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PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset
<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset</title> <!--Generated on Tue Mar 18 20:18:48 2025 by LaTeXML (version 0.8.8) http://dlmf.nist.gov/LaTeXML/.--> <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/2503.14711v1/"/></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/2503.14711v1#S1" title="In PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2" title="In PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Notation and Models</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/2503.14711v1#S2.SS1" title="In 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Fully synthetic data</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.SS2" title="In 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Partially synthetic data</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.SS3" title="In 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Generating fully synthetic data</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.SS4" title="In 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.4 </span>Generalized variance</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.SS5" title="In 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.5 </span>Sphericity test</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.SS5.SSS1" title="In 2.5 Sphericity test ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.5.1 </span>Independence test</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.SS5.SSS2" title="In 2.5 Sphericity test ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.5.2 </span>Regression of One set of Variables on other</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S3" title="In PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Description and Implementation of PSInference</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4" title="In PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Numerical Studies and Demonstration of Programming Code</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/2503.14711v1#S4.SS1" title="In 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Average coverage probability of the Generalized Variance</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4.SS2" title="In 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Average coverage probability for the Sphericity Test</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4.SS3" title="In 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Average coverage probability for the Independence Test</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4.SS4" title="In 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.4 </span>Average coverage probability for the Regression Test</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S5" title="In PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Summary</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> <span class="ltx_text" id="id4.4.4" style="font-size:90%;">Ricardo Moura<sup class="ltx_sup" id="id4.4.4.1"><span class="ltx_text ltx_font_italic" id="id4.4.4.1.1">1,2</span></sup>, Mina Norouzirad<sup class="ltx_sup" id="id4.4.4.2"><span class="ltx_text ltx_font_italic" id="id4.4.4.2.1">1</span></sup>, Vítor Augusto<sup class="ltx_sup" id="id4.4.4.3"><span class="ltx_text ltx_font_italic" id="id4.4.4.3.1">2</span></sup> and Miguel Fonseca<sup class="ltx_sup" id="id4.4.4.4"><span class="ltx_text ltx_font_italic" id="id4.4.4.4.1">2,3</span></sup> <br class="ltx_break"/></span> <sup class="ltx_sup" id="id8.8.id1"><span class="ltx_text ltx_font_italic" id="id8.8.id1.1" style="font-size:80%;">1</span></sup><span class="ltx_text" id="id7.7.6" style="font-size:80%;">Center for Mathematics and Applications (NOVA Math), NOVA School of Science and <br class="ltx_break"/>Technology (NOVA FCT), NOVA University Lisbon, Caparica, Portugal <br class="ltx_break"/><sup class="ltx_sup" id="id7.7.6.1"><span class="ltx_text ltx_font_italic" id="id7.7.6.1.1">2</span></sup>Portuguese Navy Research Center (CINAV), Naval Academy, Portuguese Navy, Alfeit, Portugal <br class="ltx_break"/><sup class="ltx_sup" id="id7.7.6.2"><span class="ltx_text ltx_font_italic" id="id7.7.6.2.1">3</span></sup>Department of Mathematics, NOVA School of Science and Technology (NOVA FCT), <br class="ltx_break"/>NOVA University Lisbon, Caparica, Portugal</span> </span><span class="ltx_author_notes"><span class="ltx_text" id="id9.9.id1" style="font-size:90%;">Corresponding author: rp.moura@fct.unl.pt</span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id10.id1">The development and generation of synthetic data are becoming increasingly vital in the field of statistical disclosure control. The PSInference package provides tools to perform exact inferential analysis on singly imputed synthetic data generated through Plug-in Sampling assuming that the original dataset follows a multivariate normal distribution. Includes functions to test the synthetic data’s covariance structure, covering aspects like generalized variance, sphericity, independence between subsets of variables, and regression of one set of variables on another. This package addresses the gap in the existing software by providing exact inferential methods suitable for cases where only a single synthetic dataset is released. <br class="ltx_break"/></p> <p class="ltx_p" id="id11.id2"><span class="ltx_text ltx_font_bold" id="id11.id2.1">Key Words:</span> Canonical test, Covariance matrix, Exact inference, Generalized variance, Multivariate normal distribution Plug-in Sampling, Single imputation, Independence test, R package, Regression test, Sphericity, Statistical disclosure control, Synthetic data.</p> </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.p1.1">Statistical disclosure control (SDC) techniques are used to balance the dual objectives of releasing statistical information from surveys while protecting the confidentiality of respondents’ data. To fulfill those two objectives, there are a range of methods, such as swapping data values, introducing random noise through addition or multiplication, and generating synthetic datasets to minimize the risk of disclosure <cite class="ltx_cite ltx_citemacro_citep">(Hundepool et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib5" title="">2010</a>)</cite>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">The development and dissemination of synthetic data are becoming increasingly vital in the field of statistical disclosure control, where preserving the privacy of sensitive information is a primary concern <cite class="ltx_cite ltx_citemacro_citep">(Klein, M., Mathew, T., & Sinha, B., <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib9" title="">2013</a>)</cite>. The main reason for this increase in attention is due to the fact that it maintains the statistical patterns of the assumed model for the original data, which is not the case for most of the other SDC techniques <cite class="ltx_cite ltx_citemacro_citep">(Drechsler, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib4" title="">2011</a>)</cite>.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">Most inferential approaches for synthetic data rely on the access to multiple imputation synthetic generated data <cite class="ltx_cite ltx_citemacro_citep">(Raghunathan, T. E., Reiter, J. P., & Rubin, D. B., <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib22" title="">2003</a>; Reiter, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib23" title="">2003</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib24" title="">2005</a>)</cite> and are based on the works of <cite class="ltx_cite ltx_citemacro_citet">Rubin (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib25" title="">1987</a>)</cite> and <cite class="ltx_cite ltx_citemacro_citet">Little (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib14" title="">1993</a>)</cite>. However in the last decade <cite class="ltx_cite ltx_citemacro_citet">Klein & Sinha (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib13" title="">2015d</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib11" title="">b</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib10" title="">a</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib12" title="">c</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Moura (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib17" title="">2016</a>); Moura et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib16" title="">2017</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib18" title="">2021</a>)</cite> and <cite class="ltx_cite ltx_citemacro_citet">Klein, Moura, & Sinha (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib8" title="">2021</a>)</cite> have developed a set of exact parametric inferential methods that are applicable to singly imputed synthetic data assuming a variety of probability models. The motivation relies on the fact that there are practical scenarios where only a single synthetic dataset <cite class="ltx_cite ltx_citemacro_citep">(Abowd et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib1" title="">2020</a>; Kinney et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib6" title="">2011</a>; Kinney, Reiter, & Miranda, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib7" title="">2014</a>; Bowen et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib3" title="">2020</a>; Alam et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib2" title="">2020</a>)</cite> is released due to constraints such as computational resources, data management policies, or privacy considerations and that the traditional asymptotic methods for combining inferences present in the literature are only suitable to draw inference when multiple synthetic versions of the original data are available. It is also crucial to note that all of the work done in the field of single imputation by those authors who were previously mentioned can be extended to multiple imputation. These inferential procedures are not asymptotic but exact, making them more robust even for small sample sizes and even when only a limited number of synthetic versions are available.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">Packages such as <span class="ltx_text ltx_font_italic" id="S1.p4.1.1">synthpop</span> <cite class="ltx_cite ltx_citemacro_citep">(Nowok, Raab, & Dibben, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib20" title="">2016</a>)</cite> and <span class="ltx_text ltx_font_italic" id="S1.p4.1.2">mice</span> <cite class="ltx_cite ltx_citemacro_citep">(van Buuren & Groothuis-Oudshoorn, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib26" title="">2011</a>)</cite> were developed to generate synthetic data, but they do not include functions specifically designed to directly perform statistical inference on synthetic datasets. Instead, users must employ standard statistical methods after synthesis, adhering to combination rules like those proposed by <cite class="ltx_cite ltx_citemacro_citet">Reiter (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib23" title="">2003</a>)</cite>. In contrast, the existing software tools that support inference, such as <span class="ltx_text ltx_font_italic" id="S1.p4.1.3">mitools</span> <cite class="ltx_cite ltx_citemacro_citep">(Lumley, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib15" title="">2020</a>)</cite>, are primarily intended for application in multiple imputation settings and are not meant to handle the unique challenges posed by synthetic data generated for disclosure control purposes.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">Building on this concept and looking to the work done in <cite class="ltx_cite ltx_citemacro_citet">Klein, Moura, & Sinha (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib8" title="">2021</a>)</cite>, our package, <a class="ltx_ref ltx_href" href="https://cran.r-project.org/web/packages/PSinference/index.html" title="">PSInference</a>, provides a suite of tools for performing inference on singly imputed synthetic data generated via Plug-in sampling (PS) when the original data is assumed to come from a multivariate normal model. The core methodologies implemented in <a class="ltx_ref ltx_href" href="https://cran.r-project.org/web/packages/PSinference/index.html" title="">PSInference</a> are grounded in the theoretical framework developed by <cite class="ltx_cite ltx_citemacro_citet">Klein, Moura, & Sinha (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib8" title="">2021</a>)</cite>. The package includes functions for testing various characteristics of the underlying population’s covariance structure, such as the generalized variance, the sphericity, the independence between two subsets of variables, and the regression of one set of variables on another.