1.m1.1.1.2" xref="S4.T1.7.7.1.1.1.1.m1.1.1.2.cmml"></mi><mo id="S4.T1.7.7.1.1.1.1.m1.1.1.1" xref="S4.T1.7.7.1.1.1.1.m1.1.1.1.cmml">∼</mo><mrow id="S4.T1.7.7.1.1.1.1.m1.1.1.3" xref="S4.T1.7.7.1.1.1.1.m1.1.1.3.cmml"><mn id="S4.T1.7.7.1.1.1.1.m1.1.1.3.2" xref="S4.T1.7.7.1.1.1.1.m1.1.1.3.2.cmml">65</mn><mo id="S4.T1.7.7.1.1.1.1.m1.1.1.3.1" xref="S4.T1.7.7.1.1.1.1.m1.1.1.3.1.cmml">%</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.7.7.1.1.1.1.m1.1b"><apply id="S4.T1.7.7.1.1.1.1.m1.1.1.cmml" xref="S4.T1.7.7.1.1.1.1.m1.1.1"><csymbol cd="latexml" id="S4.T1.7.7.1.1.1.1.m1.1.1.1.cmml" xref="S4.T1.7.7.1.1.1.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.T1.7.7.1.1.1.1.m1.1.1.2.cmml" xref="S4.T1.7.7.1.1.1.1.m1.1.1.2">absent</csymbol><apply id="S4.T1.7.7.1.1.1.1.m1.1.1.3.cmml" xref="S4.T1.7.7.1.1.1.1.m1.1.1.3"><csymbol cd="latexml" id="S4.T1.7.7.1.1.1.1.m1.1.1.3.1.cmml" xref="S4.T1.7.7.1.1.1.1.m1.1.1.3.1">percent</csymbol><cn id="S4.T1.7.7.1.1.1.1.m1.1.1.3.2.cmml" type="integer" xref="S4.T1.7.7.1.1.1.1.m1.1.1.3.2">65</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.7.7.1.1.1.1.m1.1c">\sim 65\%</annotation><annotation encoding="application/x-llamapun" id="S4.T1.7.7.1.1.1.1.m1.1d">∼ 65 %</annotation></semantics></math> cens.</span> </span></span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S4.T1.7.7.2">500</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.7.7.3">[-0.19, 3.69]</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.7.7.4">2.61</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.7.7.5">0.04</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.7.7.6">1.00</td> <td class="ltx_td ltx_border_t" id="S4.T1.7.7.7"></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.7.7.8">[-0.16, 2.99]</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.7.7.9">0.57</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.7.7.10">0.09</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.7.7.11">1.00</td> </tr> <tr class="ltx_tr" id="S4.T1.9.16.7"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T1.9.16.7.1">1000</th> <td class="ltx_td ltx_align_center" id="S4.T1.9.16.7.2">[-0.10, 2.46]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.16.7.3">0.12</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.16.7.4">0.11</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.16.7.5">1.00</td> <td class="ltx_td" id="S4.T1.9.16.7.6"></td> <td class="ltx_td ltx_align_center" id="S4.T1.9.16.7.7">[-0.08, 2.23]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.16.7.8">0.07</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.16.7.9">0.14</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.16.7.10">1.00</td> </tr> <tr class="ltx_tr" id="S4.T1.9.17.8"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T1.9.17.8.1">2000</th> <td class="ltx_td ltx_align_center" id="S4.T1.9.17.8.2">[-0.05, 2.14]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.17.8.3">0.07</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.17.8.4">0.20</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.17.8.5">1.00</td> <td class="ltx_td" id="S4.T1.9.17.8.6"></td> <td class="ltx_td ltx_align_center" id="S4.T1.9.17.8.7">[-0.03, 1.90]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.17.8.8">0.03</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.17.8.9">0.28</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.17.8.10">1.00</td> </tr> <tr class="ltx_tr" id="S4.T1.8.8"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S4.T1.8.8.1" rowspan="3"><span class="ltx_text" id="S4.T1.8.8.1.1"> <span class="ltx_inline-block ltx_align_center" id="S4.T1.8.8.1.1.1"> <span class="ltx_p" id="S4.T1.8.8.1.1.1.2">Pos. dep.</span> <span class="ltx_p" id="S4.T1.8.8.1.1.1.1"><math alttext="\sim 65\%" class="ltx_Math" display="inline" id="S4.T1.8.8.1.1.1.1.m1.1"><semantics id="S4.T1.8.8.1.1.1.1.m1.1a"><mrow id="S4.T1.8.8.1.1.1.1.m1.1.1" xref="S4.T1.8.8.1.1.1.1.m1.1.1.cmml"><mi id="S4.T1.8.8.1.1.1.1.m1.1.1.2" xref="S4.T1.8.8.1.1.1.1.m1.1.1.2.cmml"></mi><mo id="S4.T1.8.8.1.1.1.1.m1.1.1.1" xref="S4.T1.8.8.1.1.1.1.m1.1.1.1.cmml">∼</mo><mrow id="S4.T1.8.8.1.1.1.1.m1.1.1.3" xref="S4.T1.8.8.1.1.1.1.m1.1.1.3.cmml"><mn id="S4.T1.8.8.1.1.1.1.m1.1.1.3.2" xref="S4.T1.8.8.1.1.1.1.m1.1.1.3.2.cmml">65</mn><mo id="S4.T1.8.8.1.1.1.1.m1.1.1.3.1" xref="S4.T1.8.8.1.1.1.1.m1.1.1.3.1.cmml">%</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.8.8.1.1.1.1.m1.1b"><apply id="S4.T1.8.8.1.1.1.1.m1.1.1.cmml" xref="S4.T1.8.8.1.1.1.1.m1.1.1"><csymbol cd="latexml" id="S4.T1.8.8.1.1.1.1.m1.1.1.1.cmml" xref="S4.T1.8.8.1.1.1.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.T1.8.8.1.1.1.1.m1.1.1.2.cmml" xref="S4.T1.8.8.1.1.1.1.m1.1.1.2">absent</csymbol><apply id="S4.T1.8.8.1.1.1.1.m1.1.1.3.cmml" xref="S4.T1.8.8.1.1.1.1.m1.1.1.3"><csymbol cd="latexml" id="S4.T1.8.8.1.1.1.1.m1.1.1.3.1.cmml" xref="S4.T1.8.8.1.1.1.1.m1.1.1.3.1">percent</csymbol><cn id="S4.T1.8.8.1.1.1.1.m1.1.1.3.2.cmml" type="integer" xref="S4.T1.8.8.1.1.1.1.m1.1.1.3.2">65</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.8.8.1.1.1.1.m1.1c">\sim 65\%</annotation><annotation encoding="application/x-llamapun" id="S4.T1.8.8.1.1.1.1.m1.1d">∼ 65 %</annotation></semantics></math> cens.</span> </span></span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S4.T1.8.8.2">500</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.8.8.3">[0.19, 3.04]</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.8.8.4">0.61</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.8.8.5">0.96</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.8.8.6">1.00</td> <td class="ltx_td ltx_border_t" id="S4.T1.8.8.7"></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.8.8.8">[0.22, 2.75]</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.8.8.9">0.20</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.8.8.10">0.97</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.8.8.11">1.00</td> </tr> <tr class="ltx_tr" id="S4.T1.9.18.9"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T1.9.18.9.1">1000</th> <td class="ltx_td ltx_align_center" id="S4.T1.9.18.9.2">[0.23, 2.39]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.18.9.3">0.08</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.18.9.4">1.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.18.9.5">1.00</td> <td class="ltx_td" id="S4.T1.9.18.9.6"></td> <td class="ltx_td ltx_align_center" id="S4.T1.9.18.9.7">[0.27, 2.25]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.18.9.8">0.06</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.18.9.9">1.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.18.9.10">1.00</td> </tr> <tr class="ltx_tr" id="S4.T1.9.19.10"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T1.9.19.10.1">2000</th> <td class="ltx_td ltx_align_center" id="S4.T1.9.19.10.2">[0.26, 2.11]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.19.10.3">0.04</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.19.10.4">1.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.19.10.5">1.00</td> <td class="ltx_td" id="S4.T1.9.19.10.6"></td> <td class="ltx_td ltx_align_center" id="S4.T1.9.19.10.7">[0.29, 2.00]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.19.10.8">0.02</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.19.10.9">1.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.19.10.10">1.00</td> </tr> <tr class="ltx_tr" id="S4.T1.9.9"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_bb ltx_border_t" id="S4.T1.9.9.1" rowspan="3"><span class="ltx_text" id="S4.T1.9.9.1.1"> <span class="ltx_inline-block ltx_align_center" id="S4.T1.9.9.1.1.1"> <span class="ltx_p" id="S4.T1.9.9.1.1.1.2">Neg. dep.</span> <span class="ltx_p" id="S4.T1.9.9.1.1.1.1"><math alttext="\sim 65\%" class="ltx_Math" display="inline" id="S4.T1.9.9.1.1.1.1.m1.1"><semantics id="S4.T1.9.9.1.1.1.1.m1.1a"><mrow id="S4.T1.9.9.1.1.1.1.m1.1.1" xref="S4.T1.9.9.1.1.1.1.m1.1.1.cmml"><mi id="S4.T1.9.9.1.1.1.1.m1.1.1.2" xref="S4.T1.9.9.1.1.1.1.m1.1.1.2.cmml"></mi><mo id="S4.T1.9.9.1.1.1.1.m1.1.1.1" xref="S4.T1.9.9.1.1.1.1.m1.1.1.1.cmml">∼</mo><mrow id="S4.T1.9.9.1.1.1.1.m1.1.1.3" xref="S4.T1.9.9.1.1.1.1.m1.1.1.3.cmml"><mn id="S4.T1.9.9.1.1.1.1.m1.1.1.3.2" xref="S4.T1.9.9.1.1.1.1.m1.1.1.3.2.cmml">65</mn><mo id="S4.T1.9.9.1.1.1.1.m1.1.1.3.1" xref="S4.T1.9.9.1.1.1.1.m1.1.1.3.1.cmml">%</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.9.9.1.1.1.1.m1.1b"><apply id="S4.T1.9.9.1.1.1.1.m1.1.1.cmml" xref="S4.T1.9.9.1.1.1.1.m1.1.1"><csymbol cd="latexml" id="S4.T1.9.9.1.1.1.1.m1.1.1.1.cmml" xref="S4.T1.9.9.1.1.1.