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.1">It also includes a function that enables a user to build a singly PS-based synthetic version of the original data, as long as the original data comes from a multivariate normal model. Even if not created for that purpose, this function can also be used to create multiple synthetic versions.</p> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.1">The package presented in this work aims to be part of a series of tools designed to support users working with synthetic datasets. We aim to establish a comprehensive suite of functions that will expand the available analysis options. We plan to expand beyond this first effort in future packages by integrating additional functions that make it possible for flexible and robust statistical analysis of synthetic data across various applications.</p> </div> <div class="ltx_para" id="S1.p8"> <p class="ltx_p" id="S1.p8.1">The paper is structured as follows: Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2" title="2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">2</span></a> describes the notation and outlines the key mathematical concepts and assumptions that serve as the foundation for this package. A detailed explanation and implementation of <a class="ltx_ref ltx_href" href="https://cran.r-project.org/web/packages/PSinference/index.html" title="">PSInference</a> are provided in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S3" title="3 Description and Implementation of PSInference ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">3</span></a>, which also provides an overview of the characteristics of the package. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4" title="4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">4</span></a> illustrates the practical applications of the package and provides code to assist users with implementation. Finally, Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S5" title="5 Summary ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">5</span></a> summarizes the key findings and contributions of the work, suggesting potential directions for future research.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Notation and Models</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">Before looking into the models that form the basis of our package, it is important to first introduce the concepts of fully synthetic data and partially synthetic data.</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>Fully synthetic data</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.1">Fully synthetic data refers to datasets where all original data instances are replaced with synthetic values generated from a statistical parametric model or from a non-parametric model. This approach aims to maximize privacy by ensuring that no original data records are directly disclosed while still preserving the statistical properties needed for valid analysis <cite class="ltx_cite ltx_citemacro_citep">(Moura, Coelho, & Sinha, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib19" title="">2024</a>)</cite>.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.2 </span>Partially synthetic data</h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.1">Partially synthetic data, by contrast, involves replacing only a subset of the original data, usually the sensitive values, with synthetic versions generated from a model (parametric or not). One can employ several strategies for this substitution: one can replace all deemed sensitive values in a respondent’s record, replace all values for a selected subset of (sensitive) variables, or only replace specific values that could potentially reveal an individual’s identity or sensitive information. This method enhances data utility while still preserving sensitive information, as non-sensitive values remain unchanged. However, it is clear that this improvement on the data quality can potentially compromise the ability to protect confidentiality <cite class="ltx_cite ltx_citemacro_citep">(Moura, Coelho, & Sinha, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib19" title="">2024</a>)</cite>.</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>Generating fully synthetic data</h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.1">The theoretical foundation of this package is based on the assumption that the original dataset comes from a multivariate normal distribution. 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11}&\dots&x_{1n}\\ \vdots&\ddots&\vdots\\ x_{p1}&\dots&x_{pn}\end{bmatrix}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m1.3d">bold_X = ( bold_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , bold_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = [ start_ARG start_ROW start_CELL italic_x start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL start_CELL … end_CELL start_CELL italic_x start_POSTSUBSCRIPT 1 italic_n end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL ⋱ end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL italic_x start_POSTSUBSCRIPT italic_p 1 end_POSTSUBSCRIPT end_CELL start_CELL … end_CELL start_CELL italic_x start_POSTSUBSCRIPT italic_p italic_n end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p1.3">where, for all <math alttext="i=1,\ldots,n" class="ltx_Math" display="inline" 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xref="S2.SS3.p1.3.m2.3.3.2.4">top</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.3.m2.3c">\mathbf{x}_{i}=(x_{1i},\ldots,x_{pi})^{\top}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.3.m2.3d">bold_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ( italic_x start_POSTSUBSCRIPT 1 italic_i end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_p italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS3.p2"> <p class="ltx_p" id="S2.SS3.p2.1">We assume that <math alttext="\mathbf{X}" class="ltx_Math" display="inline" id="S2.SS3.p2.1.m1.1"><semantics id="S2.SS3.p2.1.m1.1a"><mi id="S2.SS3.p2.1.m1.1.1" xref="S2.SS3.p2.1.m1.1.1.cmml">𝐗</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.1.m1.1b"><ci id="S2.SS3.p2.1.m1.1.1.cmml" xref="S2.SS3.p2.1.m1.1.1">𝐗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.1.m1.1c">\mathbf{X}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.1.m1.1d">bold_X</annotation></semantics></math> will be normally distributed, <span class="ltx_text ltx_font_italic" id="S2.SS3.p2.1.1">i.e.</span>,</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="\mathbf{x}_{i}\sim\mathcal{N}_{p}(\boldsymbol{\mu},\boldsymbol{\Sigma}),i=1,% \dots,n," class="ltx_Math" display="block" id="S2.E1.m1.6"><semantics id="S2.E1.m1.6a"><mrow id="S2.E1.m1.6.6.1"><mrow id="S2.E1.m1.6.6.1.1.2" xref="S2.E1.m1.6.6.1.1.3.cmml"><mrow id="S2.E1.m1.6.6.1.1.1.1" xref="S2.E1.m1.6.6.1.1.1.1.cmml"><msub id="S2.E1.m1.6.6.1.1.1.1.2" xref="S2.E1.m1.6.6.1.1.1.1.2.cmml"><mi id="S2.E1.m1.6.6.1.1.1.1.2.2" xref="S2.E1.m1.6.6.1.1.1.1.2.2.cmml">𝐱</mi><mi id="S2.E1.m1.6.6.1.1.1.1.2.3" xref="S2.E1.m1.6.6.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S2.E1.m1.6.6.1.1.1.1.1" xref="S2.E1.m1.6.6.1.1.1.1.1.cmml">∼</mo><mrow id="S2.E1.m1.6.6.1.1.1.1.3" xref="S2.E1.m1.6.6.1.1.1.1.3.cmml"><msub id="S2.E1.m1.6.6.1.1.1.1.3.2" xref="S2.E1.m1.6.6.1.1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E1.m1.6.6.1.1.1.1.3.2.2" xref="S2.E1.m1.6.6.1.1.1.1.3.2.2.cmml">𝒩</mi><mi id="S2.E1.m1.6.6.1.1.1.1.3.2.3" xref="S2.E1.m1.6.6.1.1.1.1.3.2.3.cmml">p</mi></msub><mo id="S2.E1.m1.6.6.1.1.1.1.3.1" xref="S2.E1.m1.6.6.1.1.1.1.3.1.cmml"></mo><mrow id="S2.E1.m1.6.6.1.1.1.1.3.3.2" xref="S2.E1.m1.6.6.1.1.1.1.3.3.1.cmml"><mo id="S2.E1.m1.6.6.1.1.1.1.3.3.2.1" stretchy="false" xref="S2.E1.m1.6.6.1.1.1.1.3.3.1.cmml">(</mo><mi id="S2.E1.m1.1.1" xref="S2.E1.m1.1.1.cmml">𝝁</mi><mo id="S2.E1.m1.6.6.1.1.1.1.3.3.2.2" xref="S2.E1.m1.6.6.1.1.1.1.3.3.1.cmml">,</mo><mi id="S2.E1.m1.2.2" xref="S2.E1.m1.2.2.cmml">𝚺</mi><mo id="S2.E1.m1.6.6.1.1.1.1.3.3.2.3" stretchy="false" xref="S2.E1.m1.6.6.1.1.1.1.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E1.m1.6.6.1.1.2.3" xref="S2.E1.m1.6.6.1.1.3a.cmml">,</mo><mrow id="S2.E1.m1.6.6.1.1.2.2" xref="S2.E1.m1.6.6.1.1.2.2.cmml"><mi id="S2.E1.m1.6.6.1.1.2.2.2" xref="S2.E1.m1.6.6.1.1.2.2.2.cmml">i</mi><mo id="S2.E1.m1.6.6.1.1.2.2.1" xref="S2.E1.m1.6.6.1.1.2.2.1.cmml">=</mo><mrow id="S2.E1.m1.6.6.1.1.2.2.3.2" xref="S2.E1.m1.6.6.1.1.2.2.3.1.cmml"><mn id="S2.E1.m1.3.3" xref="S2.E1.m1.3.3.cmml">1</mn><mo id="S2.E1.m1.6.6.1.1.2.2.3.2.1" xref="S2.E1.m1.6.6.1.1.2.2.3.1.cmml">,</mo><mi id="S2.E1.m1.4.4" mathvariant="normal" xref="S2.E1.m1.4.4.cmml">…</mi><mo id="S2.E1.m1.6.6.1.1.2.2.3.2.2" xref="S2.E1.m1.6.6.1.1.2.2.3.1.cmml">,</mo><mi id="S2.E1.m1.5.5" xref="S2.E1.m1.5.5.cmml">n</mi></mrow></mrow></mrow><mo id="S2.E1.m1.6.6.1.2">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E1.m1.6b"><apply id="S2.E1.m1.6.6.1.1.3.cmml" xref="S2.E1.m1.6.6.1.1.2"><csymbol cd="ambiguous" id="S2.E1.m1.6.6.1.1.3a.cmml" 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encoding="application/x-llamapun" id="S2.E1.m1.6d">bold_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ caligraphic_N start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( bold_italic_μ , bold_Σ ) , italic_i = 1 , … , italic_n ,</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.SS3.p2.3">where <math alttext="\boldsymbol{\mu}" class="ltx_Math" display="inline" id="S2.SS3.p2.2.m1.1"><semantics id="S2.SS3.p2.2.m1.1a"><mi id="S2.SS3.p2.2.m1.1.1" xref="S2.SS3.p2.2.m1.1.1.cmml">𝝁</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.2.m1.1b"><ci id="S2.SS3.p2.2.m1.1.1.cmml" xref="S2.SS3.p2.2.m1.1.1">𝝁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.2.m1.1c">\boldsymbol{\mu}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.2.m1.1d">bold_italic_μ</annotation></semantics></math> is the mean vector, and <math alttext="\boldsymbol{\Sigma}" class="ltx_Math" display="inline" id="S2.SS3.p2.3.m2.1"><semantics id="S2.SS3.p2.3.m2.1a"><mi id="S2.SS3.p2.3.m2.1.1" xref="S2.SS3.p2.3.m2.1.1.cmml">𝚺</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.3.m2.1b"><ci id="S2.SS3.p2.3.m2.1.1.cmml" xref="S2.SS3.p2.3.m2.1.1">𝚺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.3.m2.1c">\boldsymbol{\Sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.3.m2.1d">bold_Σ</annotation></semantics></math> denotes the covariance matrix.</p> </div> <div class="ltx_para" id="S2.SS3.p3"> <p class="ltx_p" id="S2.SS3.p3.5">To generate a fully synthetic version of the original data, we will need the sample mean <math alttext="\bar{\mathbf{x}}=\frac{1}{n}\sum_{i=1}^{n}\mathbf{x}_{i}" class="ltx_Math" display="inline" 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xref="S2.SS3.p3.1.m1.1.1.3.3.2.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.1.m1.1c">\bar{\mathbf{x}}=\frac{1}{n}\sum_{i=1}^{n}\mathbf{x}_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.1.m1.1d">over¯ start_ARG bold_x end_ARG = divide start_ARG 1 end_ARG start_ARG italic_n end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT bold_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and the sample covariance matrix <math alttext="\hat{\boldsymbol{\Sigma}}=\frac{1}{n-1}{\mathbf{S}}" class="ltx_Math" display="inline" id="S2.SS3.p3.2.m2.1"><semantics id="S2.SS3.p3.2.m2.1a"><mrow id="S2.SS3.p3.2.m2.1.1" xref="S2.SS3.p3.2.m2.1.1.cmml"><mover accent="true" id="S2.SS3.p3.2.m2.1.1.2" xref="S2.SS3.p3.2.m2.1.1.2.cmml"><mi id="S2.SS3.p3.2.m2.1.1.2.2" xref="S2.SS3.p3.2.m2.1.1.2.2.cmml">𝚺</mi><mo id="S2.SS3.p3.2.m2.1.1.2.1" xref="S2.SS3.p3.2.m2.1.1.2.1.cmml">^</mo></mover><mo id="S2.SS3.p3.2.m2.1.1.1" xref="S2.SS3.p3.2.m2.1.1.1.cmml">=</mo><mrow id="S2.SS3.p3.2.m2.1.1.3" xref="S2.SS3.p3.2.m2.1.1.3.cmml"><mfrac id="S2.SS3.p3.2.m2.1.1.3.2" xref="S2.SS3.p3.2.m2.1.1.3.2.cmml"><mn id="S2.SS3.p3.2.m2.1.1.3.2.2" xref="S2.SS3.p3.2.m2.1.1.3.2.2.cmml">1</mn><mrow id="S2.SS3.p3.2.m2.1.1.3.2.3" xref="S2.SS3.p3.2.m2.1.1.3.2.3.cmml"><mi id="S2.SS3.p3.2.m2.1.1.3.2.3.2" xref="S2.SS3.p3.2.m2.1.1.3.2.3.2.cmml">n</mi><mo id="S2.SS3.p3.2.m2.1.1.3.2.3.1" xref="S2.SS3.p3.2.m2.1.1.3.2.3.1.cmml">−</mo><mn id="S2.SS3.p3.2.m2.1.1.3.2.3.3" xref="S2.SS3.p3.2.m2.1.1.3.2.3.3.cmml">1</mn></mrow></mfrac><mo id="S2.SS3.p3.2.m2.1.1.3.1" xref="S2.SS3.p3.2.m2.1.1.3.1.cmml"></mo><mi id="S2.SS3.p3.2.m2.1.1.3.3" xref="S2.SS3.p3.2.m2.1.1.3.3.cmml">𝐒</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.2.m2.1b"><apply id="S2.SS3.p3.2.m2.1.1.cmml" xref="S2.SS3.p3.2.m2.1.1"><eq id="S2.SS3.p3.2.m2.1.1.1.cmml" xref="S2.SS3.p3.2.m2.1.1.1"></eq><apply id="S2.SS3.p3.2.m2.1.1.2.cmml" xref="S2.SS3.p3.2.m2.1.1.2"><ci id="S2.SS3.p3.2.m2.1.1.2.1.cmml" xref="S2.SS3.p3.2.m2.1.1.2.1">^</ci><ci id="S2.SS3.p3.2.m2.1.1.2.2.cmml" xref="S2.SS3.p3.2.m2.1.1.2.2">𝚺</ci></apply><apply id="S2.SS3.p3.2.m2.1.1.3.cmml" xref="S2.SS3.p3.2.m2.1.1.3"><times id="S2.SS3.p3.2.m2.1.1.3.1.cmml" xref="S2.SS3.p3.2.m2.1.1.3.1"></times><apply id="S2.SS3.p3.2.m2.1.1.3.2.cmml" xref="S2.SS3.p3.2.m2.1.1.3.2"><divide id="S2.SS3.p3.2.m2.1.1.3.2.1.cmml" xref="S2.SS3.p3.2.m2.1.1.3.2"></divide><cn id="S2.SS3.p3.2.m2.1.1.3.2.2.cmml" type="integer" xref="S2.SS3.p3.2.m2.1.1.3.2.2">1</cn><apply id="S2.SS3.p3.2.m2.1.1.3.2.3.cmml" xref="S2.SS3.p3.2.m2.1.1.3.2.3"><minus id="S2.SS3.p3.2.m2.1.1.3.2.3.1.cmml" xref="S2.SS3.p3.2.m2.1.1.3.2.3.1"></minus><ci id="S2.SS3.p3.2.m2.1.1.3.2.3.2.cmml" xref="S2.SS3.p3.2.m2.1.1.3.2.3.2">𝑛</ci><cn id="S2.SS3.p3.2.m2.1.1.3.2.3.3.cmml" type="integer" xref="S2.SS3.p3.2.m2.1.1.3.2.3.3">1</cn></apply></apply><ci id="S2.SS3.p3.2.m2.1.1.3.3.cmml" xref="S2.SS3.p3.2.m2.1.1.3.3">𝐒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.2.m2.1c">\hat{\boldsymbol{\Sigma}}=\frac{1}{n-1}{\mathbf{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.2.m2.1d">over^ start_ARG bold_Σ end_ARG = divide start_ARG 1 end_ARG start_ARG italic_n - 1 end_ARG bold_S</annotation></semantics></math>, where <math alttext="\mathbf{S}=\sum_{i=1}^{n}(\mathbf{x}_{i}-\bar{\mathbf{x}})(\mathbf{x}_{i}-\bar% {\mathbf{x}})^{\top}" class="ltx_Math" display="inline" id="S2.SS3.p3.3.m3.2"><semantics id="S2.SS3.p3.3.m3.2a"><mrow id="S2.SS3.p3.3.m3.2.2" xref="S2.SS3.p3.3.m3.2.2.cmml"><mi id="S2.SS3.p3.3.m3.2.2.4" xref="S2.SS3.p3.3.m3.2.2.4.cmml">𝐒</mi><mo id="S2.SS3.p3.3.m3.2.2.3" rspace="0.111em" xref="S2.SS3.p3.3.m3.2.2.3.cmml">=</mo><mrow id="S2.SS3.p3.3.m3.2.2.2" xref="S2.SS3.p3.3.m3.2.2.2.cmml"><msubsup 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end_POSTSUPERSCRIPT ( bold_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - over¯ start_ARG bold_x end_ARG ) ( bold_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - over¯ start_ARG bold_x end_ARG ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math> is the sample Wishart matrix, where it is distributed as a Wishart distribution with <math alttext="(n-1)" class="ltx_Math" display="inline" id="S2.SS3.p3.4.m4.1"><semantics id="S2.SS3.p3.4.m4.1a"><mrow id="S2.SS3.p3.4.m4.1.1.1" xref="S2.SS3.p3.4.m4.1.1.1.1.cmml"><mo id="S2.SS3.p3.4.m4.1.1.1.2" stretchy="false" xref="S2.SS3.p3.4.m4.1.1.1.1.cmml">(</mo><mrow id="S2.SS3.p3.4.m4.1.1.1.1" xref="S2.SS3.p3.4.m4.1.1.1.1.cmml"><mi id="S2.SS3.p3.4.m4.1.1.1.1.2" xref="S2.SS3.p3.4.m4.1.1.1.1.2.cmml">n</mi><mo id="S2.SS3.p3.4.m4.1.1.1.1.1" xref="S2.SS3.p3.4.m4.1.1.1.1.1.cmml">−</mo><mn id="S2.SS3.p3.4.m4.1.1.1.1.3" xref="S2.SS3.p3.4.m4.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS3.p3.4.m4.1.1.1.3" stretchy="false" xref="S2.SS3.p3.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.4.m4.1b"><apply id="S2.SS3.p3.4.m4.1.1.1.1.cmml" xref="S2.SS3.p3.4.m4.1.1.1"><minus id="S2.SS3.p3.4.m4.1.1.1.1.1.cmml" xref="S2.SS3.p3.4.m4.1.1.1.1.1"></minus><ci id="S2.SS3.p3.4.m4.1.1.1.1.2.cmml" xref="S2.SS3.p3.4.m4.1.1.1.1.2">𝑛</ci><cn id="S2.SS3.p3.4.m4.1.1.1.1.3.cmml" type="integer" xref="S2.SS3.p3.4.m4.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.4.m4.1c">(n-1)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.4.m4.1d">( italic_n - 1 )</annotation></semantics></math> degrees of freedom and covariance matrix <math alttext="\boldsymbol{\Sigma}" class="ltx_Math" display="inline" id="S2.SS3.p3.5.m5.1"><semantics id="S2.SS3.p3.5.m5.1a"><mi id="S2.SS3.p3.5.m5.1.1" xref="S2.SS3.p3.5.m5.1.1.cmml">𝚺</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p3.5.m5.1b"><ci id="S2.SS3.p3.5.m5.1.1.cmml" xref="S2.SS3.p3.5.m5.1.1">𝚺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p3.5.m5.1c">\boldsymbol{\Sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p3.5.m5.1d">bold_Σ</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS3.p4"> <p class="ltx_p" id="S2.SS3.p4.3">The underlined idea of the PS Sampling method of generating data is that we plug-in the sample estimates in the original assumed model and randomly generate a new dataset. Specifically, we generate</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex2"> <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="\mathbf{V}=\left(\mathbf{v}_{1},\dots,\mathbf{v}_{n}\right)=\begin{bmatrix}v_{% 11}&\dots&v_{1n}\\ \vdots&\ddots&\vdots\\ v_{p1}&\dots&v_{pn}\end{bmatrix}," class="ltx_Math" display="block" id="S2.Ex2.m1.3"><semantics id="S2.Ex2.m1.3a"><mrow id="S2.Ex2.m1.3.3.1" xref="S2.Ex2.m1.3.3.1.1.cmml"><mrow id="S2.Ex2.m1.3.3.1.1" xref="S2.Ex2.m1.3.3.1.1.cmml"><mi id="S2.Ex2.m1.3.3.1.1.4" xref="S2.Ex2.m1.3.3.1.1.4.cmml">𝐕</mi><mo id="S2.Ex2.m1.3.3.1.1.5" xref="S2.Ex2.m1.3.3.1.1.5.cmml">=</mo><mrow id="S2.Ex2.m1.3.3.1.1.2.2" xref="S2.Ex2.m1.3.3.1.1.2.3.cmml"><mo id="S2.Ex2.m1.3.3.1.1.2.2.3" xref="S2.Ex2.m1.3.3.1.1.2.3.cmml">(</mo><msub id="S2.Ex2.m1.3.3.1.1.1.1.1" xref="S2.Ex2.m1.3.3.1.1.1.1.1.cmml"><mi id="S2.Ex2.m1.3.3.1.1.1.1.1.2" 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encoding="application/x-llamapun" id="S2.Ex2.m1.3d">bold_V = ( bold_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , bold_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = [ start_ARG start_ROW start_CELL italic_v start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL start_CELL … end_CELL start_CELL italic_v start_POSTSUBSCRIPT 1 italic_n end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL ⋱ end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL italic_v start_POSTSUBSCRIPT italic_p 1 end_POSTSUBSCRIPT end_CELL start_CELL … end_CELL start_CELL italic_v start_POSTSUBSCRIPT italic_p italic_n end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p4.2">where, for all <math alttext="i=1,\ldots,n" class="ltx_Math" display="inline" id="S2.SS3.p4.1.m1.3"><semantics id="S2.SS3.p4.1.m1.3a"><mrow id="S2.SS3.p4.1.m1.3.4" 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id="S2.SS3.p5.1.m1.1d">italic_M</annotation></semantics></math> times, where <math alttext="M" class="ltx_Math" display="inline" id="S2.SS3.p5.2.m2.1"><semantics id="S2.SS3.p5.2.m2.1a"><mi id="S2.SS3.p5.2.m2.1.1" xref="S2.SS3.p5.2.m2.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p5.2.m2.1b"><ci id="S2.SS3.p5.2.m2.1.1.cmml" xref="S2.SS3.p5.2.m2.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p5.2.m2.1c">M</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p5.2.m2.1d">italic_M</annotation></semantics></math> is the number of synthetic versions required.</p> </div> <div class="ltx_para" id="S2.SS3.p6"> <p class="ltx_p" id="S2.SS3.p6.2">To be perfectly clear, the original data <math alttext="\mathbf{X}" class="ltx_Math" display="inline" id="S2.SS3.p6.1.m1.1"><semantics id="S2.SS3.p6.1.m1.1a"><mi id="S2.SS3.p6.1.m1.1.1" xref="S2.SS3.p6.1.m1.1.1.cmml">𝐗</mi><annotation-xml encoding="MathML-Content" 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id="S2.SS3.p7.2.m2.1d">over¯ start_ARG bold_v end_ARG = divide start_ARG 1 end_ARG start_ARG italic_n end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT bold_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is the maximum likelihood (also unbiased) estimator of <math alttext="\boldsymbol{\mu}" class="ltx_Math" display="inline" id="S2.SS3.p7.3.m3.1"><semantics id="S2.SS3.p7.3.m3.1a"><mi id="S2.SS3.p7.3.m3.1.1" xref="S2.SS3.p7.3.m3.1.1.cmml">𝝁</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.3.m3.1b"><ci id="S2.SS3.p7.3.m3.1.1.cmml" xref="S2.SS3.p7.3.m3.1.1">𝝁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.3.m3.1c">\boldsymbol{\mu}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.3.m3.1d">bold_italic_μ</annotation></semantics></math>, that</p> <table class="ltx_equation ltx_eqn_table" id="S2.E3"> <tbody><tr class="ltx_equation 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id="S2.E3.m1.2.2.2.2.2.2.1.1.1.3.2.cmml" xref="S2.E3.m1.2.2.2.2.2.2.1.1.1.3.2">𝐯</ci></apply></apply><csymbol cd="latexml" id="S2.E3.m1.2.2.2.2.2.2.3.cmml" xref="S2.E3.m1.2.2.2.2.2.2.3">top</csymbol></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.2c">\boldsymbol{\Sigma}^{\star}=\frac{1}{n-1}\mathbf{S}^{\star}=\frac{1}{n-1}\sum_% {i=1}^{n}(\mathbf{v}_{i}-\bar{\mathbf{v}})(\mathbf{v}_{i}-\bar{\mathbf{v}})^{\top}</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.2d">bold_Σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_n - 1 end_ARG bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_n - 1 end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( bold_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - over¯ start_ARG bold_v end_ARG ) ( bold_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - over¯ start_ARG bold_v end_ARG ) start_POSTSUPERSCRIPT ⊤ 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">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p7.6">is an unbiased estimator of <math alttext="\boldsymbol{\Sigma}" class="ltx_Math" display="inline" id="S2.SS3.p7.4.m1.1"><semantics id="S2.SS3.p7.4.m1.1a"><mi id="S2.SS3.p7.4.m1.1.1" xref="S2.SS3.p7.4.m1.1.1.cmml">𝚺</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.4.m1.1b"><ci id="S2.SS3.p7.4.m1.1.1.cmml" xref="S2.SS3.p7.4.m1.1.1">𝚺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.4.m1.1c">\boldsymbol{\Sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.4.m1.1d">bold_Σ</annotation></semantics></math>, and that the distribution of <math alttext="\mathbf{S}^{\star}" class="ltx_Math" display="inline" id="S2.SS3.p7.5.m2.1"><semantics id="S2.SS3.p7.5.m2.1a"><msup id="S2.SS3.p7.5.m2.1.1" xref="S2.SS3.p7.5.m2.1.1.cmml"><mi id="S2.SS3.p7.5.m2.1.1.2" xref="S2.SS3.p7.5.m2.1.1.2.cmml">𝐒</mi><mo id="S2.SS3.p7.5.m2.1.1.3" xref="S2.SS3.p7.5.m2.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.5.m2.1b"><apply id="S2.SS3.p7.5.m2.1.1.cmml" xref="S2.SS3.p7.5.m2.1.1"><csymbol cd="ambiguous" id="S2.SS3.p7.5.m2.1.1.1.cmml" xref="S2.SS3.p7.5.m2.1.1">superscript</csymbol><ci id="S2.SS3.p7.5.m2.1.1.2.cmml" xref="S2.SS3.p7.5.m2.1.1.2">𝐒</ci><ci id="S2.SS3.p7.5.m2.1.1.3.cmml" xref="S2.SS3.p7.5.m2.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.5.m2.1c">\mathbf{S}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.5.m2.1d">bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>, conditionally on <math alttext="\mathbf{S}" class="ltx_Math" display="inline" id="S2.SS3.p7.6.m3.1"><semantics id="S2.SS3.p7.6.m3.1a"><mi id="S2.SS3.p7.6.m3.1.1" xref="S2.SS3.p7.6.m3.1.1.cmml">𝐒</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.6.m3.1b"><ci id="S2.SS3.p7.6.m3.1.1.cmml" xref="S2.SS3.p7.6.m3.1.1">𝐒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.6.m3.1c">\mathbf{S}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.6.m3.1d">bold_S</annotation></semantics></math> is <cite class="ltx_cite ltx_citemacro_citep">(Klein, Moura, & Sinha, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib8" title="">2021</a>)</cite></p> <table class="ltx_equation ltx_eqn_table" id="S2.E4"> <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="\mathbf{S}^{\star}\sim\mathcal{W}_{p}\left(n-1,\frac{1}{n-1}\mathbf{S}\right)," class="ltx_Math" display="block" 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"><msup id="S2.E4.m1.1.1.1.1.4" xref="S2.E4.m1.1.1.1.1.4.cmml"><mi id="S2.E4.m1.1.1.1.1.4.2" xref="S2.E4.m1.1.1.1.1.4.2.cmml">𝐒</mi><mo id="S2.E4.m1.1.1.1.1.4.3" xref="S2.E4.m1.1.1.1.1.4.3.cmml">⋆</mo></msup><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" xref="S2.E4.m1.1.1.1.1.2.cmml"><msub id="S2.E4.m1.1.1.1.1.2.4" xref="S2.E4.m1.1.1.1.1.2.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E4.m1.1.1.1.1.2.4.2" xref="S2.E4.m1.1.1.1.1.2.4.2.cmml">𝒲</mi><mi id="S2.E4.m1.1.1.1.1.2.4.3" xref="S2.E4.m1.1.1.1.1.2.4.3.cmml">p</mi></msub><mo id="S2.E4.m1.1.1.1.1.2.3" xref="S2.E4.m1.1.1.1.1.2.3.cmml"></mo><mrow id="S2.E4.m1.1.1.1.1.2.2.2" xref="S2.E4.m1.1.1.1.1.2.2.3.cmml"><mo id="S2.E4.m1.1.1.1.1.2.2.2.3" xref="S2.E4.m1.1.1.1.1.2.2.3.cmml">(</mo><mrow id="S2.E4.m1.1.1.1.1.1.1.1.1" xref="S2.E4.