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.T1.9.9.1.1.1.1.m1.1.1.2.cmml" xref="S4.T1.9.9.1.1.1.1.m1.1.1.2">absent</csymbol><apply id="S4.T1.9.9.1.1.1.1.m1.1.1.3.cmml" xref="S4.T1.9.9.1.1.1.1.m1.1.1.3"><csymbol cd="latexml" id="S4.T1.9.9.1.1.1.1.m1.1.1.3.1.cmml" xref="S4.T1.9.9.1.1.1.1.m1.1.1.3.1">percent</csymbol><cn id="S4.T1.9.9.1.1.1.1.m1.1.1.3.2.cmml" type="integer" xref="S4.T1.9.9.1.1.1.1.m1.1.1.3.2">65</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.9.9.1.1.1.1.m1.1c">\sim 65\%</annotation><annotation encoding="application/x-llamapun" id="S4.T1.9.9.1.1.1.1.m1.1d">∼ 65 %</annotation></semantics></math> cens.</span> </span></span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_t" id="S4.T1.9.9.2">500</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.9.9.3">[-0.50, 9.76]</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.9.9.4">0.76</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.9.9.5">0.00</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.9.9.6">1.00</td> <td class="ltx_td ltx_border_t" id="S4.T1.9.9.7"></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.9.9.8">[-0.45, 9.36]</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.9.9.9">2.21</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.9.9.10">0.00</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.9.9.11">1.00</td> </tr> <tr class="ltx_tr" id="S4.T1.9.20.11"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row" id="S4.T1.9.20.11.1">1000</th> <td class="ltx_td ltx_align_center" id="S4.T1.9.20.11.2">[-0.32, 7.52]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.20.11.3">5.18</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.20.11.4">0.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.20.11.5">1.00</td> <td class="ltx_td" id="S4.T1.9.20.11.6"></td> <td class="ltx_td ltx_align_center" id="S4.T1.9.20.11.7">[-0.31, 5.45]</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.20.11.8">4.69</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.20.11.9">0.00</td> <td class="ltx_td ltx_align_center" id="S4.T1.9.20.11.10">1.00</td> </tr> <tr class="ltx_tr" id="S4.T1.9.21.12"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_bb" id="S4.T1.9.21.12.1">2000</th> <td class="ltx_td ltx_align_center ltx_border_bb" id="S4.T1.9.21.12.2">[-0.25, 5.00]</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S4.T1.9.21.12.3">3.46</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S4.T1.9.21.12.4">0.00</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S4.T1.9.21.12.5">1.00</td> <td class="ltx_td ltx_border_bb" id="S4.T1.9.21.12.6"></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S4.T1.9.21.12.7">[-0.22, 3.35]</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S4.T1.9.21.12.8">0.41</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S4.T1.9.21.12.9">0.00</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S4.T1.9.21.12.10">1.00</td> </tr> </tbody> </table> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S4.T1.11.1.1" style="font-size:90%;">Table 1</span>: </span><span class="ltx_text" id="S4.T1.12.2" style="font-size:90%;">Results of the main simulation using the Cox link function.</span></figcaption> </figure> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>Summary of additional simulations</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">More simulations were carried out to study the performance of the proposed methodology under various settings. In this section, we will restrict to summarizing the conclusions we can draw from them. We note that some simulations were performed under modified settings with respect to the ones used above in order to ease computational burden. Detailed information on these simulations as well as the tables containing their results is deferred to Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Simulations</span>.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.5"><span class="ltx_text ltx_font_bold" id="S4.SS1.p2.5.1">More instrumental functions.</span> In a first additional analysis, we investigate the effect of further increasing the number of instrumental functions used in transforming the conditional moment inequalities to unconditional ones. We find that using <math alttext="10" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><mn id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><cn id="S4.SS1.p2.1.m1.1.1.cmml" type="integer" xref="S4.SS1.p2.1.m1.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">10</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">10</annotation></semantics></math> B-spline functions for the continuous covariate (hence <math alttext="N_{IF}=20" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><mrow id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml"><msub id="S4.SS1.p2.2.m2.1.1.2" xref="S4.SS1.p2.2.m2.1.1.2.cmml"><mi id="S4.SS1.p2.2.m2.1.1.2.2" xref="S4.SS1.p2.2.m2.1.1.2.2.cmml">N</mi><mrow id="S4.SS1.p2.2.m2.1.1.2.3" xref="S4.SS1.p2.2.m2.1.1.2.3.cmml"><mi id="S4.SS1.p2.2.m2.1.1.2.3.2" xref="S4.SS1.p2.2.m2.1.1.2.3.2.cmml">I</mi><mo id="S4.SS1.p2.2.m2.1.1.2.3.1" xref="S4.SS1.p2.2.m2.1.1.2.3.1.cmml">⁢</mo><mi id="S4.SS1.p2.2.m2.1.1.2.3.3" xref="S4.SS1.p2.2.m2.1.1.2.3.3.cmml">F</mi></mrow></msub><mo id="S4.SS1.p2.2.m2.1.1.1" xref="S4.SS1.p2.2.m2.1.1.1.cmml">=</mo><mn id="S4.SS1.p2.2.m2.1.1.3" xref="S4.SS1.p2.2.m2.1.1.3.cmml">20</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><apply id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1"><eq id="S4.SS1.p2.2.m2.1.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1.1"></eq><apply id="S4.SS1.p2.2.m2.1.1.2.cmml" xref="S4.SS1.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p2.2.m2.1.1.2.1.cmml" xref="S4.SS1.p2.2.m2.1.1.2">subscript</csymbol><ci id="S4.SS1.p2.2.m2.1.1.2.2.cmml" xref="S4.SS1.p2.2.m2.1.1.2.2">𝑁</ci><apply id="S4.SS1.p2.2.m2.1.1.2.3.cmml" xref="S4.SS1.p2.2.m2.1.1.2.3"><times id="S4.SS1.p2.2.m2.1.1.2.3.1.cmml" xref="S4.SS1.p2.2.m2.1.1.2.3.1"></times><ci id="S4.SS1.p2.2.m2.1.1.2.3.2.cmml" xref="S4.SS1.p2.2.m2.1.1.2.3.2">𝐼</ci><ci id="S4.SS1.p2.2.m2.1.1.2.3.3.cmml" xref="S4.SS1.p2.2.m2.1.1.2.3.3">𝐹</ci></apply></apply><cn id="S4.SS1.p2.2.m2.1.1.3.cmml" type="integer" xref="S4.SS1.p2.2.m2.1.1.3">20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">N_{IF}=20</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">italic_N start_POSTSUBSCRIPT italic_I italic_F end_POSTSUBSCRIPT = 20</annotation></semantics></math>) can still help to decrease the width of the bounds slightly, while a further increase to <math alttext="15" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.1"><semantics id="S4.SS1.p2.3.m3.1a"><mn id="S4.SS1.p2.3.m3.1.1" xref="S4.SS1.p2.3.m3.1.1.cmml">15</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.1b"><cn id="S4.SS1.p2.3.m3.1.1.cmml" type="integer" xref="S4.SS1.p2.3.m3.1.1">15</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.1c">15</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.1d">15</annotation></semantics></math> B-spline functions can have adverse effects. This is in line with the remark made at the end of Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.SS4" title="3.4 Discussion ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">3.4</span></a>, cautioning against taking the class <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.1"><semantics id="S4.SS1.p2.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.4.m4.1.1" xref="S4.SS1.p2.4.m4.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.1b"><ci id="S4.SS1.p2.4.m4.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m4.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m4.1d">caligraphic_G</annotation></semantics></math> too large. We also note that including more instrumental functions in the analysis will increase computational burden, as well as increase the risk of not satisfying Assumption <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.I1.i7" title="item (A7) ‣ 3.1 Modeling assumptions and theoretical results ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">(A7)</span></a>. Higher numbers of instrumental functions should therefore be motivated by the availability of sufficient sample size and computing power. In light of these results, we suggest to use around <math alttext="5-10" class="ltx_Math" display="inline" id="S4.SS1.p2.5.m5.1"><semantics id="S4.SS1.p2.5.m5.1a"><mrow id="S4.SS1.p2.5.m5.1.1" xref="S4.SS1.p2.5.m5.1.1.cmml"><mn id="S4.SS1.p2.5.m5.1.1.2" xref="S4.SS1.p2.5.m5.1.1.2.cmml">5</mn><mo id="S4.SS1.p2.5.m5.1.1.1" xref="S4.SS1.p2.5.m5.1.1.1.cmml">−</mo><mn id="S4.SS1.p2.5.m5.1.1.3" xref="S4.SS1.p2.5.m5.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.5.m5.1b"><apply id="S4.SS1.p2.5.m5.1.1.cmml" xref="S4.SS1.p2.5.m5.1.1"><minus id="S4.SS1.p2.5.m5.1.1.1.cmml" xref="S4.SS1.p2.5.m5.1.1.1"></minus><cn id="S4.SS1.p2.5.m5.1.1.2.cmml" type="integer" xref="S4.SS1.p2.5.m5.1.1.2">5</cn><cn id="S4.SS1.p2.5.m5.1.1.3.cmml" type="integer" xref="S4.SS1.p2.5.m5.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.5.m5.1c">5-10</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m5.1d">5 - 10</annotation></semantics></math> instrumental functions per continuous covariate, and to use the indicator family of instrumental functions for the categorical covariates, irrespective of the link function used.</p> </div> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.1"><span class="ltx_text ltx_font_bold" id="S4.SS1.p3.1.1">Almost no censoring.</span> In a second analysis, we consider a setting in which there is only <math alttext="2\%" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.1"><semantics id="S4.SS1.p3.1.m1.1a"><mrow id="S4.SS1.p3.1.m1.1.1" xref="S4.SS1.p3.1.m1.1.1.cmml"><mn id="S4.SS1.p3.1.m1.1.1.2" xref="S4.SS1.p3.1.m1.1.1.2.cmml">2</mn><mo id="S4.SS1.p3.1.m1.1.1.1" xref="S4.SS1.p3.1.m1.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.1b"><apply id="S4.SS1.p3.1.m1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1"><csymbol cd="latexml" id="S4.SS1.p3.1.m1.1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1.1">percent</csymbol><cn id="S4.SS1.p3.1.m1.1.1.2.cmml" type="integer" xref="S4.SS1.p3.1.m1.1.1.2">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.1c">2\%</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.1d">2 %</annotation></semantics></math> censoring. As a consequence, this setting is close to one without censoring, in which case the parameters in model (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E1" title="In 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">1</span></a>) are identified. In line with this observation, we see that the estimated identified intervals are very narrow in all cases. For small sample sizes, the coverage can be below its nominal value, though this problem disappears as the sample size increases.</p> </div> <div class="ltx_para" id="S4.SS1.p4"> <p class="ltx_p" id="S4.SS1.p4.1"><span class="ltx_text ltx_font_bold" id="S4.SS1.p4.1.1">Time-independent effects of covariates.</span> Lastly, we investigate the modification of our approach to time-independent effects of covariates, proposed in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.SS3" title="3.3 Time-independent effects of covariates ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">3.3</span></a>. We find that both the intersection and majority vote method can improve upon the estimation algorithm solely applied to a single point, though we emphasize that this improvement comes at the cost of having to assume that <math alttext="\beta_{\text{true},1}" class="ltx_Math" display="inline" id="S4.SS1.p4.1.m1.2"><semantics id="S4.SS1.p4.1.m1.2a"><msub id="S4.SS1.p4.1.m1.2.3" xref="S4.SS1.p4.1.m1.2.3.cmml"><mi id="S4.SS1.p4.1.m1.2.3.2" xref="S4.SS1.p4.1.m1.2.3.2.cmml">β</mi><mrow id="S4.SS1.p4.1.m1.2.2.2.4" xref="S4.SS1.p4.1.m1.2.2.2.3.cmml"><mtext id="S4.SS1.p4.1.m1.1.1.1.1" xref="S4.SS1.p4.1.m1.1.1.1.1a.cmml">true</mtext><mo id="S4.SS1.p4.1.m1.2.2.2.4.1" xref="S4.SS1.p4.1.m1.2.2.2.3.cmml">,</mo><mn id="S4.SS1.p4.1.m1.2.2.2.2" xref="S4.SS1.p4.1.m1.2.2.2.2.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.1.m1.2b"><apply id="S4.SS1.p4.1.m1.2.3.cmml" xref="S4.SS1.p4.1.m1.2.3"><csymbol cd="ambiguous" id="S4.SS1.p4.1.m1.2.3.1.cmml" xref="S4.SS1.p4.1.m1.2.3">subscript</csymbol><ci id="S4.SS1.p4.1.m1.2.3.2.cmml" xref="S4.SS1.p4.1.m1.2.3.2">𝛽</ci><list id="S4.SS1.p4.1.m1.2.2.2.3.cmml" xref="S4.SS1.p4.1.m1.2.2.2.4"><ci id="S4.SS1.p4.1.m1.1.1.1.1a.cmml" xref="S4.SS1.p4.1.m1.1.1.1.1"><mtext id="S4.SS1.p4.1.m1.1.1.1.1.cmml" mathsize="70%" xref="S4.SS1.p4.1.m1.1.1.1.1">true</mtext></ci><cn id="S4.SS1.p4.1.m1.2.2.2.2.cmml" type="integer" xref="S4.SS1.p4.1.m1.2.2.2.2">1</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.1.m1.2c">\beta_{\text{true},1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.1.m1.2d">italic_β start_POSTSUBSCRIPT true , 1 end_POSTSUBSCRIPT</annotation></semantics></math> is time-independent. The intersection method outperforms majority vote when the number of points in the considered grid of time points is small, whereas this relation reverses when the number of grid points increases, as indicated by the additional analysis based on the data application (Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Comparison_of_combination_methods</span>).</p> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Data applications</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.3">This section applies the developed methodology to two data sets. Throughout, the class of instrumental functions is constructed by using five spline functions for each continuous covariate, using indicator functions for the categorical covariates and combining them via Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S3.E10" title="In 3.2 Instrumental functions ‣ 3 Estimation procedure ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">10</span></a>). Analogous to the simulation study, the lower and upper bounds for the regression parameters are set at <math alttext="-10" class="ltx_Math" display="inline" id="S5.p1.1.m1.1"><semantics id="S5.p1.1.m1.1a"><mrow id="S5.p1.1.m1.1.1" xref="S5.p1.1.m1.1.1.cmml"><mo id="S5.p1.1.m1.1.1a" xref="S5.p1.1.m1.1.1.cmml">−</mo><mn id="S5.p1.1.m1.1.1.2" xref="S5.p1.1.m1.1.1.2.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.1b"><apply id="S5.p1.1.m1.1.1.cmml" xref="S5.p1.1.m1.1.1"><minus id="S5.p1.1.m1.1.1.1.cmml" xref="S5.p1.1.m1.1.1"></minus><cn id="S5.p1.1.m1.1.1.2.cmml" type="integer" xref="S5.p1.1.m1.1.1.2">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.1c">-10</annotation><annotation encoding="application/x-llamapun" id="S5.p1.1.m1.1d">- 10</annotation></semantics></math> and <math alttext="10" class="ltx_Math" display="inline" id="S5.p1.2.m2.1"><semantics id="S5.p1.2.m2.1a"><mn id="S5.p1.2.m2.1.1" xref="S5.p1.2.m2.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="S5.p1.2.m2.1b"><cn id="S5.p1.2.m2.1.1.cmml" type="integer" xref="S5.p1.2.m2.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.2.m2.1c">10</annotation><annotation encoding="application/x-llamapun" id="S5.p1.2.m2.1d">10</annotation></semantics></math>, respectively. All results correspond to a <math alttext="95\%" class="ltx_Math" display="inline" id="S5.p1.3.m3.1"><semantics id="S5.p1.3.m3.1a"><mrow id="S5.p1.3.m3.1.1" xref="S5.p1.3.m3.1.1.cmml"><mn id="S5.p1.3.m3.1.1.2" xref="S5.p1.3.m3.1.1.2.cmml">95</mn><mo id="S5.p1.3.m3.1.1.1" xref="S5.p1.3.m3.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.3.m3.1b"><apply id="S5.p1.3.m3.1.1.cmml" xref="S5.p1.3.m3.1.1"><csymbol cd="latexml" id="S5.p1.3.m3.1.1.1.cmml" xref="S5.p1.3.m3.1.1.1">percent</csymbol><cn id="S5.p1.3.m3.1.1.2.cmml" type="integer" xref="S5.p1.3.m3.1.1.2">95</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.3.m3.1c">95\%</annotation><annotation encoding="application/x-llamapun" id="S5.p1.3.m3.1d">95 %</annotation></semantics></math> confidence level.</p> </div> <section class="ltx_subsection" id="S5.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.1 </span>Pancreas cancer data</h3> <div class="ltx_para" id="S5.SS1.p1"> <p class="ltx_p" id="S5.SS1.p1.1">First, we consider a data set on pancreatic cancer retrieved from the Surveillance, Epidemiology and End Results (SEER) data base. In the construction of this data set, patients who were diagnosed with pancreatic cancer were followed up and their death time or possible censoring time was recorded. Moreover, for each patient in the study, demographic variables as well as variables pertaining to their disease were obtained. These data have previously been analyzed by <cite class="ltx_cite ltx_citemacro_cite">Czado and Van Keilegom, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib9" title="">2023</a>)</cite>, who focus on the subpopulation of black patients. For this data application, we will narrow the target group further by only considering black males. In this way, we obtain a data set of <math alttext="4490" class="ltx_markedasmath" display="inline" id="S5.SS1.p1.1.m1.1.1.m1.1"><semantics id="S5.SS1.p1.1.m1.1.1.m1.1a"><mn id="S5.SS1.p1.1.m1.1.1.m1.1.1" xref="S5.SS1.p1.1.m1.1.1.m1.1.1.cmml">4490</mn><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.1.m1.1.1.m1.1b"><cn id="S5.SS1.p1.1.m1.1.1.m1.1.1.cmml" type="integer" xref="S5.SS1.p1.1.m1.1.1.m1.1.1">4490</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.1.m1.1.1.m1.1c">4490</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.1.m1.1.1.m1.1d">4490</annotation></semantics></math> observations.