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.E4.m1.1.1.1.1.1.1.1.1.2" xref="S2.E4.m1.1.1.1.1.1.1.1.1.2.cmml">n</mi><mo id="S2.E4.m1.1.1.1.1.1.1.1.1.1" xref="S2.E4.m1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S2.E4.m1.1.1.1.1.1.1.1.1.3" xref="S2.E4.m1.1.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.E4.m1.1.1.1.1.2.2.2.4" xref="S2.E4.m1.1.1.1.1.2.2.3.cmml">,</mo><mrow id="S2.E4.m1.1.1.1.1.2.2.2.2" xref="S2.E4.m1.1.1.1.1.2.2.2.2.cmml"><mfrac id="S2.E4.m1.1.1.1.1.2.2.2.2.2" xref="S2.E4.m1.1.1.1.1.2.2.2.2.2.cmml"><mn id="S2.E4.m1.1.1.1.1.2.2.2.2.2.2" xref="S2.E4.m1.1.1.1.1.2.2.2.2.2.2.cmml">1</mn><mrow id="S2.E4.m1.1.1.1.1.2.2.2.2.2.3" xref="S2.E4.m1.1.1.1.1.2.2.2.2.2.3.cmml"><mi id="S2.E4.m1.1.1.1.1.2.2.2.2.2.3.2" xref="S2.E4.m1.1.1.1.1.2.2.2.2.2.3.2.cmml">n</mi><mo id="S2.E4.m1.1.1.1.1.2.2.2.2.2.3.1" xref="S2.E4.m1.1.1.1.1.2.2.2.2.2.3.1.cmml">−</mo><mn id="S2.E4.m1.1.1.1.1.2.2.2.2.2.3.3" xref="S2.E4.m1.1.1.1.1.2.2.2.2.2.3.3.cmml">1</mn></mrow></mfrac><mo id="S2.E4.m1.1.1.1.1.2.2.2.2.1" xref="S2.E4.m1.1.1.1.1.2.2.2.2.1.cmml"></mo><mi id="S2.E4.m1.1.1.1.1.2.2.2.2.3" xref="S2.E4.m1.1.1.1.1.2.2.2.2.3.cmml">𝐒</mi></mrow><mo id="S2.E4.m1.1.1.1.1.2.2.2.5" xref="S2.E4.m1.1.1.1.1.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E4.m1.1.1.1.2" xref="S2.E4.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E4.m1.1b"><apply id="S2.E4.m1.1.1.1.1.cmml" xref="S2.E4.m1.1.1.1"><csymbol cd="latexml" id="S2.E4.m1.1.1.1.1.3.cmml" xref="S2.E4.m1.1.1.1.1.3">similar-to</csymbol><apply id="S2.E4.m1.1.1.1.1.4.cmml" xref="S2.E4.m1.1.1.1.1.4"><csymbol cd="ambiguous" id="S2.E4.m1.1.1.1.1.4.1.cmml" xref="S2.E4.m1.1.1.1.1.4">superscript</csymbol><ci id="S2.E4.m1.1.1.1.1.4.2.cmml" xref="S2.E4.m1.1.1.1.1.4.2">𝐒</ci><ci id="S2.E4.m1.1.1.1.1.4.3.cmml" xref="S2.E4.m1.1.1.1.1.4.3">⋆</ci></apply><apply id="S2.E4.m1.1.1.1.1.2.cmml" xref="S2.E4.m1.1.1.1.1.2"><times id="S2.E4.m1.1.1.1.1.2.3.cmml" xref="S2.E4.m1.1.1.1.1.2.3"></times><apply id="S2.E4.m1.1.1.1.1.2.4.cmml" xref="S2.E4.m1.1.1.1.1.2.4"><csymbol cd="ambiguous" id="S2.E4.m1.1.1.1.1.2.4.1.cmml" xref="S2.E4.m1.1.1.1.1.2.4">subscript</csymbol><ci id="S2.E4.m1.1.1.1.1.2.4.2.cmml" xref="S2.E4.m1.1.1.1.1.2.4.2">𝒲</ci><ci id="S2.E4.m1.1.1.1.1.2.4.3.cmml" xref="S2.E4.m1.1.1.1.1.2.4.3">𝑝</ci></apply><interval closure="open" id="S2.E4.m1.1.1.1.1.2.2.3.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2"><apply id="S2.E4.m1.1.1.1.1.1.1.1.1.cmml" xref="S2.E4.m1.1.1.1.1.1.1.1.1"><minus id="S2.E4.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E4.m1.1.1.1.1.1.1.1.1.1"></minus><ci id="S2.E4.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.E4.m1.1.1.1.1.1.1.1.1.2">𝑛</ci><cn id="S2.E4.m1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.E4.m1.1.1.1.1.1.1.1.1.3">1</cn></apply><apply id="S2.E4.m1.1.1.1.1.2.2.2.2.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2.2"><times id="S2.E4.m1.1.1.1.1.2.2.2.2.1.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2.2.1"></times><apply id="S2.E4.m1.1.1.1.1.2.2.2.2.2.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2.2.2"><divide id="S2.E4.m1.1.1.1.1.2.2.2.2.2.1.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2.2.2"></divide><cn id="S2.E4.m1.1.1.1.1.2.2.2.2.2.2.cmml" type="integer" xref="S2.E4.m1.1.1.1.1.2.2.2.2.2.2">1</cn><apply id="S2.E4.m1.1.1.1.1.2.2.2.2.2.3.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2.2.2.3"><minus id="S2.E4.m1.1.1.1.1.2.2.2.2.2.3.1.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2.2.2.3.1"></minus><ci id="S2.E4.m1.1.1.1.1.2.2.2.2.2.3.2.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2.2.2.3.2">𝑛</ci><cn id="S2.E4.m1.1.1.1.1.2.2.2.2.2.3.3.cmml" type="integer" xref="S2.E4.m1.1.1.1.1.2.2.2.2.2.3.3">1</cn></apply></apply><ci id="S2.E4.m1.1.1.1.1.2.2.2.2.3.cmml" xref="S2.E4.m1.1.1.1.1.2.2.2.2.3">𝐒</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.m1.1c">\mathbf{S}^{\star}\sim\mathcal{W}_{p}\left(n-1,\frac{1}{n-1}\mathbf{S}\right),</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.1d">bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ∼ caligraphic_W start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_n - 1 , divide start_ARG 1 end_ARG start_ARG italic_n - 1 end_ARG bold_S ) ,</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.SS3.p7.10">where <math alttext="\mathcal{W}_{p}\left(\nu,\mathbf{A}\right)" class="ltx_Math" display="inline" id="S2.SS3.p7.7.m1.2"><semantics id="S2.SS3.p7.7.m1.2a"><mrow id="S2.SS3.p7.7.m1.2.3" xref="S2.SS3.p7.7.m1.2.3.cmml"><msub id="S2.SS3.p7.7.m1.2.3.2" xref="S2.SS3.p7.7.m1.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p7.7.m1.2.3.2.2" xref="S2.SS3.p7.7.m1.2.3.2.2.cmml">𝒲</mi><mi id="S2.SS3.p7.7.m1.2.3.2.3" xref="S2.SS3.p7.7.m1.2.3.2.3.cmml">p</mi></msub><mo id="S2.SS3.p7.7.m1.2.3.1" xref="S2.SS3.p7.7.m1.2.3.1.cmml"></mo><mrow id="S2.SS3.p7.7.m1.2.3.3.2" xref="S2.SS3.p7.7.m1.2.3.3.1.cmml"><mo id="S2.SS3.p7.7.m1.2.3.3.2.1" xref="S2.SS3.p7.7.m1.2.3.3.1.cmml">(</mo><mi id="S2.SS3.p7.7.m1.1.1" xref="S2.SS3.p7.7.m1.1.1.cmml">ν</mi><mo id="S2.SS3.p7.7.m1.2.3.3.2.2" xref="S2.SS3.p7.7.m1.2.3.3.1.cmml">,</mo><mi id="S2.SS3.p7.7.m1.2.2" xref="S2.SS3.p7.7.m1.2.2.cmml">𝐀</mi><mo id="S2.SS3.p7.7.m1.2.3.3.2.3" xref="S2.SS3.p7.7.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.7.m1.2b"><apply id="S2.SS3.p7.7.m1.2.3.cmml" xref="S2.SS3.p7.7.m1.2.3"><times id="S2.SS3.p7.7.m1.2.3.1.cmml" xref="S2.SS3.p7.7.m1.2.3.1"></times><apply id="S2.SS3.p7.7.m1.2.3.2.cmml" xref="S2.SS3.p7.7.m1.2.3.2"><csymbol cd="ambiguous" id="S2.SS3.p7.7.m1.2.3.2.1.cmml" xref="S2.SS3.p7.7.m1.2.3.2">subscript</csymbol><ci id="S2.SS3.p7.7.m1.2.3.2.2.cmml" xref="S2.SS3.p7.7.m1.2.3.2.2">𝒲</ci><ci id="S2.SS3.p7.7.m1.2.3.2.3.cmml" xref="S2.SS3.p7.7.m1.2.3.2.3">𝑝</ci></apply><interval closure="open" id="S2.SS3.p7.7.m1.2.3.3.1.cmml" xref="S2.SS3.p7.7.m1.2.3.3.2"><ci id="S2.SS3.p7.7.m1.1.1.cmml" xref="S2.SS3.p7.7.m1.1.1">𝜈</ci><ci id="S2.SS3.p7.7.m1.2.2.cmml" xref="S2.SS3.p7.7.m1.2.2">𝐀</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.7.m1.2c">\mathcal{W}_{p}\left(\nu,\mathbf{A}\right)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.7.m1.2d">caligraphic_W start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_ν , bold_A )</annotation></semantics></math> is the Wishart distribution with <math alttext="\nu" class="ltx_Math" display="inline" id="S2.SS3.p7.8.m2.1"><semantics id="S2.SS3.p7.8.m2.1a"><mi id="S2.SS3.p7.8.m2.1.1" xref="S2.SS3.p7.8.m2.1.1.cmml">ν</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.8.m2.1b"><ci id="S2.SS3.p7.8.m2.1.1.cmml" xref="S2.SS3.p7.8.m2.1.1">𝜈</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.8.m2.1c">\nu</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.8.m2.1d">italic_ν</annotation></semantics></math> degrees of freedom and a <math alttext="p\times p" class="ltx_Math" display="inline" id="S2.SS3.p7.9.m3.1"><semantics id="S2.SS3.p7.9.m3.1a"><mrow id="S2.SS3.p7.9.m3.1.1" xref="S2.SS3.p7.9.m3.1.1.cmml"><mi id="S2.SS3.p7.9.m3.1.1.2" xref="S2.SS3.p7.9.m3.1.1.2.cmml">p</mi><mo id="S2.SS3.p7.9.m3.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS3.p7.9.m3.1.1.1.cmml">×</mo><mi id="S2.SS3.p7.9.m3.1.1.3" xref="S2.SS3.p7.9.m3.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.9.m3.1b"><apply id="S2.SS3.p7.9.m3.1.1.cmml" xref="S2.SS3.p7.9.m3.1.1"><times id="S2.SS3.p7.9.m3.1.1.1.cmml" xref="S2.SS3.p7.9.m3.1.1.1"></times><ci id="S2.SS3.p7.9.m3.1.1.2.cmml" xref="S2.SS3.p7.9.m3.1.1.2">𝑝</ci><ci id="S2.SS3.p7.9.m3.1.1.3.cmml" xref="S2.SS3.p7.9.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.9.m3.1c">p\times p</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.9.m3.1d">italic_p × italic_p</annotation></semantics></math> scale matrix <math alttext="\mathbf{A}" class="ltx_Math" display="inline" id="S2.SS3.p7.10.m4.1"><semantics id="S2.SS3.p7.10.m4.1a"><mi id="S2.SS3.p7.10.m4.1.1" xref="S2.SS3.p7.10.m4.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p7.10.m4.1b"><ci id="S2.SS3.p7.10.m4.1.1.cmml" xref="S2.SS3.p7.10.m4.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p7.10.m4.1c">\mathbf{A}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p7.10.m4.1d">bold_A</annotation></semantics></math>.</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>Generalized variance</h3> <div class="ltx_para" id="S2.SS4.p1"> <p class="ltx_p" id="S2.SS4.p1.6">Based on Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E4" title="In 2.3 Generating fully synthetic data ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">4</span></a>), <cite class="ltx_cite ltx_citemacro_citet">Klein, Moura, & Sinha (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib8" title="">2021</a>)</cite> concluded that</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="T_{1}^{\star}=(n-1)\frac{|\boldsymbol{S}^{*}|}{|\boldsymbol{\Sigma}|}" class="ltx_Math" display="block" id="S2.E5.m1.3"><semantics id="S2.E5.m1.3a"><mrow id="S2.E5.m1.3.3" xref="S2.E5.m1.3.3.cmml"><msubsup id="S2.E5.m1.3.3.3" xref="S2.E5.m1.3.3.3.cmml"><mi id="S2.E5.m1.3.3.3.2.2" xref="S2.E5.m1.3.3.3.2.2.cmml">T</mi><mn id="S2.E5.m1.3.3.3.2.3" xref="S2.E5.m1.3.3.3.2.3.cmml">1</mn><mo id="S2.E5.m1.3.3.3.3" xref="S2.E5.m1.3.3.3.3.cmml">⋆</mo></msubsup><mo id="S2.E5.m1.3.3.2" xref="S2.E5.m1.3.3.2.cmml">=</mo><mrow id="S2.E5.m1.3.3.1" xref="S2.E5.m1.3.3.1.cmml"><mrow id="S2.E5.m1.3.3.1.1.1" xref="S2.E5.m1.3.3.1.1.1.1.cmml"><mo id="S2.E5.m1.3.3.1.1.1.2" stretchy="false" xref="S2.E5.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S2.E5.m1.3.3.1.1.1.1" xref="S2.E5.m1.3.3.1.1.1.1.cmml"><mi id="S2.E5.m1.3.3.1.1.1.1.2" xref="S2.E5.m1.3.3.1.1.1.1.2.cmml">n</mi><mo id="S2.E5.m1.3.3.1.1.1.1.1" xref="S2.E5.m1.3.3.1.1.1.1.1.cmml">−</mo><mn id="S2.E5.m1.3.3.1.1.1.1.3" xref="S2.E5.m1.3.3.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.E5.m1.3.3.1.1.1.3" stretchy="false" xref="S2.E5.m1.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S2.E5.m1.3.3.1.2" xref="S2.E5.m1.3.3.1.2.cmml"></mo><mfrac id="S2.E5.m1.2.2" xref="S2.E5.m1.2.2.cmml"><mrow id="S2.E5.m1.1.1.1.1" xref="S2.E5.m1.1.1.1.2.cmml"><mo id="S2.E5.m1.1.1.1.1.2" stretchy="false" xref="S2.E5.m1.1.1.1.2.1.cmml">|</mo><msup id="S2.E5.m1.1.1.1.1.1" xref="S2.E5.m1.1.1.1.1.1.cmml"><mi id="S2.E5.m1.1.1.1.1.1.2" xref="S2.E5.m1.1.1.1.1.1.2.cmml">𝑺</mi><mo id="S2.E5.m1.1.1.1.1.1.3" xref="S2.E5.m1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S2.E5.m1.1.1.1.1.3" stretchy="false" xref="S2.E5.m1.1.1.1.2.1.cmml">|</mo></mrow><mrow id="S2.E5.m1.2.2.2.3" xref="S2.E5.m1.2.2.2.2.cmml"><mo id="S2.E5.m1.2.2.2.3.1" stretchy="false" xref="S2.E5.m1.2.2.2.2.1.cmml">|</mo><mi id="S2.E5.m1.2.2.2.1" xref="S2.E5.m1.2.2.2.1.cmml">𝚺</mi><mo id="S2.E5.m1.2.2.2.3.2" stretchy="false" xref="S2.E5.m1.2.2.2.2.1.cmml">|</mo></mrow></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E5.m1.3b"><apply id="S2.E5.m1.3.3.cmml" xref="S2.E5.m1.3.3"><eq id="S2.E5.m1.3.3.2.cmml" xref="S2.E5.m1.3.3.2"></eq><apply id="S2.E5.m1.3.3.3.cmml" xref="S2.E5.m1.3.3.3"><csymbol cd="ambiguous" id="S2.E5.m1.3.3.3.1.cmml" xref="S2.E5.m1.3.3.3">superscript</csymbol><apply id="S2.E5.m1.3.3.3.2.cmml" xref="S2.E5.m1.3.3.3"><csymbol cd="ambiguous" id="S2.E5.m1.3.3.3.2.1.cmml" xref="S2.E5.m1.3.3.3">subscript</csymbol><ci id="S2.E5.m1.3.3.3.2.2.cmml" xref="S2.E5.m1.3.3.3.2.2">𝑇</ci><cn id="S2.E5.m1.3.3.3.2.3.cmml" type="integer" xref="S2.E5.m1.3.3.3.2.3">1</cn></apply><ci id="S2.E5.m1.3.3.3.3.cmml" xref="S2.E5.m1.3.3.3.3">⋆</ci></apply><apply id="S2.E5.m1.3.3.1.cmml" xref="S2.E5.m1.3.3.1"><times id="S2.E5.m1.3.3.1.2.cmml" xref="S2.E5.m1.3.3.1.2"></times><apply id="S2.E5.m1.3.3.1.1.1.1.cmml" xref="S2.E5.m1.3.3.1.1.1"><minus id="S2.E5.m1.3.3.1.1.1.1.1.cmml" xref="S2.E5.m1.3.3.1.1.1.1.1"></minus><ci id="S2.E5.m1.3.3.1.1.1.1.2.cmml" xref="S2.E5.m1.3.3.1.1.1.1.2">𝑛</ci><cn id="S2.E5.m1.3.3.1.1.1.1.3.cmml" type="integer" xref="S2.E5.m1.3.3.1.1.1.1.3">1</cn></apply><apply id="S2.E5.m1.2.2.cmml" xref="S2.E5.m1.2.2"><divide id="S2.E5.m1.2.2.3.cmml" xref="S2.E5.m1.2.2"></divide><apply id="S2.E5.m1.1.1.1.2.cmml" xref="S2.E5.m1.1.1.1.1"><abs id="S2.E5.m1.1.1.1.2.1.cmml" xref="S2.E5.m1.1.1.1.1.2"></abs><apply id="S2.E5.m1.1.1.1.1.1.cmml" xref="S2.E5.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.E5.m1.1.1.1.1.1.1.cmml" xref="S2.E5.m1.1.1.1.1.1">superscript</csymbol><ci id="S2.E5.m1.1.1.1.1.1.2.cmml" xref="S2.E5.m1.1.1.1.1.1.2">𝑺</ci><times id="S2.E5.m1.1.1.1.1.1.3.cmml" xref="S2.E5.m1.1.1.1.1.1.3"></times></apply></apply><apply id="S2.E5.m1.2.2.2.2.cmml" xref="S2.E5.m1.2.2.2.3"><abs id="S2.E5.m1.2.2.2.2.1.cmml" xref="S2.E5.m1.2.2.2.3.1"></abs><ci id="S2.E5.m1.2.2.2.1.cmml" xref="S2.E5.m1.2.2.2.1">𝚺</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E5.m1.3c">T_{1}^{\star}=(n-1)\frac{|\boldsymbol{S}^{*}|}{|\boldsymbol{\Sigma}|}</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m1.3d">italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT = ( italic_n - 1 ) divide start_ARG | bold_italic_S start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | end_ARG start_ARG | bold_Σ | 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">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.p1.7">would be stochastically equivalent to</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="\left(\prod_{j=1}^{p}A_{j}\right)\left(\prod_{j=1}^{p}B_{j}\right)" class="ltx_Math" display="block" id="S2.E6.m1.2"><semantics id="S2.E6.m1.2a"><mrow id="S2.E6.m1.2.2" xref="S2.E6.m1.2.2.cmml"><mrow id="S2.E6.m1.1.1.1.1" xref="S2.E6.m1.1.1.1.1.1.cmml"><mo id="S2.E6.m1.1.1.1.1.2" xref="S2.E6.m1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E6.m1.1.1.1.1.1" xref="S2.E6.m1.1.1.1.1.1.cmml"><munderover id="S2.E6.m1.1.1.1.1.1.1" xref="S2.E6.m1.1.1.1.1.1.1.cmml"><mo id="S2.E6.m1.1.1.1.1.1.1.2.2" lspace="0em" movablelimits="false" xref="S2.E6.m1.1.1.1.1.1.1.2.2.cmml">∏</mo><mrow id="S2.E6.m1.1.1.1.1.1.1.2.3" xref="S2.E6.m1.1.1.1.1.1.1.2.3.cmml"><mi id="S2.E6.m1.1.1.1.1.1.1.2.3.2" xref="S2.E6.m1.1.1.1.1.1.1.2.3.2.cmml">j</mi><mo id="S2.E6.m1.1.1.1.1.1.1.2.3.1" xref="S2.E6.m1.1.1.1.1.1.1.2.3.1.cmml">=</mo><mn id="S2.E6.m1.1.1.1.1.1.1.2.3.3" xref="S2.E6.m1.1.1.1.1.1.1.2.3.3.cmml">1</mn></mrow><mi id="S2.E6.m1.1.1.1.1.1.1.3" xref="S2.E6.m1.1.1.1.1.1.1.3.cmml">p</mi></munderover><msub id="S2.E6.m1.1.1.1.1.1.2" xref="S2.E6.m1.1.1.1.1.1.2.cmml"><mi id="S2.E6.m1.1.1.1.1.1.2.2" xref="S2.E6.m1.1.1.1.1.1.2.2.cmml">A</mi><mi id="S2.E6.m1.1.1.1.1.1.2.3" xref="S2.E6.m1.1.1.1.1.1.2.3.cmml">j</mi></msub></mrow><mo id="S2.E6.m1.1.1.1.1.3" xref="S2.E6.m1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.E6.m1.2.2.3" xref="S2.E6.m1.2.2.3.cmml"></mo><mrow id="S2.E6.m1.2.2.2.1" xref="S2.E6.m1.2.2.2.1.1.cmml"><mo id="S2.E6.m1.2.2.2.1.2" xref="S2.E6.m1.2.2.2.1.1.cmml">(</mo><mrow id="S2.E6.m1.2.2.2.1.1" xref="S2.E6.m1.2.2.2.1.1.cmml"><munderover id="S2.E6.m1.2.2.2.1.1.1" xref="S2.E6.m1.2.2.2.1.1.1.cmml"><mo id="S2.E6.m1.2.2.2.1.1.1.2.2" lspace="0em" movablelimits="false" xref="S2.E6.m1.2.2.2.1.1.1.2.2.cmml">∏</mo><mrow id="S2.E6.m1.2.2.2.1.1.1.2.3" xref="S2.E6.m1.2.2.2.1.1.1.2.3.cmml"><mi id="S2.E6.m1.2.2.2.1.1.1.2.3.2" xref="S2.E6.m1.2.2.2.1.1.1.2.3.2.cmml">j</mi><mo id="S2.E6.m1.2.2.2.1.1.1.2.3.1" xref="S2.E6.m1.2.2.2.1.1.1.2.3.1.cmml">=</mo><mn id="S2.E6.m1.2.2.2.1.1.1.2.3.3" xref="S2.E6.m1.2.2.2.1.1.1.2.3.3.cmml">1</mn></mrow><mi id="S2.E6.m1.2.2.2.1.1.1.3" xref="S2.E6.m1.2.2.2.1.1.1.3.cmml">p</mi></munderover><msub id="S2.E6.m1.2.2.2.1.1.2" xref="S2.E6.m1.2.2.2.1.1.2.cmml"><mi id="S2.E6.m1.2.2.2.1.1.2.2" xref="S2.E6.m1.2.2.2.1.1.2.2.cmml">B</mi><mi id="S2.E6.m1.2.2.2.1.1.2.3" xref="S2.E6.m1.2.2.2.1.1.2.3.cmml">j</mi></msub></mrow><mo id="S2.E6.m1.2.2.2.1.3" xref="S2.E6.m1.2.2.2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E6.m1.2b"><apply id="S2.E6.m1.2.2.cmml" xref="S2.E6.m1.2.2"><times id="S2.E6.m1.2.2.3.cmml" xref="S2.E6.m1.2.2.3"></times><apply id="S2.E6.m1.1.1.1.1.1.cmml" xref="S2.E6.m1.1.1.1.1"><apply id="S2.E6.m1.1.1.1.1.1.1.cmml" xref="S2.E6.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.E6.m1.1.1.1.1.1.1.1.cmml" xref="S2.E6.m1.1.1.1.1.1.1">superscript</csymbol><apply id="S2.E6.m1.1.1.1.1.1.1.2.cmml" xref="S2.E6.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.E6.m1.1.1.1.1.1.1.2.1.cmml" xref="S2.E6.m1.1.1.1.1.1.1">subscript</csymbol><csymbol cd="latexml" id="S2.E6.m1.1.1.1.1.1.1.2.2.cmml" xref="S2.E6.m1.1.1.1.1.1.1.2.2">product</csymbol><apply id="S2.E6.m1.1.1.1.1.1.1.2.3.cmml" xref="S2.E6.m1.1.1.1.1.1.1.2.3"><eq id="S2.E6.m1.1.1.1.1.1.1.2.3.1.cmml" xref="S2.E6.m1.1.1.1.1.1.1.2.3.1"></eq><ci id="S2.E6.m1.1.1.1.1.1.1.2.3.2.cmml" xref="S2.E6.m1.1.1.1.1.1.1.2.3.2">𝑗</ci><cn id="S2.E6.m1.1.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S2.E6.m1.1.1.1.1.1.1.2.3.3">1</cn></apply></apply><ci id="S2.E6.m1.1.1.1.1.1.1.3.cmml" xref="S2.E6.m1.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="S2.E6.m1.1.1.1.1.1.2.cmml" xref="S2.E6.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E6.m1.1.1.1.1.1.2.1.cmml" xref="S2.E6.m1.1.1.1.1.1.2">subscript</csymbol><ci id="S2.E6.m1.1.1.1.1.1.2.2.cmml" xref="S2.E6.m1.1.1.1.1.1.2.2">𝐴</ci><ci id="S2.E6.m1.1.1.1.1.1.2.3.cmml" xref="S2.E6.m1.1.1.1.1.1.2.3">𝑗</ci></apply></apply><apply id="S2.E6.m1.2.2.2.1.1.cmml" xref="S2.E6.m1.2.2.2.1"><apply id="S2.E6.m1.2.2.2.1.1.1.cmml" xref="S2.E6.m1.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.E6.m1.2.2.2.1.1.1.1.cmml" xref="S2.E6.m1.2.2.2.1.1.1">superscript</csymbol><apply id="S2.E6.m1.2.2.2.1.1.1.2.cmml" xref="S2.E6.m1.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.E6.m1.2.2.2.1.1.1.2.1.cmml" xref="S2.E6.m1.2.2.2.1.1.1">subscript</csymbol><csymbol cd="latexml" id="S2.E6.m1.2.2.2.1.1.1.2.2.cmml" xref="S2.E6.m1.2.2.2.1.1.1.2.2">product</csymbol><apply id="S2.E6.m1.2.2.2.1.1.1.2.3.cmml" xref="S2.E6.m1.2.2.2.1.1.1.2.3"><eq id="S2.E6.m1.2.2.2.1.1.1.2.3.1.cmml" xref="S2.E6.m1.2.2.2.1.1.1.2.3.1"></eq><ci id="S2.E6.m1.2.2.2.1.1.1.2.3.2.cmml" xref="S2.E6.m1.2.2.2.1.1.1.2.3.2">𝑗</ci><cn id="S2.E6.m1.2.2.2.1.1.1.2.3.3.cmml" type="integer" xref="S2.E6.m1.2.2.2.1.1.1.2.3.3">1</cn></apply></apply><ci id="S2.E6.m1.2.2.2.1.1.1.3.cmml" xref="S2.E6.m1.2.2.2.1.1.1.3">𝑝</ci></apply><apply id="S2.E6.m1.2.2.2.1.1.2.cmml" xref="S2.E6.m1.2.2.2.1.1.2"><csymbol cd="ambiguous" id="S2.E6.m1.2.2.2.1.1.2.1.cmml" xref="S2.E6.m1.2.2.2.1.1.2">subscript</csymbol><ci id="S2.E6.m1.2.2.2.1.1.2.2.cmml" xref="S2.E6.m1.2.2.2.1.1.2.2">𝐵</ci><ci id="S2.E6.m1.2.2.2.1.1.2.3.cmml" xref="S2.E6.m1.2.2.2.1.1.2.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m1.2c">\left(\prod_{j=1}^{p}A_{j}\right)\left(\prod_{j=1}^{p}B_{j}\right)</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m1.2d">( ∏ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ( ∏ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_B start_POSTSUBSCRIPT italic_j 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">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.p1.5">where <math alttext="A_{j}" class="ltx_Math" display="inline" id="S2.SS4.p1.1.m1.1"><semantics id="S2.SS4.p1.1.m1.1a"><msub id="S2.SS4.p1.1.m1.1.1" xref="S2.SS4.p1.1.m1.1.1.cmml"><mi id="S2.SS4.p1.1.m1.1.1.2" xref="S2.SS4.p1.1.m1.1.1.2.cmml">A</mi><mi id="S2.SS4.p1.1.m1.1.1.3" xref="S2.SS4.p1.1.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.1.m1.1b"><apply id="S2.SS4.p1.1.m1.1.1.cmml" xref="S2.SS4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.1.m1.1.1.1.cmml" xref="S2.SS4.p1.1.m1.1.1">subscript</csymbol><ci id="S2.SS4.p1.1.m1.1.1.2.cmml" xref="S2.SS4.p1.1.m1.1.1.2">𝐴</ci><ci id="S2.SS4.p1.1.m1.1.1.3.cmml" xref="S2.SS4.p1.1.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.1.m1.1c">A_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.1.m1.1d">italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="B_{j}" class="ltx_Math" display="inline" id="S2.SS4.p1.2.m2.1"><semantics id="S2.SS4.p1.2.m2.1a"><msub id="S2.SS4.p1.2.m2.1.1" xref="S2.SS4.p1.2.m2.1.1.cmml"><mi id="S2.SS4.p1.2.m2.1.1.2" xref="S2.SS4.p1.2.m2.1.1.2.cmml">B</mi><mi id="S2.SS4.p1.2.m2.1.1.3" xref="S2.SS4.p1.2.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.2.m2.1b"><apply id="S2.SS4.p1.2.m2.1.1.cmml" xref="S2.SS4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.2.m2.1.1.1.cmml" xref="S2.SS4.p1.2.m2.1.1">subscript</csymbol><ci id="S2.SS4.p1.2.m2.1.1.2.cmml" xref="S2.SS4.p1.2.m2.1.1.2">𝐵</ci><ci id="S2.SS4.p1.2.m2.1.1.3.cmml" xref="S2.SS4.p1.2.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.2.m2.1c">B_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.2.m2.1d">italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, for <math alttext="j=1,\dots,p" class="ltx_Math" display="inline" id="S2.SS4.p1.3.m3.3"><semantics id="S2.SS4.p1.3.m3.3a"><mrow id="S2.SS4.p1.3.m3.3.4" xref="S2.SS4.p1.3.m3.3.4.cmml"><mi id="S2.SS4.p1.3.m3.3.4.2" xref="S2.SS4.p1.3.m3.3.4.2.cmml">j</mi><mo id="S2.SS4.p1.3.m3.3.4.1" xref="S2.SS4.p1.3.m3.3.4.1.cmml">=</mo><mrow id="S2.SS4.p1.3.m3.3.4.3.2" xref="S2.SS4.p1.3.m3.3.4.3.1.cmml"><mn id="S2.SS4.p1.3.m3.1.1" xref="S2.SS4.p1.3.m3.1.1.cmml">1</mn><mo id="S2.SS4.p1.3.m3.3.4.3.2.1" xref="S2.SS4.p1.3.m3.3.4.3.1.cmml">,</mo><mi id="S2.SS4.p1.3.m3.2.2" mathvariant="normal" xref="S2.SS4.p1.3.m3.2.2.cmml">…</mi><mo id="S2.SS4.p1.3.m3.3.4.3.2.2" xref="S2.SS4.p1.3.m3.3.4.3.1.cmml">,</mo><mi id="S2.SS4.p1.3.m3.3.3" xref="S2.SS4.p1.3.m3.3.3.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.3.m3.3b"><apply id="S2.SS4.p1.3.m3.3.4.cmml" xref="S2.SS4.p1.3.m3.3.4"><eq id="S2.SS4.p1.3.m3.3.4.1.cmml" xref="S2.SS4.p1.3.m3.3.4.1"></eq><ci id="S2.SS4.p1.3.m3.3.4.2.cmml" xref="S2.SS4.p1.3.m3.3.4.2">𝑗</ci><list id="S2.SS4.p1.3.m3.3.4.3.1.cmml" xref="S2.SS4.p1.3.m3.3.4.3.2"><cn id="S2.SS4.p1.3.m3.1.1.cmml" type="integer" xref="S2.SS4.p1.3.m3.1.1">1</cn><ci id="S2.SS4.p1.3.m3.2.2.cmml" xref="S2.SS4.p1.3.m3.2.2">…</ci><ci id="S2.SS4.p1.3.m3.3.3.cmml" xref="S2.SS4.p1.3.m3.3.3">𝑝</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.3.m3.3c">j=1,\dots,p</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.3.m3.3d">italic_j = 1 , … , italic_p</annotation></semantics></math>, are all independently distributed as <math alttext="\chi^{2}_{n-j}" class="ltx_Math" display="inline" id="S2.SS4.p1.4.m4.1"><semantics id="S2.SS4.p1.4.m4.1a"><msubsup id="S2.SS4.p1.4.m4.1.1" xref="S2.SS4.p1.4.m4.1.1.cmml"><mi id="S2.SS4.p1.4.m4.1.1.2.2" xref="S2.SS4.p1.4.m4.1.1.2.2.cmml">χ</mi><mrow id="S2.SS4.p1.4.m4.1.1.3" xref="S2.SS4.p1.4.m4.1.1.3.cmml"><mi id="S2.SS4.p1.4.m4.1.1.3.2" xref="S2.SS4.p1.4.m4.1.1.3.2.cmml">n</mi><mo id="S2.SS4.p1.4.m4.1.1.3.1" xref="S2.SS4.p1.4.m4.1.1.3.1.cmml">−</mo><mi id="S2.SS4.p1.4.m4.1.1.3.3" xref="S2.SS4.p1.4.m4.1.1.3.3.cmml">j</mi></mrow><mn id="S2.SS4.p1.4.m4.1.1.2.3" xref="S2.SS4.p1.4.m4.1.1.2.3.cmml">2</mn></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.4.m4.1b"><apply id="S2.SS4.p1.4.m4.1.1.cmml" xref="S2.SS4.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.4.m4.1.1.1.cmml" xref="S2.SS4.p1.4.m4.1.1">subscript</csymbol><apply id="S2.SS4.p1.4.m4.1.1.2.cmml" xref="S2.SS4.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.4.m4.1.1.2.1.cmml" xref="S2.SS4.p1.4.m4.1.1">superscript</csymbol><ci id="S2.SS4.p1.4.m4.1.1.2.2.cmml" xref="S2.SS4.p1.4.m4.1.1.2.2">𝜒</ci><cn id="S2.SS4.p1.4.m4.1.1.2.3.cmml" type="integer" xref="S2.SS4.p1.4.m4.1.1.2.3">2</cn></apply><apply id="S2.SS4.p1.4.m4.1.1.3.cmml" xref="S2.SS4.p1.4.m4.1.1.3"><minus id="S2.SS4.p1.4.m4.1.1.3.1.cmml" xref="S2.SS4.p1.4.m4.1.1.3.1"></minus><ci id="S2.SS4.p1.4.m4.1.1.3.2.cmml" xref="S2.SS4.p1.4.m4.1.1.3.2">𝑛</ci><ci id="S2.SS4.p1.4.m4.1.1.3.3.cmml" xref="S2.SS4.p1.4.m4.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.4.m4.1c">\chi^{2}_{n-j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.4.m4.1d">italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, following a chi-square distribution with <math alttext="n-j" class="ltx_Math" display="inline" id="S2.SS4.p1.5.m5.1"><semantics id="S2.SS4.p1.5.m5.1a"><mrow id="S2.SS4.p1.5.m5.1.1" xref="S2.SS4.p1.5.m5.1.1.cmml"><mi id="S2.SS4.p1.5.m5.1.1.2" xref="S2.SS4.p1.5.m5.1.1.2.cmml">n</mi><mo id="S2.SS4.p1.5.m5.1.1.1" xref="S2.SS4.p1.5.m5.1.1.1.cmml">−</mo><mi id="S2.SS4.p1.5.m5.1.1.3" xref="S2.SS4.p1.5.m5.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.5.m5.1b"><apply id="S2.SS4.p1.5.m5.1.1.cmml" xref="S2.SS4.p1.5.m5.1.1"><minus id="S2.SS4.p1.5.m5.1.1.1.cmml" xref="S2.SS4.p1.5.m5.1.1.1"></minus><ci id="S2.SS4.p1.5.m5.1.1.2.cmml" xref="S2.SS4.p1.5.m5.1.1.2">𝑛</ci><ci id="S2.SS4.p1.5.m5.1.1.3.cmml" xref="S2.SS4.p1.5.m5.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.5.m5.1c">n-j</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.5.m5.1d">italic_n - italic_j</annotation></semantics></math> degrees of freedom.