</p> </div> <div class="ltx_para" id="S5.SS1.p2"> <p class="ltx_p" id="S5.SS1.p2.3">We will model the time until a patient succumbs to their disease <math alttext="(T)" class="ltx_Math" display="inline" id="S5.SS1.p2.1.m1.1"><semantics id="S5.SS1.p2.1.m1.1a"><mrow id="S5.SS1.p2.1.m1.1.2.2"><mo id="S5.SS1.p2.1.m1.1.2.2.1" stretchy="false">(</mo><mi id="S5.SS1.p2.1.m1.1.1" xref="S5.SS1.p2.1.m1.1.1.cmml">T</mi><mo id="S5.SS1.p2.1.m1.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.1.m1.1b"><ci id="S5.SS1.p2.1.m1.1.1.cmml" xref="S5.SS1.p2.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.1.m1.1c">(T)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.1.m1.1d">( italic_T )</annotation></semantics></math> and view all other events as censoring. One possible event that is therefore viewed as censoring occurs when a patient receives a transplant. Since only those patients who are in the worst medical condition will be eligible for this treatment, the time until a patient undergoes transplantation and the time until they would have otherwise died is likely positively related. By extension, the validity of an independence assumption between <math alttext="T" class="ltx_Math" display="inline" id="S5.SS1.p2.2.m2.1"><semantics id="S5.SS1.p2.2.m2.1a"><mi id="S5.SS1.p2.2.m2.1.1" xref="S5.SS1.p2.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.2.m2.1b"><ci id="S5.SS1.p2.2.m2.1.1.cmml" xref="S5.SS1.p2.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.2.m2.1d">italic_T</annotation></semantics></math> and <math alttext="C" class="ltx_Math" display="inline" id="S5.SS1.p2.3.m3.1"><semantics id="S5.SS1.p2.3.m3.1a"><mi id="S5.SS1.p2.3.m3.1.1" xref="S5.SS1.p2.3.m3.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.3.m3.1b"><ci id="S5.SS1.p2.3.m3.1.1.cmml" xref="S5.SS1.p2.3.m3.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.3.m3.1c">C</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.3.m3.1d">italic_C</annotation></semantics></math> seems questionable.</p> </div> <div class="ltx_para" id="S5.SS1.p3"> <p class="ltx_p" id="S5.SS1.p3.10">We consider model (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2.E1" title="In 2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">1</span></a>) using the Cox link function as discussed in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S2" title="2 Model and methodology ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">2</span></a> and provide three separate analyses based on the stage of the cancer, which can either be local <math alttext="(n=$476$)" class="ltx_Math" display="inline" id="S5.SS1.p3.1.m1.2"><semantics id="S5.SS1.p3.1.m1.2a"><mrow id="S5.SS1.p3.1.m1.2.2.1" 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id="S5.SS1.p3.1.m1.2c">(n=$476$)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.1.m1.2d">( italic_n = 476 )</annotation></semantics></math>, regional <math alttext="(n=$1404$)" class="ltx_Math" display="inline" id="S5.SS1.p3.2.m2.2"><semantics id="S5.SS1.p3.2.m2.2a"><mrow id="S5.SS1.p3.2.m2.2.2.1" xref="S5.SS1.p3.2.m2.2.2.1.1.cmml"><mo id="S5.SS1.p3.2.m2.2.2.1.2" stretchy="false" xref="S5.SS1.p3.2.m2.2.2.1.1.cmml">(</mo><mrow id="S5.SS1.p3.2.m2.2.2.1.1" xref="S5.SS1.p3.2.m2.2.2.1.1.cmml"><mi id="S5.SS1.p3.2.m2.2.2.1.1.2" xref="S5.SS1.p3.2.m2.2.2.1.1.2.cmml">n</mi><mo id="S5.SS1.p3.2.m2.2.2.1.1.1" xref="S5.SS1.p3.2.m2.2.2.1.1.1.cmml">=</mo><mn id="S5.SS1.p3.2.m2.2.2.1.1.3" xref="S5.SS1.p3.2.m2.2.2.1.1.3.cmml">1404</mn></mrow><mo id="S5.SS1.p3.2.m2.2.2.1.3" stretchy="false" xref="S5.SS1.p3.2.m2.2.2.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.2.m2.2b"><apply id="S5.SS1.p3.2.m2.2.2.1.1.cmml" xref="S5.SS1.p3.2.m2.2.2.1"><eq 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id="S5.SS1.p3.3.m3.2.2.1.1.3" xref="S5.SS1.p3.3.m3.2.2.1.1.3.cmml">2610</mn></mrow><mo id="S5.SS1.p3.3.m3.2.2.1.3" stretchy="false" xref="S5.SS1.p3.3.m3.2.2.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.3.m3.2b"><apply id="S5.SS1.p3.3.m3.2.2.1.1.cmml" xref="S5.SS1.p3.3.m3.2.2.1"><eq id="S5.SS1.p3.3.m3.2.2.1.1.1.cmml" xref="S5.SS1.p3.3.m3.2.2.1.1.1"></eq><ci id="S5.SS1.p3.3.m3.2.2.1.1.2.cmml" xref="S5.SS1.p3.3.m3.2.2.1.1.2">𝑛</ci><cn id="S5.SS1.p3.3.m3.2.2.1.1.3.cmml" type="integer" xref="S5.SS1.p3.3.m3.2.2.1.1.3">2610</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p3.3.m3.2c">(n=$2610$)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.3.m3.2d">( italic_n = 2610 )</annotation></semantics></math>. Each analysis further considers two continuous covariates, namely the size of the tumor <math alttext="(X_{1})" class="ltx_Math" display="inline" id="S5.SS1.p3.4.m4.1"><semantics id="S5.SS1.p3.4.m4.1a"><mrow id="S5.SS1.p3.4.m4.1.1.1" xref="S5.SS1.p3.4.m4.1.1.1.1.cmml"><mo id="S5.SS1.p3.4.m4.1.1.1.2" stretchy="false" xref="S5.SS1.p3.4.m4.1.1.1.1.cmml">(</mo><msub id="S5.SS1.p3.4.m4.1.1.1.1" xref="S5.SS1.p3.4.m4.1.1.1.1.cmml"><mi id="S5.SS1.p3.4.m4.1.1.1.1.2" xref="S5.SS1.p3.4.m4.1.1.1.1.2.cmml">X</mi><mn id="S5.SS1.p3.4.m4.1.1.1.1.3" xref="S5.SS1.p3.4.m4.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.SS1.p3.4.m4.1.1.1.3" stretchy="false" xref="S5.SS1.p3.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.4.m4.1b"><apply id="S5.SS1.p3.4.m4.1.1.1.1.cmml" xref="S5.SS1.p3.4.m4.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p3.4.m4.1.1.1.1.1.cmml" xref="S5.SS1.p3.4.m4.1.1.1">subscript</csymbol><ci id="S5.SS1.p3.4.m4.1.1.1.1.2.cmml" xref="S5.SS1.p3.4.m4.1.1.1.1.2">𝑋</ci><cn id="S5.SS1.p3.4.m4.1.1.1.1.3.cmml" type="integer" xref="S5.SS1.p3.4.m4.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p3.4.m4.1c">(X_{1})</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.4.m4.1d">( italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> and the age of the patient at diagnosis <math alttext="(X_{2})" class="ltx_Math" display="inline" id="S5.SS1.p3.5.m5.1"><semantics id="S5.SS1.p3.5.m5.1a"><mrow id="S5.SS1.p3.5.m5.1.1.1" xref="S5.SS1.p3.5.m5.1.1.1.1.cmml"><mo id="S5.SS1.p3.5.m5.1.1.1.2" stretchy="false" xref="S5.SS1.p3.5.m5.1.1.1.1.cmml">(</mo><msub id="S5.SS1.p3.5.m5.1.1.1.1" xref="S5.SS1.p3.5.m5.1.1.1.1.cmml"><mi id="S5.SS1.p3.5.m5.1.1.1.1.2" xref="S5.SS1.p3.5.m5.1.1.1.1.2.cmml">X</mi><mn id="S5.SS1.p3.5.m5.1.1.1.1.3" xref="S5.SS1.p3.5.m5.1.1.1.1.3.cmml">2</mn></msub><mo id="S5.SS1.p3.5.m5.1.1.1.3" stretchy="false" xref="S5.SS1.p3.5.m5.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.5.m5.1b"><apply id="S5.SS1.p3.5.m5.1.1.1.1.cmml" xref="S5.SS1.p3.5.m5.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p3.5.m5.1.1.1.1.1.cmml" xref="S5.SS1.p3.5.m5.1.1.1">subscript</csymbol><ci id="S5.SS1.p3.5.m5.1.1.1.1.2.cmml" xref="S5.SS1.p3.5.m5.1.1.1.1.2">𝑋</ci><cn id="S5.SS1.p3.5.m5.1.1.1.1.3.cmml" type="integer" xref="S5.SS1.p3.5.m5.1.1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p3.5.m5.1c">(X_{2})</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.5.m5.1d">( italic_X start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>. Both were standardized and had outliers – defined as values outside the interval between the <math alttext="0.025" class="ltx_Math" display="inline" id="S5.SS1.p3.6.m6.1"><semantics id="S5.SS1.p3.6.m6.1a"><mn id="S5.SS1.p3.6.m6.1.1" xref="S5.SS1.p3.6.m6.1.1.cmml">0.025</mn><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.6.m6.1b"><cn id="S5.SS1.p3.6.m6.1.1.cmml" type="float" xref="S5.SS1.p3.6.m6.1.1">0.025</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p3.6.m6.1c">0.025</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.6.m6.1d">0.025</annotation></semantics></math>-th and <math alttext="0.975" class="ltx_Math" display="inline" id="S5.SS1.p3.7.m7.1"><semantics id="S5.SS1.p3.7.m7.1a"><mn id="S5.SS1.p3.7.m7.1.1" xref="S5.SS1.p3.7.m7.1.1.cmml">0.975</mn><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.7.m7.1b"><cn id="S5.SS1.p3.7.m7.1.1.cmml" type="float" xref="S5.SS1.p3.7.m7.1.1">0.975</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p3.7.m7.1c">0.975</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.7.m7.1d">0.975</annotation></semantics></math>-th quantile – and missing values removed prior to the analysis. Interest will be in the survival at either <math alttext="6" class="ltx_Math" display="inline" id="S5.SS1.p3.8.m8.1"><semantics id="S5.SS1.p3.8.m8.1a"><mn id="S5.SS1.p3.8.m8.1.1" xref="S5.SS1.p3.8.m8.1.1.cmml">6</mn><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.8.m8.1b"><cn id="S5.SS1.p3.8.m8.1.1.cmml" type="integer" xref="S5.SS1.p3.8.m8.1.1">6</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p3.8.m8.1c">6</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.8.m8.1d">6</annotation></semantics></math>, <math alttext="12" class="ltx_Math" display="inline" id="S5.SS1.p3.9.m9.1"><semantics id="S5.SS1.p3.9.m9.1a"><mn id="S5.SS1.p3.9.m9.1.1" xref="S5.SS1.p3.9.m9.1.1.cmml">12</mn><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.9.m9.1b"><cn id="S5.SS1.p3.9.m9.1.1.cmml" type="integer" xref="S5.SS1.p3.9.m9.1.1">12</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p3.9.m9.1c">12</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.9.m9.1d">12</annotation></semantics></math> and <math alttext="18" class="ltx_Math" display="inline" id="S5.SS1.p3.10.m10.1"><semantics id="S5.SS1.p3.10.m10.1a"><mn id="S5.SS1.p3.10.m10.1.1" xref="S5.SS1.p3.10.m10.1.1.cmml">18</mn><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.10.m10.1b"><cn id="S5.SS1.p3.10.m10.1.1.cmml" type="integer" xref="S5.SS1.p3.10.m10.1.1">18</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p3.10.m10.1c">18</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.10.m10.1d">18</annotation></semantics></math> months after the initial diagnosis. We also consider the case in which covariate effects are assumed to be time-independent by combining identified intervals at each considered time point, estimated at a Bonferroni corrected level, by means of intersection.</p> </div> <div class="ltx_para" id="S5.SS1.p4"> <p class="ltx_p" id="S5.SS1.p4.6">The results are shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S5.F2" title="Figure 2 ‣ 5.1 Pancreas cancer data ‣ 5 Data applications ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">2</span></a>. From the left panel, it can be seen that when the cancer is local, the identified intervals for the effects of <math alttext="X_{1}" class="ltx_Math" display="inline" id="S5.SS1.p4.1.m1.1"><semantics id="S5.SS1.p4.1.m1.1a"><msub id="S5.SS1.p4.1.m1.1.1" xref="S5.SS1.p4.1.m1.1.1.cmml"><mi id="S5.SS1.p4.1.m1.1.1.2" xref="S5.SS1.p4.1.m1.1.1.2.cmml">X</mi><mn id="S5.SS1.p4.1.m1.1.1.3" xref="S5.SS1.p4.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.1.m1.1b"><apply id="S5.SS1.p4.1.m1.1.1.cmml" xref="S5.SS1.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p4.1.m1.1.1.1.cmml" xref="S5.SS1.p4.1.m1.1.1">subscript</csymbol><ci id="S5.SS1.p4.1.m1.1.1.2.cmml" xref="S5.SS1.p4.1.m1.1.1.2">𝑋</ci><cn id="S5.SS1.p4.1.m1.1.1.3.cmml" type="integer" xref="S5.SS1.p4.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.1.m1.1c">X_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.1.m1.1d">italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="X_{2}" class="ltx_Math" display="inline" id="S5.SS1.p4.2.m2.1"><semantics id="S5.SS1.p4.2.m2.1a"><msub id="S5.SS1.p4.2.m2.1.1" xref="S5.SS1.p4.2.m2.1.1.cmml"><mi id="S5.SS1.p4.2.m2.1.1.2" xref="S5.SS1.p4.2.m2.1.1.2.cmml">X</mi><mn id="S5.SS1.p4.2.m2.1.1.3" xref="S5.SS1.p4.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.2.m2.1b"><apply id="S5.SS1.p4.2.m2.1.1.cmml" xref="S5.SS1.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS1.p4.2.m2.1.1.1.cmml" xref="S5.SS1.p4.2.m2.1.1">subscript</csymbol><ci id="S5.SS1.p4.2.m2.1.1.2.cmml" xref="S5.SS1.p4.2.m2.1.1.2">𝑋</ci><cn id="S5.SS1.p4.2.m2.1.1.3.cmml" type="integer" xref="S5.SS1.p4.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.2.m2.1c">X_{2}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.2.m2.1d">italic_X start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are entirely comprised of positive values when <math alttext="t=6" class="ltx_Math" display="inline" id="S5.SS1.p4.3.m3.1"><semantics id="S5.SS1.p4.3.m3.1a"><mrow id="S5.SS1.p4.3.m3.1.1" xref="S5.SS1.p4.3.m3.1.1.cmml"><mi id="S5.SS1.p4.3.m3.1.1.2" xref="S5.SS1.p4.3.m3.1.1.2.cmml">t</mi><mo id="S5.SS1.p4.3.m3.1.1.1" xref="S5.SS1.p4.3.m3.1.1.1.cmml">=</mo><mn id="S5.SS1.p4.3.m3.1.1.3" xref="S5.SS1.p4.3.m3.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.3.m3.1b"><apply id="S5.SS1.p4.3.m3.1.1.cmml" xref="S5.SS1.p4.3.m3.1.1"><eq id="S5.SS1.p4.3.m3.1.1.1.cmml" xref="S5.SS1.p4.3.m3.1.1.1"></eq><ci id="S5.SS1.p4.3.m3.1.1.2.cmml" xref="S5.SS1.p4.3.m3.1.1.2">𝑡</ci><cn id="S5.SS1.p4.3.m3.1.1.3.cmml" type="integer" xref="S5.SS1.p4.3.m3.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.3.m3.1c">t=6</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.3.m3.1d">italic_t = 6</annotation></semantics></math>. As such, for this time point, even though we cannot be certain of the exact covariate effects on the probability of dying before <math alttext="6" class="ltx_Math" display="inline" id="S5.SS1.p4.4.m4.1"><semantics id="S5.SS1.p4.4.m4.1a"><mn id="S5.SS1.p4.4.m4.1.1" xref="S5.SS1.p4.4.m4.1.1.cmml">6</mn><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.4.m4.1b"><cn id="S5.SS1.p4.4.m4.1.1.cmml" type="integer" xref="S5.SS1.p4.4.m4.1.1">6</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.4.m4.1c">6</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.4.m4.1d">6</annotation></semantics></math> months, we can be confident that the effect sizes are greater than zero. Furthermore, we can see that the effect of age remains significant also at the later time points, while the identified interval for the effect of the size of the tumor contains zero when <math alttext="t=12" class="ltx_Math" display="inline" id="S5.SS1.p4.5.m5.1"><semantics id="S5.SS1.p4.5.m5.1a"><mrow id="S5.SS1.p4.5.m5.1.1" xref="S5.SS1.p4.5.m5.1.1.cmml"><mi id="S5.SS1.p4.5.m5.1.1.2" xref="S5.SS1.p4.5.m5.1.1.2.cmml">t</mi><mo id="S5.SS1.p4.5.m5.1.1.1" xref="S5.SS1.p4.5.m5.1.1.1.cmml">=</mo><mn id="S5.SS1.p4.5.m5.1.1.3" xref="S5.SS1.p4.5.m5.1.1.3.cmml">12</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.5.m5.1b"><apply id="S5.SS1.p4.5.m5.1.1.cmml" xref="S5.SS1.p4.5.m5.1.1"><eq id="S5.SS1.p4.5.m5.1.1.1.cmml" xref="S5.SS1.p4.5.m5.1.1.1"></eq><ci id="S5.SS1.p4.5.m5.1.1.2.cmml" xref="S5.SS1.p4.5.m5.1.1.2">𝑡</ci><cn id="S5.SS1.p4.5.m5.1.1.3.cmml" type="integer" xref="S5.SS1.p4.5.m5.1.1.3">12</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.5.m5.1c">t=12</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.5.m5.1d">italic_t = 12</annotation></semantics></math> or <math alttext="t=18" class="ltx_Math" display="inline" id="S5.SS1.p4.6.m6.1"><semantics id="S5.SS1.p4.6.m6.1a"><mrow id="S5.SS1.p4.6.m6.1.1" xref="S5.SS1.p4.6.m6.1.1.cmml"><mi id="S5.SS1.p4.6.m6.1.1.2" xref="S5.SS1.p4.6.m6.1.1.2.cmml">t</mi><mo id="S5.SS1.p4.6.m6.1.1.1" xref="S5.SS1.p4.6.m6.1.1.1.cmml">=</mo><mn id="S5.SS1.p4.6.m6.1.1.3" xref="S5.SS1.p4.6.m6.1.1.3.cmml">18</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.6.m6.1b"><apply id="S5.SS1.p4.6.m6.1.1.cmml" xref="S5.SS1.p4.6.m6.1.1"><eq id="S5.SS1.p4.6.m6.1.1.1.cmml" xref="S5.SS1.p4.6.m6.1.1.1"></eq><ci id="S5.SS1.p4.6.m6.1.1.2.cmml" xref="S5.SS1.p4.6.m6.1.1.2">𝑡</ci><cn id="S5.SS1.p4.6.m6.1.1.3.cmml" type="integer" xref="S5.SS1.p4.6.m6.1.1.3">18</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.6.m6.1c">t=18</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.6.m6.1d">italic_t = 18</annotation></semantics></math>. If one would be willing to assume that covariate effects do not change over time, the identified interval does become significant, albeit only barely so. The plots containing the results of the analyses for regional and distant cancer can be interpreted similarly. For reference, the results of a classical Cox model assuming independence are overlaid.</p> </div> <div class="ltx_para" id="S5.SS1.p5"> <p class="ltx_p" id="S5.SS1.p5.1">From a similar analysis using the AFT link function (delegated to Supplementary material <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:supp:_Pancreas_data_application</span>), similar conclusions may be drawn.</p> </div> <figure class="ltx_figure" id="S5.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="223" id="S5.F2.g1" src="extracted/6289908/Figures/results_all_Cox.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S5.F2.4.2.1" style="font-size:90%;">Figure 2</span>: </span><span class="ltx_text" id="S5.F2.2.1" style="font-size:90%;">The results of the model using a Cox link function applied to the SEER Pancreas data set. The vertical bars indicate the estimated identified intervals at each time point of interest, while the dashed horizontal lines represent the point estimates of a Cox model assuming independence. The vertical bars at <math alttext="t=0" class="ltx_Math" display="inline" id="S5.F2.2.1.m1.1"><semantics id="S5.F2.2.1.m1.1b"><mrow id="S5.F2.2.1.m1.1.1" xref="S5.F2.2.1.m1.1.1.cmml"><mi id="S5.F2.2.1.m1.1.1.2" xref="S5.F2.2.1.m1.1.1.2.cmml">t</mi><mo id="S5.F2.2.1.m1.1.1.1" xref="S5.F2.2.1.m1.1.1.1.cmml">=</mo><mn id="S5.F2.2.1.m1.1.1.3" xref="S5.F2.2.