</p> </div> <div class="ltx_para" id="S2.SS4.p2"> <p class="ltx_p" id="S2.SS4.p2.2">From Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E5" title="In 2.4 Generalized variance ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">5</span></a>), we can construct the <math alttext="(1-\alpha)" class="ltx_Math" display="inline" id="S2.SS4.p2.1.m1.1"><semantics id="S2.SS4.p2.1.m1.1a"><mrow id="S2.SS4.p2.1.m1.1.1.1" xref="S2.SS4.p2.1.m1.1.1.1.1.cmml"><mo id="S2.SS4.p2.1.m1.1.1.1.2" stretchy="false" xref="S2.SS4.p2.1.m1.1.1.1.1.cmml">(</mo><mrow id="S2.SS4.p2.1.m1.1.1.1.1" xref="S2.SS4.p2.1.m1.1.1.1.1.cmml"><mn id="S2.SS4.p2.1.m1.1.1.1.1.2" xref="S2.SS4.p2.1.m1.1.1.1.1.2.cmml">1</mn><mo id="S2.SS4.p2.1.m1.1.1.1.1.1" xref="S2.SS4.p2.1.m1.1.1.1.1.1.cmml">−</mo><mi id="S2.SS4.p2.1.m1.1.1.1.1.3" xref="S2.SS4.p2.1.m1.1.1.1.1.3.cmml">α</mi></mrow><mo id="S2.SS4.p2.1.m1.1.1.1.3" stretchy="false" xref="S2.SS4.p2.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml 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id="S2.SS4.p2.2.m2.1.2.2.2" stretchy="false" xref="S2.SS4.p2.2.m2.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.2.m2.1b"><apply id="S2.SS4.p2.2.m2.1.2.1.cmml" xref="S2.SS4.p2.2.m2.1.2.2"><abs id="S2.SS4.p2.2.m2.1.2.1.1.cmml" xref="S2.SS4.p2.2.m2.1.2.2.1"></abs><ci id="S2.SS4.p2.2.m2.1.1.cmml" xref="S2.SS4.p2.2.m2.1.1">𝚺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.2.m2.1c">|\boldsymbol{\Sigma}|</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.2.m2.1d">| bold_Σ |</annotation></semantics></math> which will be</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="\left(\frac{(n-1)^{p}|\mathbf{S}^{\star}|}{t^{\star}_{1,1-\alpha/2}},\frac{(n-% 1)^{p}|\mathbf{S}^{\star}|}{t^{\star}_{1,\alpha/2}}\right)" 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id="S2.Ex3.m1.8c">\left(\frac{(n-1)^{p}|\mathbf{S}^{\star}|}{t^{\star}_{1,1-\alpha/2}},\frac{(n-% 1)^{p}|\mathbf{S}^{\star}|}{t^{\star}_{1,\alpha/2}}\right)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m1.8d">( divide start_ARG ( italic_n - 1 ) start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT | bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | end_ARG start_ARG italic_t start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 , 1 - italic_α / 2 end_POSTSUBSCRIPT end_ARG , divide start_ARG ( italic_n - 1 ) start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT | bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | end_ARG start_ARG italic_t start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 , italic_α / 2 end_POSTSUBSCRIPT end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.p2.5">where <math alttext="t^{\star}_{1,\gamma}" class="ltx_Math" display="inline" id="S2.SS4.p2.3.m1.2"><semantics id="S2.SS4.p2.3.m1.2a"><msubsup id="S2.SS4.p2.3.m1.2.3" xref="S2.SS4.p2.3.m1.2.3.cmml"><mi id="S2.SS4.p2.3.m1.2.3.2.2" xref="S2.SS4.p2.3.m1.2.3.2.2.cmml">t</mi><mrow id="S2.SS4.p2.3.m1.2.2.2.4" xref="S2.SS4.p2.3.m1.2.2.2.3.cmml"><mn id="S2.SS4.p2.3.m1.1.1.1.1" xref="S2.SS4.p2.3.m1.1.1.1.1.cmml">1</mn><mo id="S2.SS4.p2.3.m1.2.2.2.4.1" xref="S2.SS4.p2.3.m1.2.2.2.3.cmml">,</mo><mi id="S2.SS4.p2.3.m1.2.2.2.2" xref="S2.SS4.p2.3.m1.2.2.2.2.cmml">γ</mi></mrow><mo id="S2.SS4.p2.3.m1.2.3.2.3" xref="S2.SS4.p2.3.m1.2.3.2.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.3.m1.2b"><apply id="S2.SS4.p2.3.m1.2.3.cmml" xref="S2.SS4.p2.3.m1.2.3"><csymbol cd="ambiguous" id="S2.SS4.p2.3.m1.2.3.1.cmml" xref="S2.SS4.p2.3.m1.2.3">subscript</csymbol><apply id="S2.SS4.p2.3.m1.2.3.2.cmml" xref="S2.SS4.p2.3.m1.2.3"><csymbol cd="ambiguous" id="S2.SS4.p2.3.m1.2.3.2.1.cmml" xref="S2.SS4.p2.3.m1.2.3">superscript</csymbol><ci id="S2.SS4.p2.3.m1.2.3.2.2.cmml" xref="S2.SS4.p2.3.m1.2.3.2.2">𝑡</ci><ci id="S2.SS4.p2.3.m1.2.3.2.3.cmml" xref="S2.SS4.p2.3.m1.2.3.2.3">⋆</ci></apply><list id="S2.SS4.p2.3.m1.2.2.2.3.cmml" xref="S2.SS4.p2.3.m1.2.2.2.4"><cn id="S2.SS4.p2.3.m1.1.1.1.1.cmml" type="integer" xref="S2.SS4.p2.3.m1.1.1.1.1">1</cn><ci id="S2.SS4.p2.3.m1.2.2.2.2.cmml" xref="S2.SS4.p2.3.m1.2.2.2.2">𝛾</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.3.m1.2c">t^{\star}_{1,\gamma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.3.m1.2d">italic_t start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 , italic_γ end_POSTSUBSCRIPT</annotation></semantics></math> is the <math alttext="\gamma" class="ltx_Math" display="inline" id="S2.SS4.p2.4.m2.1"><semantics id="S2.SS4.p2.4.m2.1a"><mi id="S2.SS4.p2.4.m2.1.1" xref="S2.SS4.p2.4.m2.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.4.m2.1b"><ci id="S2.SS4.p2.4.m2.1.1.cmml" xref="S2.SS4.p2.4.m2.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.4.m2.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.4.m2.1d">italic_γ</annotation></semantics></math>th percentile of <math alttext="T_{1}" class="ltx_Math" display="inline" id="S2.SS4.p2.5.m3.1"><semantics id="S2.SS4.p2.5.m3.1a"><msub id="S2.SS4.p2.5.m3.1.1" xref="S2.SS4.p2.5.m3.1.1.cmml"><mi id="S2.SS4.p2.5.m3.1.1.2" xref="S2.SS4.p2.5.m3.1.1.2.cmml">T</mi><mn id="S2.SS4.p2.5.m3.1.1.3" xref="S2.SS4.p2.5.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.5.m3.1b"><apply id="S2.SS4.p2.5.m3.1.1.cmml" xref="S2.SS4.p2.5.m3.1.1"><csymbol cd="ambiguous" id="S2.SS4.p2.5.m3.1.1.1.cmml" xref="S2.SS4.p2.5.m3.1.1">subscript</csymbol><ci id="S2.SS4.p2.5.m3.1.1.2.cmml" xref="S2.SS4.p2.5.m3.1.1.2">𝑇</ci><cn id="S2.SS4.p2.5.m3.1.1.3.cmml" type="integer" xref="S2.SS4.p2.5.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.5.m3.1c">T_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.5.m3.1d">italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, obtained from (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E6" title="In 2.4 Generalized variance ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">6</span></a>).</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>Sphericity test</h3> <div class="ltx_para" id="S2.SS5.p1"> <p class="ltx_p" id="S2.SS5.p1.1">The assumption of sphericity structure of the covariance matrix is an important condition for the validity of many inferential tests, including multivariate analysis of variance (MANOVA), repeated measures ANOVA, and multivariate analysis of covariance (MANCOVA) <cite class="ltx_cite ltx_citemacro_citep">(Moura, Coelho, & Sinha, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib19" title="">2024</a>)</cite>. This assumption implies that the variances of the differences between all possible pairs of within-subject conditions are equal.</p> </div> <div class="ltx_para" id="S2.SS5.p2"> <p class="ltx_p" id="S2.SS5.p2.1">The sphericity test consists on testing the hypotheses</p> <table class="ltx_equation ltx_eqn_table" id="S2.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{H}_{0}:\boldsymbol{\Sigma}=\sigma^{2}\mathbf{I}_{p}~{}~{}\text{vs.}~{% }~{}\mathcal{H}_{1}:\boldsymbol{\Sigma}\neq\sigma^{2}\mathbf{I}_{p}." class="ltx_Math" display="block" id="S2.Ex4.m1.1"><semantics id="S2.Ex4.m1.1a"><mrow id="S2.Ex4.m1.1.1.1" xref="S2.Ex4.m1.1.1.1.1.cmml"><mrow id="S2.Ex4.m1.1.1.1.1" xref="S2.Ex4.m1.1.1.1.1.cmml"><msub id="S2.Ex4.m1.1.1.1.1.2" xref="S2.Ex4.m1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex4.m1.1.1.1.1.2.2" xref="S2.Ex4.m1.1.1.1.1.2.2.cmml">ℋ</mi><mn id="S2.Ex4.m1.1.1.1.1.2.3" xref="S2.Ex4.m1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S2.Ex4.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S2.Ex4.m1.1.1.1.1.3.cmml">:</mo><mrow id="S2.Ex4.m1.1.1.1.1.4" xref="S2.Ex4.m1.1.1.1.1.4.cmml"><mi id="S2.Ex4.m1.1.1.1.1.4.2" xref="S2.Ex4.m1.1.1.1.1.4.2.cmml">𝚺</mi><mo id="S2.Ex4.m1.1.1.1.1.4.1" xref="S2.Ex4.m1.1.1.1.1.4.1.cmml">=</mo><mrow id="S2.Ex4.m1.1.1.1.1.4.3" xref="S2.Ex4.m1.1.1.1.1.4.3.cmml"><msup id="S2.Ex4.m1.1.1.1.1.4.3.2" xref="S2.Ex4.m1.1.1.1.1.4.3.2.cmml"><mi id="S2.Ex4.m1.1.1.1.1.4.3.2.2" xref="S2.Ex4.m1.1.1.1.1.4.3.2.2.cmml">σ</mi><mn id="S2.Ex4.m1.1.1.1.1.4.3.2.3" xref="S2.Ex4.m1.1.1.1.1.4.3.2.3.cmml">2</mn></msup><mo id="S2.Ex4.m1.1.1.1.1.4.3.1" xref="S2.Ex4.m1.1.1.1.1.4.3.1.cmml"></mo><msub id="S2.Ex4.m1.1.1.1.1.4.3.3" xref="S2.Ex4.m1.1.1.1.1.4.3.3.cmml"><mi id="S2.Ex4.m1.1.1.1.1.4.3.3.2" xref="S2.Ex4.m1.1.1.1.1.4.3.3.2.cmml">𝐈</mi><mi id="S2.Ex4.m1.1.1.1.1.4.3.3.3" 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ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS5.p3"> <p class="ltx_p" id="S2.SS5.p3.9">Considering that</p> <table class="ltx_equation ltx_eqn_table" id="S2.E7"> <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="T^{\star}_{2}=\frac{\left|\mathbf{S}^{\star}\right|^{1/p}}{tr\left(\mathbf{S}^% {\star}\right)/p}," class="ltx_Math" display="block" id="S2.E7.m1.3"><semantics id="S2.E7.m1.3a"><mrow id="S2.E7.m1.3.3.1" xref="S2.E7.m1.3.3.1.1.cmml"><mrow id="S2.E7.m1.3.3.1.1" xref="S2.E7.m1.3.3.1.1.cmml"><msubsup id="S2.E7.m1.3.3.1.1.2" xref="S2.E7.m1.3.3.1.1.2.cmml"><mi id="S2.E7.m1.3.3.1.1.2.2.2" xref="S2.E7.m1.3.3.1.1.2.2.2.cmml">T</mi><mn id="S2.E7.m1.3.3.1.1.2.3" xref="S2.E7.m1.3.3.1.1.2.3.cmml">2</mn><mo id="S2.E7.m1.3.3.1.1.2.2.3" xref="S2.E7.m1.3.3.1.1.2.2.3.cmml">⋆</mo></msubsup><mo id="S2.E7.m1.3.3.1.1.1" 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end_ARG start_ARG italic_t italic_r ( bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ) / italic_p 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">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS5.p3.1"><cite class="ltx_cite ltx_citemacro_citet">Klein, Moura, & Sinha (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib8" title="">2021</a>)</cite> demonstrated that its distribution, under <math alttext="\mathcal{H}_{0}" class="ltx_Math" display="inline" id="S2.SS5.p3.1.m1.1"><semantics id="S2.SS5.p3.1.m1.1a"><msub id="S2.SS5.p3.1.m1.1.1" xref="S2.SS5.p3.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p3.1.m1.1.1.2" xref="S2.SS5.p3.1.m1.1.1.2.cmml">ℋ</mi><mn id="S2.SS5.p3.1.m1.1.1.3" xref="S2.SS5.p3.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml 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end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 1 / italic_p end_POSTSUPERSCRIPT end_ARG start_ARG tr ( bold_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT bold_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) / italic_p 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">(8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS5.p3.8">where <math alttext="\mathbf{W}_{1}\sim\mathcal{W}_{p}\left(n-1,(1\,/\,(n-1))\mathbf{I}_{p}\right)" class="ltx_Math" display="inline" id="S2.SS5.p3.2.m1.2"><semantics id="S2.SS5.p3.2.m1.2a"><mrow id="S2.SS5.p3.2.m1.2.2" xref="S2.SS5.p3.2.m1.2.2.cmml"><msub id="S2.SS5.p3.2.m1.2.2.4" xref="S2.SS5.p3.2.m1.2.2.4.cmml"><mi id="S2.SS5.p3.2.m1.2.2.4.2" xref="S2.SS5.p3.2.m1.2.2.4.2.cmml">𝐖</mi><mn id="S2.SS5.p3.2.m1.2.2.4.3" xref="S2.SS5.p3.2.m1.2.2.4.3.cmml">1</mn></msub><mo id="S2.SS5.p3.2.m1.2.2.3" xref="S2.SS5.p3.2.m1.2.2.3.cmml">∼</mo><mrow id="S2.SS5.p3.2.m1.2.2.2" xref="S2.SS5.p3.2.m1.2.2.2.cmml"><msub id="S2.SS5.p3.2.m1.2.2.2.4" xref="S2.SS5.p3.2.m1.2.2.2.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p3.2.m1.2.2.2.4.2" xref="S2.SS5.p3.2.m1.2.2.2.4.2.cmml">𝒲</mi><mi id="S2.SS5.p3.2.m1.2.2.2.4.3" xref="S2.SS5.p3.2.m1.2.2.2.4.3.cmml">p</mi></msub><mo id="S2.SS5.p3.2.m1.2.2.2.3" xref="S2.SS5.p3.2.m1.2.2.2.3.cmml"></mo><mrow id="S2.SS5.p3.2.m1.2.2.2.2.2" xref="S2.SS5.p3.2.m1.2.2.2.2.3.cmml"><mo id="S2.SS5.p3.2.m1.2.2.2.2.2.3" xref="S2.SS5.p3.2.m1.2.2.2.2.3.cmml">(</mo><mrow id="S2.SS5.p3.2.m1.1.1.1.1.1.1" xref="S2.SS5.p3.2.m1.1.1.1.1.1.1.cmml"><mi id="S2.SS5.p3.2.m1.1.1.1.1.1.1.2" xref="S2.SS5.p3.2.m1.1.1.1.1.1.1.2.cmml">n</mi><mo id="S2.SS5.p3.2.m1.1.1.1.1.1.1.1" xref="S2.SS5.p3.2.m1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S2.SS5.p3.2.m1.1.1.1.1.1.1.3" xref="S2.SS5.p3.2.m1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS5.p3.2.m1.2.2.2.2.2.4" xref="S2.SS5.p3.2.m1.2.2.2.2.3.cmml">,</mo><mrow id="S2.SS5.p3.2.m1.2.2.2.2.2.2" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.cmml"><mrow id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.cmml"><mo id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.2" stretchy="false" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.cmml">(</mo><mrow id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.cmml"><mn id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.3" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.3.cmml">1</mn><mo id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.2" lspace="0.392em" rspace="0.392em" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.2.cmml">/</mo><mrow id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1.cmml"><mo id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.2" stretchy="false" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1.cmml"><mi id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1.2" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.cmml">n</mi><mo id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1.1" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.cmml">−</mo><mn id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1.3" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.3" stretchy="false" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.3" stretchy="false" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="S2.SS5.p3.2.m1.2.2.2.2.2.2.2" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.2.cmml"></mo><msub id="S2.SS5.p3.2.m1.2.2.2.2.2.2.3" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.3.cmml"><mi id="S2.SS5.p3.2.m1.2.2.2.2.2.2.3.2" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.3.2.cmml">𝐈</mi><mi id="S2.SS5.p3.2.m1.2.2.2.2.2.2.3.3" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.3.3.cmml">p</mi></msub></mrow><mo id="S2.SS5.p3.2.m1.2.2.2.2.2.5" xref="S2.SS5.p3.2.