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.F2.2.1.m1.1c"><apply id="S5.F2.2.1.m1.1.1.cmml" xref="S5.F2.2.1.m1.1.1"><eq id="S5.F2.2.1.m1.1.1.1.cmml" xref="S5.F2.2.1.m1.1.1.1"></eq><ci id="S5.F2.2.1.m1.1.1.2.cmml" xref="S5.F2.2.1.m1.1.1.2">𝑡</ci><cn id="S5.F2.2.1.m1.1.1.3.cmml" type="integer" xref="S5.F2.2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F2.2.1.m1.1d">t=0</annotation><annotation encoding="application/x-llamapun" id="S5.F2.2.1.m1.1e">italic_t = 0</annotation></semantics></math> pertain to the model assuming time-independent regression parameters.</span></figcaption> </figure> </section> <section class="ltx_subsection" id="S5.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.2 </span>NLSY data</h3> <div class="ltx_para" id="S5.SS2.p1"> <p class="ltx_p" id="S5.SS2.p1.4">Next, we study the National Longitudinal Surveys Year 1979 (NLSY79) data, which is made available by the U.S. Bureau of Labour Statistics. Specifically, we are interested in unemployment spells, i.e. the time until people who enter the job market find a job. Censoring of these spells occurs at the end of the considered time period, or when subjects stop responding to the surveys. This latter cause may lead to a dependence between <math alttext="T" class="ltx_Math" display="inline" id="S5.SS2.p1.1.m1.1"><semantics id="S5.SS2.p1.1.m1.1a"><mi id="S5.SS2.p1.1.m1.1.1" xref="S5.SS2.p1.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.1.m1.1b"><ci id="S5.SS2.p1.1.m1.1.1.cmml" xref="S5.SS2.p1.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.1.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.1.m1.1d">italic_T</annotation></semantics></math> and <math alttext="C" class="ltx_Math" display="inline" id="S5.SS2.p1.2.m2.1"><semantics id="S5.SS2.p1.2.m2.1a"><mi id="S5.SS2.p1.2.m2.1.1" xref="S5.SS2.p1.2.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.2.m2.1b"><ci id="S5.SS2.p1.2.m2.1.1.cmml" xref="S5.SS2.p1.2.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.2.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.2.m2.1d">italic_C</annotation></semantics></math>, as subjects who either found a job or have lost motivation to look for one might be more inclined to discontinue their participation in the study. This point is further corroborated by <cite class="ltx_cite ltx_citemacro_cite">Frandsen, (<a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#bib.bib16" title="">2019</a>)</cite>. Analogous to the aforementioned paper, we obtain data sets of a workable size by narrowing our focus to the period <math alttext="1989-1993" class="ltx_Math" display="inline" id="S5.SS2.p1.3.m3.1"><semantics id="S5.SS2.p1.3.m3.1a"><mrow id="S5.SS2.p1.3.m3.1.1" xref="S5.SS2.p1.3.m3.1.1.cmml"><mn id="S5.SS2.p1.3.m3.1.1.2" xref="S5.SS2.p1.3.m3.1.1.2.cmml">1989</mn><mo id="S5.SS2.p1.3.m3.1.1.1" xref="S5.SS2.p1.3.m3.1.1.1.cmml">−</mo><mn id="S5.SS2.p1.3.m3.1.1.3" xref="S5.SS2.p1.3.m3.1.1.3.cmml">1993</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.3.m3.1b"><apply id="S5.SS2.p1.3.m3.1.1.cmml" xref="S5.SS2.p1.3.m3.1.1"><minus id="S5.SS2.p1.3.m3.1.1.1.cmml" xref="S5.SS2.p1.3.m3.1.1.1"></minus><cn id="S5.SS2.p1.3.m3.1.1.2.cmml" type="integer" xref="S5.SS2.p1.3.m3.1.1.2">1989</cn><cn id="S5.SS2.p1.3.m3.1.1.3.cmml" type="integer" xref="S5.SS2.p1.3.m3.1.1.3">1993</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.3.m3.1c">1989-1993</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.3.m3.1d">1989 - 1993</annotation></semantics></math>. We further divide this subset into different strata and run the method on each of them. Specifically, six strata are constructed based on the combinations of sex (male or female) and race (hispanic, black or other). Two variables are considered: a binary variable indicating whether an unemployed subject is highly educated (defined as having obtained a high school diploma before or during the unemployment spell), and a continuous covariate representing the age of the subject at the start of their unemployment spell. For the former, missing values were imputed based on the nearest-in-time observed values. The latter was standardized prior to the analysis. Interest is in the covariate effects on the probability of having an unemployment spell longer than <math alttext="6" class="ltx_Math" display="inline" id="S5.SS2.p1.4.m4.1"><semantics id="S5.SS2.p1.4.m4.1a"><mn id="S5.SS2.p1.4.m4.1.1" xref="S5.SS2.p1.4.m4.1.1.cmml">6</mn><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.4.m4.1b"><cn id="S5.SS2.p1.4.m4.1.1.cmml" type="integer" xref="S5.SS2.p1.4.m4.1.1">6</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.4.m4.1c">6</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.4.m4.1d">6</annotation></semantics></math> months. The results of the analysis are shown in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.11210v2#S5.T2" title="Table 2 ‣ 5.2 NLSY data ‣ 5 Data applications ‣ Bounds for the regression parameters in dependently censored survival models"><span class="ltx_text ltx_ref_tag">2</span></a>.</p> </div> <div class="ltx_para" id="S5.SS2.p2"> <p class="ltx_p" id="S5.SS2.p2.1">For all strata pertaining to females, the identified intervals for the effect of education on the unemployment spell are strictly positive. We observe the same for black males. Hence for these subgroups in the population we conclude that having obtained a high school diploma significantly decreases the probability of being unemployed for longer than <math alttext="6" class="ltx_Math" display="inline" id="S5.SS2.p2.1.m1.1"><semantics id="S5.SS2.p2.1.m1.1a"><mn id="S5.SS2.p2.1.m1.1.1" xref="S5.SS2.p2.1.m1.1.1.cmml">6</mn><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.1.m1.1b"><cn id="S5.SS2.p2.1.m1.1.1.cmml" type="integer" xref="S5.SS2.p2.1.m1.1.1">6</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.1.m1.1c">6</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.1.m1.1d">6</annotation></semantics></math> months. Notably, the model using the Cox link function was determined to be misspecified in the stratum “Male–Other” due to an empty identified interval.</p> </div> <figure class="ltx_table" id="S5.T2"> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S5.T2.2"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S5.T2.2.1.1"> <th class="ltx_td ltx_th ltx_th_row ltx_border_tt" id="S5.T2.2.1.1.1"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="2" id="S5.T2.2.1.1.2">AFT link</th> <th class="ltx_td ltx_th ltx_th_column ltx_border_tt" id="S5.T2.2.1.1.3"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="2" id="S5.T2.2.1.1.4">Cox link</th> </tr> <tr class="ltx_tr" id="S5.T2.2.2.2"> <th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row" id="S5.T2.2.2.2.1">Stratum</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.2.2.2.2">Age</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.2.2.2.3">education</th> <th class="ltx_td ltx_th ltx_th_column" id="S5.T2.2.2.2.4"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.2.2.2.5">Age</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.2.2.2.6">Education</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S5.T2.2.3.1"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_t" id="S5.T2.2.3.1.1">Male–Hispanic</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.2.3.1.2">[-0.48, 0.09]</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.2.3.1.3">[-0.08, 0.88]</td> <td class="ltx_td ltx_border_t" id="S5.T2.2.3.1.4"></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.2.3.1.5">[-0.27, 0.04]</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.2.3.1.6">[-0.05, 0.42]</td> </tr> <tr class="ltx_tr" id="S5.T2.2.4.2"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row" id="S5.T2.2.4.2.1">Male–Black</th> <td class="ltx_td ltx_align_center" id="S5.T2.2.4.2.2">[-0.32, 0.05]</td> <td class="ltx_td ltx_align_center" id="S5.T2.2.4.2.3">[0.13, 0.74]</td> <td class="ltx_td" id="S5.T2.2.4.2.4"></td> <td class="ltx_td ltx_align_center" id="S5.T2.2.4.2.5">[-0.19, 0.01]</td> <td class="ltx_td ltx_align_center" id="S5.T2.2.4.2.6">[0.07, 0.42]</td> </tr> <tr class="ltx_tr" id="S5.T2.2.5.3"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row" id="S5.T2.2.5.3.1">Male–Other</th> <td class="ltx_td ltx_align_center" id="S5.T2.2.5.3.2">[-0.39, 0.24]</td> <td class="ltx_td ltx_align_center" id="S5.T2.2.5.3.3">[-0.15, 0.58]</td> <td class="ltx_td" id="S5.T2.2.5.3.4"></td> <td class="ltx_td ltx_align_center" id="S5.T2.2.5.3.5">/</td> <td class="ltx_td ltx_align_center" id="S5.T2.2.5.3.6">/</td> </tr> <tr class="ltx_tr" id="S5.T2.2.6.4"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row" id="S5.T2.2.6.4.1">Female–Hispanic</th> <td class="ltx_td ltx_align_center" id="S5.T2.2.6.4.2">[-0.22, 0.34]</td> <td class="ltx_td ltx_align_center" id="S5.T2.2.6.4.3">[0.34, 0.98]</td> <td class="ltx_td" id="S5.T2.2.6.4.4"></td> <td