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.2.m1.2b"><apply id="S2.SS5.p3.2.m1.2.2.cmml" xref="S2.SS5.p3.2.m1.2.2"><csymbol cd="latexml" id="S2.SS5.p3.2.m1.2.2.3.cmml" xref="S2.SS5.p3.2.m1.2.2.3">similar-to</csymbol><apply id="S2.SS5.p3.2.m1.2.2.4.cmml" xref="S2.SS5.p3.2.m1.2.2.4"><csymbol cd="ambiguous" id="S2.SS5.p3.2.m1.2.2.4.1.cmml" xref="S2.SS5.p3.2.m1.2.2.4">subscript</csymbol><ci id="S2.SS5.p3.2.m1.2.2.4.2.cmml" xref="S2.SS5.p3.2.m1.2.2.4.2">𝐖</ci><cn id="S2.SS5.p3.2.m1.2.2.4.3.cmml" type="integer" xref="S2.SS5.p3.2.m1.2.2.4.3">1</cn></apply><apply id="S2.SS5.p3.2.m1.2.2.2.cmml" xref="S2.SS5.p3.2.m1.2.2.2"><times id="S2.SS5.p3.2.m1.2.2.2.3.cmml" xref="S2.SS5.p3.2.m1.2.2.2.3"></times><apply id="S2.SS5.p3.2.m1.2.2.2.4.cmml" xref="S2.SS5.p3.2.m1.2.2.2.4"><csymbol cd="ambiguous" id="S2.SS5.p3.2.m1.2.2.2.4.1.cmml" xref="S2.SS5.p3.2.m1.2.2.2.4">subscript</csymbol><ci id="S2.SS5.p3.2.m1.2.2.2.4.2.cmml" xref="S2.SS5.p3.2.m1.2.2.2.4.2">𝒲</ci><ci id="S2.SS5.p3.2.m1.2.2.2.4.3.cmml" xref="S2.SS5.p3.2.m1.2.2.2.4.3">𝑝</ci></apply><interval closure="open" id="S2.SS5.p3.2.m1.2.2.2.2.3.cmml" xref="S2.SS5.p3.2.m1.2.2.2.2.2"><apply id="S2.SS5.p3.2.m1.1.1.1.1.1.1.cmml" xref="S2.SS5.p3.2.m1.1.1.1.1.1.1"><minus id="S2.SS5.p3.2.m1.1.1.1.1.1.1.1.cmml" xref="S2.SS5.p3.2.m1.1.1.1.1.1.1.1"></minus><ci id="S2.SS5.p3.2.m1.1.1.1.1.1.1.2.cmml" xref="S2.SS5.p3.2.m1.1.1.1.1.1.1.2">𝑛</ci><cn id="S2.SS5.p3.2.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS5.p3.2.m1.1.1.1.1.1.1.3">1</cn></apply><apply id="S2.SS5.p3.2.m1.2.2.2.2.2.2.cmml" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2"><times id="S2.SS5.p3.2.m1.2.2.2.2.2.2.2.cmml" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.2"></times><apply id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.cmml" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1"><divide id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.2.cmml" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.2"></divide><cn id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.3">1</cn><apply id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1.cmml" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1"><minus id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1.1.cmml" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1.1"></minus><ci id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1.2.cmml" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1.2">𝑛</ci><cn id="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.1.1.1.1.1.1.3">1</cn></apply></apply><apply id="S2.SS5.p3.2.m1.2.2.2.2.2.2.3.cmml" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S2.SS5.p3.2.m1.2.2.2.2.2.2.3.1.cmml" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.3">subscript</csymbol><ci id="S2.SS5.p3.2.m1.2.2.2.2.2.2.3.2.cmml" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.3.2">𝐈</ci><ci id="S2.SS5.p3.2.m1.2.2.2.2.2.2.3.3.cmml" xref="S2.SS5.p3.2.m1.2.2.2.2.2.2.3.3">𝑝</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.2.m1.2c">\mathbf{W}_{1}\sim\mathcal{W}_{p}\left(n-1,(1\,/\,(n-1))\mathbf{I}_{p}\right)</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.2.m1.2d">bold_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∼ caligraphic_W start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_n - 1 , ( 1 / ( italic_n - 1 ) ) bold_I start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="\mathbf{W}_{2}\sim W_{p}\left(n-1,\mathbf{I}_{p}\right)" class="ltx_Math" display="inline" id="S2.SS5.p3.3.m2.2"><semantics id="S2.SS5.p3.3.m2.2a"><mrow id="S2.SS5.p3.3.m2.2.2" xref="S2.SS5.p3.3.m2.2.2.cmml"><msub id="S2.SS5.p3.3.m2.2.2.4" xref="S2.SS5.p3.3.m2.2.2.4.cmml"><mi id="S2.SS5.p3.3.m2.2.2.4.2" xref="S2.SS5.p3.3.m2.2.2.4.2.cmml">𝐖</mi><mn id="S2.SS5.p3.3.m2.2.2.4.3" xref="S2.SS5.p3.3.m2.2.2.4.3.cmml">2</mn></msub><mo id="S2.SS5.p3.3.m2.2.2.3" xref="S2.SS5.p3.3.m2.2.2.3.cmml">∼</mo><mrow id="S2.SS5.p3.3.m2.2.2.2" xref="S2.SS5.p3.3.m2.2.2.2.cmml"><msub id="S2.SS5.p3.3.m2.2.2.2.4" xref="S2.SS5.p3.3.m2.2.2.2.4.cmml"><mi id="S2.SS5.p3.3.m2.2.2.2.4.2" xref="S2.SS5.p3.3.m2.2.2.2.4.2.cmml">W</mi><mi id="S2.SS5.p3.3.m2.2.2.2.4.3" xref="S2.SS5.p3.3.m2.2.2.2.4.3.cmml">p</mi></msub><mo id="S2.SS5.p3.3.m2.2.2.2.3" xref="S2.SS5.p3.3.m2.2.2.2.3.cmml"></mo><mrow id="S2.SS5.p3.3.m2.2.2.2.2.2" xref="S2.SS5.p3.3.m2.2.2.2.2.3.cmml"><mo id="S2.SS5.p3.3.m2.2.2.2.2.2.3" xref="S2.SS5.p3.3.m2.2.2.2.2.3.cmml">(</mo><mrow id="S2.SS5.p3.3.m2.1.1.1.1.1.1" xref="S2.SS5.p3.3.m2.1.1.1.1.1.1.cmml"><mi id="S2.SS5.p3.3.m2.1.1.1.1.1.1.2" xref="S2.SS5.p3.3.m2.1.1.1.1.1.1.2.cmml">n</mi><mo id="S2.SS5.p3.3.m2.1.1.1.1.1.1.1" xref="S2.SS5.p3.3.m2.1.1.1.1.1.1.1.cmml">−</mo><mn id="S2.SS5.p3.3.m2.1.1.1.1.1.1.3" xref="S2.SS5.p3.3.m2.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS5.p3.3.m2.2.2.2.2.2.4" xref="S2.SS5.p3.3.m2.2.2.2.2.3.cmml">,</mo><msub id="S2.SS5.p3.3.m2.2.2.2.2.2.2" xref="S2.SS5.p3.3.m2.2.2.2.2.2.2.cmml"><mi 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id="S2.SS5.p3.3.m2.2.2.2.2.2.2.2.cmml" xref="S2.SS5.p3.3.m2.2.2.2.2.2.2.2">𝐈</ci><ci id="S2.SS5.p3.3.m2.2.2.2.2.2.2.3.cmml" xref="S2.SS5.p3.3.m2.2.2.2.2.2.2.3">𝑝</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.3.m2.2c">\mathbf{W}_{2}\sim W_{p}\left(n-1,\mathbf{I}_{p}\right)</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.3.m2.2d">bold_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ italic_W start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_n - 1 , bold_I start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT )</annotation></semantics></math>, independent between each other. From (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E8" title="In 2.5 Sphericity test ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">8</span></a>) we may construct the empirical distribution of <math alttext="T_{2}^{\star}" class="ltx_Math" display="inline" id="S2.SS5.p3.4.m3.1"><semantics id="S2.SS5.p3.4.m3.1a"><msubsup id="S2.SS5.p3.4.m3.1.1" xref="S2.SS5.p3.4.m3.1.1.cmml"><mi id="S2.SS5.p3.4.m3.1.1.2.2" xref="S2.SS5.p3.4.m3.1.1.2.2.cmml">T</mi><mn id="S2.SS5.p3.4.m3.1.1.2.3" xref="S2.SS5.p3.4.m3.1.1.2.3.cmml">2</mn><mo id="S2.SS5.p3.4.m3.1.1.3" xref="S2.SS5.p3.4.m3.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.4.m3.1b"><apply id="S2.SS5.p3.4.m3.1.1.cmml" xref="S2.SS5.p3.4.m3.1.1"><csymbol cd="ambiguous" id="S2.SS5.p3.4.m3.1.1.1.cmml" xref="S2.SS5.p3.4.m3.1.1">superscript</csymbol><apply id="S2.SS5.p3.4.m3.1.1.2.cmml" xref="S2.SS5.p3.4.m3.1.1"><csymbol cd="ambiguous" id="S2.SS5.p3.4.m3.1.1.2.1.cmml" xref="S2.SS5.p3.4.m3.1.1">subscript</csymbol><ci id="S2.SS5.p3.4.m3.1.1.2.2.cmml" xref="S2.SS5.p3.4.m3.1.1.2.2">𝑇</ci><cn id="S2.SS5.p3.4.m3.1.1.2.3.cmml" type="integer" xref="S2.SS5.p3.4.m3.1.1.2.3">2</cn></apply><ci id="S2.SS5.p3.4.m3.1.1.3.cmml" xref="S2.SS5.p3.4.m3.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.4.m3.1c">T_{2}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.4.m3.1d">italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> and reject the null hypothesis, for a level of significance <math alttext="\alpha" class="ltx_Math" display="inline" id="S2.SS5.p3.5.m4.1"><semantics id="S2.SS5.p3.5.m4.1a"><mi id="S2.SS5.p3.5.m4.1.1" xref="S2.SS5.p3.5.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.5.m4.1b"><ci id="S2.SS5.p3.5.m4.1.1.cmml" xref="S2.SS5.p3.5.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.5.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.5.m4.1d">italic_α</annotation></semantics></math>, if the observed value of <math alttext="T_{2}^{\star}" class="ltx_Math" display="inline" id="S2.SS5.p3.6.m5.1"><semantics id="S2.SS5.p3.6.m5.1a"><msubsup id="S2.SS5.p3.6.m5.1.1" xref="S2.SS5.p3.6.m5.1.1.cmml"><mi id="S2.SS5.p3.6.m5.1.1.2.2" xref="S2.SS5.p3.6.m5.1.1.2.2.cmml">T</mi><mn id="S2.SS5.p3.6.m5.1.1.2.3" xref="S2.SS5.p3.6.m5.1.1.2.3.cmml">2</mn><mo id="S2.SS5.p3.6.m5.1.1.3" xref="S2.SS5.p3.6.m5.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.6.m5.1b"><apply id="S2.SS5.p3.6.m5.1.1.cmml" xref="S2.SS5.p3.6.m5.1.1"><csymbol cd="ambiguous" id="S2.SS5.p3.6.m5.1.1.1.cmml" xref="S2.SS5.p3.6.m5.1.1">superscript</csymbol><apply id="S2.SS5.p3.6.m5.1.1.2.cmml" xref="S2.SS5.p3.6.m5.1.1"><csymbol cd="ambiguous" id="S2.SS5.p3.6.m5.1.1.2.1.cmml" xref="S2.SS5.p3.6.m5.1.1">subscript</csymbol><ci id="S2.SS5.p3.6.m5.1.1.2.2.cmml" xref="S2.SS5.p3.6.m5.1.1.2.2">𝑇</ci><cn id="S2.SS5.p3.6.m5.1.1.2.3.cmml" type="integer" xref="S2.SS5.p3.6.m5.1.1.2.3">2</cn></apply><ci id="S2.SS5.p3.6.m5.1.1.3.cmml" xref="S2.SS5.p3.6.m5.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.6.m5.1c">T_{2}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.6.m5.1d">italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> is less than <math alttext="t_{2;\alpha}^{\star}" class="ltx_Math" display="inline" id="S2.SS5.p3.7.m6.2"><semantics id="S2.SS5.p3.7.m6.2a"><msubsup id="S2.SS5.p3.7.m6.2.3" xref="S2.SS5.p3.7.m6.2.3.cmml"><mi id="S2.SS5.p3.7.m6.2.3.2.2" xref="S2.SS5.p3.7.m6.2.3.2.2.cmml">t</mi><mrow id="S2.SS5.p3.7.m6.2.2.2.4" xref="S2.SS5.p3.7.m6.2.2.2.3.cmml"><mn id="S2.SS5.p3.7.m6.1.1.1.1" xref="S2.SS5.p3.7.m6.1.1.1.1.cmml">2</mn><mo id="S2.SS5.p3.7.m6.2.2.2.4.1" xref="S2.SS5.p3.7.m6.2.2.2.3.cmml">;</mo><mi id="S2.SS5.p3.7.m6.2.2.2.2" xref="S2.SS5.p3.7.m6.2.2.2.2.cmml">α</mi></mrow><mo id="S2.SS5.p3.7.m6.2.3.3" xref="S2.SS5.p3.7.m6.2.3.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.7.m6.2b"><apply id="S2.SS5.p3.7.m6.2.3.cmml" xref="S2.SS5.p3.7.m6.2.3"><csymbol cd="ambiguous" id="S2.SS5.p3.7.m6.2.3.1.cmml" xref="S2.SS5.p3.7.m6.2.3">superscript</csymbol><apply id="S2.SS5.p3.7.m6.2.3.2.cmml" xref="S2.SS5.p3.7.m6.2.3"><csymbol cd="ambiguous" id="S2.SS5.p3.7.m6.2.3.2.1.cmml" xref="S2.SS5.p3.7.m6.2.3">subscript</csymbol><ci id="S2.SS5.p3.7.m6.2.3.2.2.cmml" xref="S2.SS5.p3.7.m6.2.3.2.2">𝑡</ci><list id="S2.SS5.p3.7.m6.2.2.2.3.cmml" xref="S2.SS5.p3.7.m6.2.2.2.4"><cn id="S2.SS5.p3.7.m6.1.1.1.1.cmml" type="integer" xref="S2.SS5.p3.7.m6.1.1.1.1">2</cn><ci id="S2.SS5.p3.7.m6.2.2.2.2.cmml" xref="S2.SS5.p3.7.m6.2.2.2.2">𝛼</ci></list></apply><ci id="S2.SS5.p3.7.m6.2.3.3.cmml" xref="S2.SS5.p3.7.m6.2.3.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.7.m6.2c">t_{2;\alpha}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.7.m6.2d">italic_t start_POSTSUBSCRIPT 2 ; italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>, the <math alttext="\alpha" class="ltx_Math" display="inline" id="S2.SS5.p3.8.m7.1"><semantics id="S2.SS5.p3.8.m7.1a"><mi id="S2.SS5.p3.8.m7.1.1" xref="S2.SS5.p3.8.m7.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.8.m7.1b"><ci id="S2.SS5.p3.8.m7.1.1.cmml" xref="S2.SS5.p3.8.m7.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.8.m7.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.8.m7.1d">italic_α</annotation></semantics></math>th percentile of (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E8" title="In 2.5 Sphericity test ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">8</span></a>).</p> </div> <section class="ltx_subsubsection" id="S2.SS5.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">2.5.1 </span>Independence test</h4> <div class="ltx_para" id="S2.SS5.SSS1.p1"> <p class="ltx_p" id="S2.SS5.SSS1.p1.1">Analyzing the independence between two subsets of variables is critical to understanding the variables’ structure. It is applicable for feature selection, for dimensionality reduction, and various other processes. The independence test serves to determine whether two subsets of variables behave independently or if there are any statistical relations between the two subsets <cite class="ltx_cite ltx_citemacro_citep">(Moura, Coelho, & Sinha, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib19" title="">2024</a>)</cite>.