class="ltx_td ltx_align_center" id="S5.T2.2.6.4.5">[-0.14, 0.18]</td> <td class="ltx_td ltx_align_center" id="S5.T2.2.6.4.6">[0.21, 0.59]</td> </tr> <tr class="ltx_tr" id="S5.T2.2.7.5"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row" id="S5.T2.2.7.5.1">Female–Black</th> <td class="ltx_td ltx_align_center" id="S5.T2.2.7.5.2">[-0.19, 0.24]</td> <td class="ltx_td ltx_align_center" id="S5.T2.2.7.5.3">[0.39, 0.91]</td> <td class="ltx_td" id="S5.T2.2.7.5.4"></td> <td class="ltx_td ltx_align_center" id="S5.T2.2.7.5.5">[-0.12, 0.15]</td> <td class="ltx_td ltx_align_center" id="S5.T2.2.7.5.6">[0.24, 0.55]</td> </tr> <tr class="ltx_tr" id="S5.T2.2.8.6"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_bb" id="S5.T2.2.8.6.1">Female–Other</th> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T2.2.8.6.2">[-0.16, 0.36]</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T2.2.8.6.3">[0.01, 0.67]</td> <td class="ltx_td ltx_border_bb" id="S5.T2.2.8.6.4"></td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T2.2.8.6.5">[-0.09, 0.19]</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S5.T2.2.8.6.6">[0.01, 0.37]</td> </tr> </tbody> </table> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S5.T2.3.1.1" style="font-size:90%;">Table 2</span>: </span><span class="ltx_text" id="S5.T2.4.2" style="font-size:90%;">Results pertaining to the NLSY data application</span></figcaption> </figure> </section> <section class="ltx_subsection" id="S5.SSx1"> <h3 class="ltx_title ltx_title_subsection">Acknowledgements</h3> <div class="ltx_para" id="S5.SSx1.p1"> <p class="ltx_p" id="S5.SSx1.p1.1">The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation Flanders (FWO) and the Flemish Government department EWI. Lastly, the authors thank Gilles Crommen and several attentive attendees at conferences where this research was presented for their useful advice and comments.</p> </div> </section> <section class="ltx_subsection" id="S5.SSx2"> <h3 class="ltx_title ltx_title_subsection">Disclosure statement</h3> <div class="ltx_para" id="S5.SSx2.p1"> <p class="ltx_p" id="S5.SSx2.p1.1">The authors report that there are no competing interests to declare.</p> </div> </section> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Andrews and Shi, (2013)</span> <span class="ltx_bibblock"> Andrews, D. W. K. and Shi, X. (2013). </span> <span class="ltx_bibblock">Inference based on conditional moment inequalities. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib1.1.1">Econometrica</span>, 81(2):609–666. </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Bei, (2024)</span> <span class="ltx_bibblock"> Bei, X. (2024). </span> <span class="ltx_bibblock">Local linearization based subvector inference in moment inequality models. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib2.1.1">Journal of Econometrics</span>, 238(1). </span> <span class="ltx_bibblock">Article 105549. </span> </li> <li class="ltx_bibitem" id="bib.bib3"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Braekers and Veraverbeke, (2005)</span> <span class="ltx_bibblock"> Braekers, R. and Veraverbeke, N. (2005). </span> <span class="ltx_bibblock">A copula-graphic estimator for the conditional survival function under dependent censoring. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib3.1.1">Canadian Journal of Statistics</span>, 33(3):429–447. </span> </li> <li class="ltx_bibitem" id="bib.bib4"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Bugni et al., (2015)</span> <span class="ltx_bibblock"> Bugni, F. A., Canay, I. A., and Shi, X. (2015). </span> <span class="ltx_bibblock">Specification tests for partially identified models defined by moment inequalities. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib4.1.1">Journal of Econometrics</span>, 185(1):259–282. </span> </li> <li class="ltx_bibitem" id="bib.bib5"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Canay and Shaikh, (2017)</span> <span class="ltx_bibblock"> Canay, I. A. and Shaikh, A. M. (2017). </span> <span class="ltx_bibblock">Practical and theoretical advances in inference for partially identified models. </span> <span class="ltx_bibblock">In Honoré, B., Pakes, A., Piazzesi, M., and Samuelson, L., editors, <span class="ltx_text ltx_font_italic" id="bib.bib5.1.1">Advances in Economics and Econometrics: Eleventh World Congress</span>, Econometric Society Monographs, page 271–306. Cambridge University Press. </span> </li> <li class="ltx_bibitem" id="bib.bib6"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Cox, (1972)</span> <span class="ltx_bibblock"> Cox, D. R. (1972). </span> <span class="ltx_bibblock">Regression models and life-tables. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib6.1.1">Journal of the Royal Statistical Society: Series B (Methodological)</span>, 34(2):187–202. </span> </li> <li class="ltx_bibitem" id="bib.bib7"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">(7)</span> <span class="ltx_bibblock"> Crommen, G., Beyhum, J., and Van Keilegom, I. (2024a). </span> <span class="ltx_bibblock">An instrumental variable approach under dependent censoring. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib7.1.1">TEST</span>, 33:473–495. </span> </li> <li class="ltx_bibitem" id="bib.bib8"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">(8)</span> <span class="ltx_bibblock"> Crommen, G., Deresa, N. W., D’Haen, M., Ding, J., Willems, I., and Van Keilegom, I. (2024b). </span> <span class="ltx_bibblock">Dependent censoring based on copulas. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib8.1.1">SORT</span>. </span> <span class="ltx_bibblock">(Submitted). </span> </li> <li class="ltx_bibitem" id="bib.bib9"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Czado and Van Keilegom, (2023)</span> <span class="ltx_bibblock"> Czado, C. and Van Keilegom, I. (2023). </span> <span class="ltx_bibblock">Dependent censoring based on parametric copulas. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib9.1.1">Biometrika</span>, 110(3):721–738. </span> </li> <li class="ltx_bibitem" id="bib.bib10"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Delgado et al., (2022)</span> <span class="ltx_bibblock"> Delgado, M. A., García-Suaza, A., and Sant’Anna, P. H. C. (2022). </span> <span class="ltx_bibblock">Distribution regression in duration analysis: an application to unemployment spells. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib10.1.1">The Econometrics Journal</span>, 25(3):675–698. </span> </li> <li class="ltx_bibitem" id="bib.bib11"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Deresa and Van Keilegom, (2020)</span> <span class="ltx_bibblock"> Deresa, N. W. and Van Keilegom, I. (2020). </span> <span class="ltx_bibblock">Flexible parametric model for survival data subject to dependent censoring. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib11.1.1">Biometrical Journal</span>, 62(1):136–156. </span> </li> <li class="ltx_bibitem" id="bib.bib12"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Deresa and Van Keilegom, (2021)</span> <span class="ltx_bibblock"> Deresa, N. W. and Van Keilegom, I. (2021). </span> <span class="ltx_bibblock">On semiparametric modelling, estimation and inference for survival data subject to dependent censoring. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib12.1.1">Biometrika</span>, 108(4):965–979. </span> </li> <li class="ltx_bibitem" id="bib.bib13"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Deresa and Van Keilegom, (2024)</span> <span class="ltx_bibblock"> Deresa, N. W. and Van Keilegom, I. (2024). </span> <span class="ltx_bibblock">Copula based Cox proportional hazards models for dependent censoring. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib13.1.1">Journal of the American Statistical Association</span>, 119:1044–1054. </span> </li> <li class="ltx_bibitem" id="bib.bib14"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Deresa and Van Keilegom, (2025)</span> <span class="ltx_bibblock"> Deresa, N. W. and Van Keilegom, I. (2025). </span> <span class="ltx_bibblock">Semiparametric transformation models for survival data with dependent censoring. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib14.1.1">Ann. Instit. Statist. Math.</span> </span> <span class="ltx_bibblock">(To appear). </span> </li> <li class="ltx_bibitem" id="bib.bib15"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Ding and Van Keilegom, (2025)</span> <span class="ltx_bibblock"> Ding, J. and Van Keilegom, I. (2025). </span> <span class="ltx_bibblock">A copula-based extension of the kaplan-meier estimator under dependent censoring with unknown association. </span> <span class="ltx_bibblock">(In preparation). </span> </li> <li class="ltx_bibitem" id="bib.bib16"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Frandsen, (2019)</span> <span class="ltx_bibblock"> Frandsen, B. R. (2019). </span> <span class="ltx_bibblock">Testing censoring point independence. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib16.1.1">Journal of Business & Economic Statistics</span>, 37(3):496–505. </span> </li> <li class="ltx_bibitem" id="bib.bib17"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Gasparin and Ramdas, (2024)</span> <span class="ltx_bibblock"> Gasparin, M. and Ramdas, A. (2024). </span> <span class="ltx_bibblock">Merging uncertainty sets via majority vote. </span> <span class="ltx_bibblock">ArXiv preprint (https://arxiv.org/abs/2401.09379). </span> </li> <li class="ltx_bibitem" id="bib.bib18"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Horowitz and Manski, (1998)</span> <span class="ltx_bibblock"> Horowitz, J. L. and Manski, C. F. (1998). </span> <span class="ltx_bibblock">Censoring of outcomes and regressors due to survey nonresponse: Identification and estimation using weights and imputations. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib18.1.1">Journal of Econometrics</span>, 84(1):37–58. </span> </li> <li class="ltx_bibitem" id="bib.bib19"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Huang and Zhang, (2008)</span> <span class="ltx_bibblock"> Huang, X. and Zhang, N. (2008). </span> <span class="ltx_bibblock">Regression survival analysis with an assumed copula for dependent censoring: A sensitivity analysis approach. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib19.1.1">Biometrics</span>, 64(4):1090–1099. </span> </li> <li class="ltx_bibitem" id="bib.bib20"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Kaido et al., (2022)</span> <span class="ltx_bibblock"> Kaido, H., Molinari, F., and Stoye, J. (2022). </span> <span class="ltx_bibblock">Constraint qualifications in partial identification. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib20.1.1">Econometric Theory</span>, 38(3):596–619. </span> </li> <li class="ltx_bibitem" id="bib.bib21"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Kaplan and Meier, (1958)</span> <span class="ltx_bibblock"> Kaplan, E. L. and Meier, P. (1958). </span> <span class="ltx_bibblock">Nonparametric estimation from incomplete observations. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib21.1.1">Journal of the American Statistical Association</span>, 53(282):457–481. </span> </li> <li class="ltx_bibitem" id="bib.bib22"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Kim, (2023)</span> <span class="ltx_bibblock"> Kim, D. (2023). </span> <span class="ltx_bibblock">Partially identifying competing risks models: An application to the war on cancer. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib22.1.1">Journal of Econometrics</span>, 234(2):536–564. </span> </li> <li class="ltx_bibitem" id="bib.bib23"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Lewbel, (2019)</span> <span class="ltx_bibblock"> Lewbel, A. (2019). </span> <span class="ltx_bibblock">The identification zoo: Meanings of identification in econometrics. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib23.1.1">Journal of Economic Literature</span>, 57(4):835–903. </span> </li> <li class="ltx_bibitem" id="bib.bib24"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Lo and Wilke, (2023)</span> <span class="ltx_bibblock"> Lo, S. M. and Wilke, R. A. (2023). </span> <span class="ltx_bibblock">A parametric competing risks regression model with unknown dependent censoring. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib24.1.1">Journal of the Royal Statistical Society – Series C</span>, 72(4):1079–1093. </span> </li> <li class="ltx_bibitem" id="bib.bib25"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Molinari, (2020)</span> <span class="ltx_bibblock"> Molinari, F. (2020). </span> <span class="ltx_bibblock">Microeconometrics with partial identification. </span> <span class="ltx_bibblock">In Durlauf, S. N., Hansen, L. P., Heckman, J. J., and Matzkin, R. L., editors, <span class="ltx_text ltx_font_italic" id="bib.bib25.1.1">Handbook of Econometrics</span>, volume 7, pages 355–486. Elsevier. </span> </li> <li class="ltx_bibitem" id="bib.bib26"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Peterson, (1976)</span> <span class="ltx_bibblock"> Peterson, A. V. (1976). </span> <span class="ltx_bibblock">Bounds for a joint distribution function with fixed sub-distribution functions: Application to competing risks. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib26.1.1">Proceedings of the National Academy of Sciences - PNAS</span>, 73(1):11–13. </span> </li> <li class="ltx_bibitem" id="bib.bib27"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Rivest and Wells, (2001)</span> <span class="ltx_bibblock"> Rivest, L.-P. and Wells, M. T. (2001). </span> <span class="ltx_bibblock">A martingale approach to the copula-graphic estimator for the survival function under dependent censoring. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib27.1.1">Journal of Multivariate Analysis</span>, 79(1):138–155. </span> </li> <li class="ltx_bibitem" id="bib.bib28"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Rutten et al., (2024)</span> <span class="ltx_bibblock"> Rutten, S., Willems, I., Crommen, G., and Van Keilegom, I. (2024). </span> <span class="ltx_bibblock">Flexible control function approach under different types of censoring. </span> <span class="ltx_bibblock">ArXiv preprint (https://arxiv.org/abs/2403.11860). (Submitted). </span> </li> <li class="ltx_bibitem" id="bib.bib29"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Sakaguchi, (2024)</span> <span class="ltx_bibblock"> Sakaguchi, S. (2024). </span> <span class="ltx_bibblock">Partial identification and inference in duration models with endogenous censoring. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib29.1.1">Journal of Applied Econometrics</span>, 39:308–326. </span> </li> <li class="ltx_bibitem" id="bib.bib30"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Sklar, (1959)</span> <span class="ltx_bibblock"> Sklar, M. (1959). </span> <span class="ltx_bibblock">Fonctions de répartition à n dimensions et leurs marges. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib30.1.1">Annales de l’ISUP</span>, 8(3):229–231. </span> </li> <li class="ltx_bibitem" id="bib.bib31"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Sujica and Van Keilegom, (2018)</span> <span class="ltx_bibblock"> Sujica, A. and Van Keilegom, I. (2018). </span> <span class="ltx_bibblock">The copula-graphic estimator in censored nonparametric location-scale regression models. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib31.1.1">Econometrics and Statistics</span>, 7:89 – 114. </span> </li> <li class="ltx_bibitem" id="bib.bib32"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Szydlowski, (2019)</span> <span class="ltx_bibblock"> Szydlowski, A. (2019). </span> <span class="ltx_bibblock">Endogenous censoring in the mixed proportional hazard model with an application to optimal unemployment insurance. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib32.1.1">Journal of Applied Econometrics</span>, 34(7):1086–1101. </span> </li> <li class="ltx_bibitem" id="bib.bib33"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Tsiatis, (1975)</span> <span class="ltx_bibblock"> Tsiatis, A. (1975). </span> <span class="ltx_bibblock">A nonidentifiability aspect of the problem of competing risks. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib33.1.1">Proceedings of the National Academy of Sciences - PNAS</span>, 72(1):20–22. </span> </li> <li class="ltx_bibitem" id="bib.bib34"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Zheng and Klein, (1995)</span> <span class="ltx_bibblock"> Zheng, M. and Klein, J. P. (1995). </span> <span class="ltx_bibblock">Estimates of marginal survival for dependent competing risks based on an assumed copula. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib34.1.1">Biometrika</span>, 82(1):127–138. </span> </li> </ul> </section> <div class="ltx_pagination ltx_role_newpage"></div> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Tue Mar 18 10:42:21 2025 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span class="ltx_font_smallcaps" style="position:relative; bottom:2.2pt;">a</span>T<span class="ltx_font_smallcaps" style="font-size:120%;position:relative; bottom:-0.2ex;">e</span></span><span style="font-size:90%; position:relative; bottom:-0.2ex;">XML</span><img alt="Mascot Sammy" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>