</p> </div> <div class="ltx_para" id="S2.SS5.SSS1.p2"> <p class="ltx_p" id="S2.SS5.SSS1.p2.8">For the application of the independence test, <math alttext="\mathbf{x}_{i}" class="ltx_Math" display="inline" id="S2.SS5.SSS1.p2.1.m1.1"><semantics id="S2.SS5.SSS1.p2.1.m1.1a"><msub id="S2.SS5.SSS1.p2.1.m1.1.1" xref="S2.SS5.SSS1.p2.1.m1.1.1.cmml"><mi id="S2.SS5.SSS1.p2.1.m1.1.1.2" xref="S2.SS5.SSS1.p2.1.m1.1.1.2.cmml">𝐱</mi><mi id="S2.SS5.SSS1.p2.1.m1.1.1.3" xref="S2.SS5.SSS1.p2.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS1.p2.1.m1.1b"><apply id="S2.SS5.SSS1.p2.1.m1.1.1.cmml" xref="S2.SS5.SSS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p2.1.m1.1.1.1.cmml" xref="S2.SS5.SSS1.p2.1.m1.1.1">subscript</csymbol><ci id="S2.SS5.SSS1.p2.1.m1.1.1.2.cmml" xref="S2.SS5.SSS1.p2.1.m1.1.1.2">𝐱</ci><ci id="S2.SS5.SSS1.p2.1.m1.1.1.3.cmml" xref="S2.SS5.SSS1.p2.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS1.p2.1.m1.1c">\mathbf{x}_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS1.p2.1.m1.1d">bold_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E1" title="In 2.3 Generating fully synthetic data ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">1</span></a>) must be divided into two subvectors or subsets of variables <math alttext="\mathbf{x}_{i}=\begin{bmatrix}\mathbf{x}_{1i}\\ \mathbf{x}_{2i}\end{bmatrix}" class="ltx_Math" display="inline" id="S2.SS5.SSS1.p2.2.m2.1"><semantics id="S2.SS5.SSS1.p2.2.m2.1a"><mrow id="S2.SS5.SSS1.p2.2.m2.1.2" xref="S2.SS5.SSS1.p2.2.m2.1.2.cmml"><msub id="S2.SS5.SSS1.p2.2.m2.1.2.2" xref="S2.SS5.SSS1.p2.2.m2.1.2.2.cmml"><mi id="S2.SS5.SSS1.p2.2.m2.1.2.2.2" xref="S2.SS5.SSS1.p2.2.m2.1.2.2.2.cmml">𝐱</mi><mi id="S2.SS5.SSS1.p2.2.m2.1.2.2.3" xref="S2.SS5.SSS1.p2.2.m2.1.2.2.3.cmml">i</mi></msub><mo id="S2.SS5.SSS1.p2.2.m2.1.2.1" xref="S2.SS5.SSS1.p2.2.m2.1.2.1.cmml">=</mo><mrow id="S2.SS5.SSS1.p2.2.m2.1.1.3" xref="S2.SS5.SSS1.p2.2.m2.1.1.2.cmml"><mo id="S2.SS5.SSS1.p2.2.m2.1.1.3.1" xref="S2.SS5.SSS1.p2.2.m2.1.1.2.1.cmml">[</mo><mtable id="S2.SS5.SSS1.p2.2.m2.1.1.1.1" rowspacing="0pt" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.cmml"><mtr id="S2.SS5.SSS1.p2.2.m2.1.1.1.1a" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.cmml"><mtd id="S2.SS5.SSS1.p2.2.m2.1.1.1.1b" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.cmml"><msub id="S2.SS5.SSS1.p2.2.m2.1.1.1.1.1.1.1" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.1.1.1.cmml"><mi 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xref="S2.SS5.SSS1.p2.2.m2.1.2.2.3">𝑖</ci></apply><apply id="S2.SS5.SSS1.p2.2.m2.1.1.2.cmml" xref="S2.SS5.SSS1.p2.2.m2.1.1.3"><csymbol cd="latexml" id="S2.SS5.SSS1.p2.2.m2.1.1.2.1.cmml" xref="S2.SS5.SSS1.p2.2.m2.1.1.3.1">matrix</csymbol><matrix id="S2.SS5.SSS1.p2.2.m2.1.1.1.1.cmml" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1"><matrixrow id="S2.SS5.SSS1.p2.2.m2.1.1.1.1a.cmml" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1"><apply id="S2.SS5.SSS1.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p2.2.m2.1.1.1.1.1.1.1.1.cmml" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS5.SSS1.p2.2.m2.1.1.1.1.1.1.1.2.cmml" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.1.1.1.2">𝐱</ci><apply id="S2.SS5.SSS1.p2.2.m2.1.1.1.1.1.1.1.3.cmml" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.1.1.1.3"><times id="S2.SS5.SSS1.p2.2.m2.1.1.1.1.1.1.1.3.1.cmml" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.1.1.1.3.1"></times><cn id="S2.SS5.SSS1.p2.2.m2.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.1.1.1.3.2">1</cn><ci id="S2.SS5.SSS1.p2.2.m2.1.1.1.1.1.1.1.3.3.cmml" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.1.1.1.3.3">𝑖</ci></apply></apply></matrixrow><matrixrow id="S2.SS5.SSS1.p2.2.m2.1.1.1.1b.cmml" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1"><apply id="S2.SS5.SSS1.p2.2.m2.1.1.1.1.2.1.1.cmml" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.2.1.1"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p2.2.m2.1.1.1.1.2.1.1.1.cmml" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.2.1.1">subscript</csymbol><ci id="S2.SS5.SSS1.p2.2.m2.1.1.1.1.2.1.1.2.cmml" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.2.1.1.2">𝐱</ci><apply id="S2.SS5.SSS1.p2.2.m2.1.1.1.1.2.1.1.3.cmml" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.2.1.1.3"><times id="S2.SS5.SSS1.p2.2.m2.1.1.1.1.2.1.1.3.1.cmml" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.2.1.1.3.1"></times><cn id="S2.SS5.SSS1.p2.2.m2.1.1.1.1.2.1.1.3.2.cmml" type="integer" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.2.1.1.3.2">2</cn><ci id="S2.SS5.SSS1.p2.2.m2.1.1.1.1.2.1.1.3.3.cmml" xref="S2.SS5.SSS1.p2.2.m2.1.1.1.1.2.1.1.3.3">𝑖</ci></apply></apply></matrixrow></matrix></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS1.p2.2.m2.1c">\mathbf{x}_{i}=\begin{bmatrix}\mathbf{x}_{1i}\\ \mathbf{x}_{2i}\end{bmatrix}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS1.p2.2.m2.1d">bold_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = [ start_ARG start_ROW start_CELL bold_x start_POSTSUBSCRIPT 1 italic_i end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL bold_x start_POSTSUBSCRIPT 2 italic_i end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ]</annotation></semantics></math> where <math alttext="\mathbf{x}_{1i}" class="ltx_Math" display="inline" id="S2.SS5.SSS1.p2.3.m3.1"><semantics id="S2.SS5.SSS1.p2.3.m3.1a"><msub id="S2.SS5.SSS1.p2.3.m3.1.1" xref="S2.SS5.SSS1.p2.3.m3.1.1.cmml"><mi id="S2.SS5.SSS1.p2.3.m3.1.1.2" xref="S2.SS5.SSS1.p2.3.m3.1.1.2.cmml">𝐱</mi><mrow id="S2.SS5.SSS1.p2.3.m3.1.1.3" xref="S2.SS5.SSS1.p2.3.m3.1.1.3.cmml"><mn id="S2.SS5.SSS1.p2.3.m3.1.1.3.2" xref="S2.SS5.SSS1.p2.3.m3.1.1.3.2.cmml">1</mn><mo id="S2.SS5.SSS1.p2.3.m3.1.1.3.1" xref="S2.SS5.SSS1.p2.3.m3.1.1.3.1.cmml"></mo><mi id="S2.SS5.SSS1.p2.3.m3.1.1.3.3" xref="S2.SS5.SSS1.p2.3.m3.1.1.3.3.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS1.p2.3.m3.1b"><apply id="S2.SS5.SSS1.p2.3.m3.1.1.cmml" xref="S2.SS5.SSS1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p2.3.m3.1.1.1.cmml" xref="S2.SS5.SSS1.p2.3.m3.1.1">subscript</csymbol><ci id="S2.SS5.SSS1.p2.3.m3.1.1.2.cmml" xref="S2.SS5.SSS1.p2.3.m3.1.1.2">𝐱</ci><apply id="S2.SS5.SSS1.p2.3.m3.1.1.3.cmml" xref="S2.SS5.SSS1.p2.3.m3.1.1.3"><times id="S2.SS5.SSS1.p2.3.m3.1.1.3.1.cmml" xref="S2.SS5.SSS1.p2.3.m3.1.1.3.1"></times><cn id="S2.SS5.SSS1.p2.3.m3.1.1.3.2.cmml" type="integer" xref="S2.SS5.SSS1.p2.3.m3.1.1.3.2">1</cn><ci id="S2.SS5.SSS1.p2.3.m3.1.1.3.3.cmml" xref="S2.SS5.SSS1.p2.3.m3.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS1.p2.3.m3.1c">\mathbf{x}_{1i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS1.p2.3.m3.1d">bold_x start_POSTSUBSCRIPT 1 italic_i end_POSTSUBSCRIPT</annotation></semantics></math> will be of size <math alttext="p_{1}" class="ltx_Math" display="inline" id="S2.SS5.SSS1.p2.4.m4.1"><semantics id="S2.SS5.SSS1.p2.4.m4.1a"><msub id="S2.SS5.SSS1.p2.4.m4.1.1" xref="S2.SS5.SSS1.p2.4.m4.1.1.cmml"><mi id="S2.SS5.SSS1.p2.4.m4.1.1.2" xref="S2.SS5.SSS1.p2.4.m4.1.1.2.cmml">p</mi><mn id="S2.SS5.SSS1.p2.4.m4.1.1.3" xref="S2.SS5.SSS1.p2.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS1.p2.4.m4.1b"><apply id="S2.SS5.SSS1.p2.4.m4.1.1.cmml" xref="S2.SS5.SSS1.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p2.4.m4.1.1.1.cmml" xref="S2.SS5.SSS1.p2.4.m4.1.1">subscript</csymbol><ci id="S2.SS5.SSS1.p2.4.m4.1.1.2.cmml" xref="S2.SS5.SSS1.p2.4.m4.1.1.2">𝑝</ci><cn id="S2.SS5.SSS1.p2.4.m4.1.1.3.cmml" type="integer" xref="S2.SS5.SSS1.p2.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS1.p2.4.m4.1c">p_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS1.p2.4.m4.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\mathbf{x}_{2i}" class="ltx_Math" display="inline" id="S2.SS5.SSS1.p2.5.m5.1"><semantics id="S2.SS5.SSS1.p2.5.m5.1a"><msub id="S2.SS5.SSS1.p2.5.m5.1.1" xref="S2.SS5.SSS1.p2.5.m5.1.1.cmml"><mi id="S2.SS5.SSS1.p2.5.m5.1.1.2" xref="S2.SS5.SSS1.p2.5.m5.1.1.2.cmml">𝐱</mi><mrow id="S2.SS5.SSS1.p2.5.m5.1.1.3" xref="S2.SS5.SSS1.p2.5.m5.1.1.3.cmml"><mn id="S2.SS5.SSS1.p2.5.m5.1.1.3.2" xref="S2.SS5.SSS1.p2.5.m5.1.1.3.2.cmml">2</mn><mo id="S2.SS5.SSS1.p2.5.m5.1.1.3.1" xref="S2.SS5.SSS1.p2.5.m5.1.1.3.1.cmml"></mo><mi id="S2.SS5.SSS1.p2.5.m5.1.1.3.3" xref="S2.SS5.SSS1.p2.5.m5.1.1.3.3.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS1.p2.5.m5.1b"><apply id="S2.SS5.SSS1.p2.5.m5.1.1.cmml" xref="S2.SS5.SSS1.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p2.5.m5.1.1.1.cmml" xref="S2.SS5.SSS1.p2.5.m5.1.1">subscript</csymbol><ci id="S2.SS5.SSS1.p2.5.m5.1.1.2.cmml" xref="S2.SS5.SSS1.p2.5.m5.1.1.2">𝐱</ci><apply id="S2.SS5.SSS1.p2.5.m5.1.1.3.cmml" xref="S2.SS5.SSS1.p2.5.m5.1.1.3"><times id="S2.SS5.SSS1.p2.5.m5.1.1.3.1.cmml" xref="S2.SS5.SSS1.p2.5.m5.1.1.3.1"></times><cn id="S2.SS5.SSS1.p2.5.m5.1.1.3.2.cmml" type="integer" xref="S2.SS5.SSS1.p2.5.m5.1.1.3.2">2</cn><ci id="S2.SS5.SSS1.p2.5.m5.1.1.3.3.cmml" xref="S2.SS5.SSS1.p2.5.m5.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS1.p2.5.m5.1c">\mathbf{x}_{2i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS1.p2.5.m5.1d">bold_x start_POSTSUBSCRIPT 2 italic_i end_POSTSUBSCRIPT</annotation></semantics></math> will be of size <math alttext="p-p_{1}" class="ltx_Math" display="inline" id="S2.SS5.SSS1.p2.6.m6.1"><semantics id="S2.SS5.SSS1.p2.6.m6.1a"><mrow id="S2.SS5.SSS1.p2.6.m6.1.1" xref="S2.SS5.SSS1.p2.6.m6.1.1.cmml"><mi id="S2.SS5.SSS1.p2.6.m6.1.1.2" xref="S2.SS5.SSS1.p2.6.m6.1.1.2.cmml">p</mi><mo id="S2.SS5.SSS1.p2.6.m6.1.1.1" xref="S2.SS5.SSS1.p2.6.m6.1.1.1.cmml">−</mo><msub id="S2.SS5.SSS1.p2.6.m6.1.1.3" xref="S2.SS5.SSS1.p2.6.m6.1.1.3.cmml"><mi id="S2.SS5.SSS1.p2.6.m6.1.1.3.2" xref="S2.SS5.SSS1.p2.6.m6.1.1.3.2.cmml">p</mi><mn id="S2.SS5.SSS1.p2.6.m6.1.1.3.3" xref="S2.SS5.SSS1.p2.6.m6.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS1.p2.6.m6.1b"><apply id="S2.SS5.SSS1.p2.6.m6.1.1.cmml" xref="S2.SS5.SSS1.p2.6.m6.1.1"><minus id="S2.SS5.SSS1.p2.6.m6.1.1.1.cmml" xref="S2.SS5.SSS1.p2.6.m6.1.1.1"></minus><ci id="S2.SS5.SSS1.p2.6.m6.1.1.2.cmml" xref="S2.SS5.SSS1.p2.6.m6.1.1.2">𝑝</ci><apply id="S2.SS5.SSS1.p2.6.m6.1.1.3.cmml" xref="S2.SS5.SSS1.p2.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p2.6.m6.1.1.3.1.cmml" xref="S2.SS5.SSS1.p2.6.m6.1.1.3">subscript</csymbol><ci id="S2.SS5.SSS1.p2.6.m6.1.1.3.2.cmml" xref="S2.SS5.SSS1.p2.6.m6.1.1.3.2">𝑝</ci><cn id="S2.SS5.SSS1.p2.6.m6.1.1.3.3.cmml" type="integer" xref="S2.SS5.SSS1.p2.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS1.p2.6.m6.1c">p-p_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS1.p2.6.m6.1d">italic_p - italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Consequently, <math alttext="\boldsymbol{\Sigma}" class="ltx_Math" display="inline" id="S2.SS5.SSS1.p2.7.m7.1"><semantics id="S2.SS5.SSS1.p2.7.m7.1a"><mi id="S2.SS5.SSS1.p2.7.m7.1.1" xref="S2.SS5.SSS1.p2.7.m7.1.1.cmml">𝚺</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS1.p2.7.m7.1b"><ci id="S2.SS5.SSS1.p2.7.m7.1.1.cmml" xref="S2.SS5.SSS1.p2.7.m7.1.1">𝚺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS1.p2.7.m7.1c">\boldsymbol{\Sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS1.p2.7.m7.1d">bold_Σ</annotation></semantics></math> and <math alttext="\mathbf{S}^{\star}" class="ltx_Math" display="inline" id="S2.SS5.SSS1.p2.8.m8.1"><semantics id="S2.SS5.SSS1.p2.8.m8.1a"><msup id="S2.SS5.SSS1.p2.8.m8.1.1" xref="S2.SS5.SSS1.p2.8.m8.1.1.cmml"><mi id="S2.SS5.SSS1.p2.8.m8.1.1.2" xref="S2.SS5.SSS1.p2.8.m8.1.1.2.cmml">𝐒</mi><mo id="S2.SS5.SSS1.p2.8.m8.1.1.3" xref="S2.SS5.SSS1.p2.8.m8.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS1.p2.8.m8.1b"><apply id="S2.SS5.SSS1.p2.8.m8.1.1.cmml" xref="S2.SS5.SSS1.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p2.8.m8.1.1.1.cmml" xref="S2.SS5.SSS1.p2.8.m8.1.1">superscript</csymbol><ci id="S2.SS5.SSS1.p2.8.m8.1.1.2.cmml" xref="S2.SS5.SSS1.p2.8.m8.1.1.2">𝐒</ci><ci id="S2.SS5.SSS1.p2.8.m8.1.1.3.cmml" xref="S2.SS5.SSS1.p2.8.m8.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS1.p2.8.m8.1c">\mathbf{S}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS1.p2.8.m8.1d">bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> will be partitioned as follows:</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="\boldsymbol{\Sigma}=\begin{bmatrix}\boldsymbol{\Sigma}_{11}&\boldsymbol{\Sigma% }_{12}\\ \boldsymbol{\Sigma}_{21}&\boldsymbol{\Sigma}_{22}\end{bmatrix},\quad\mathbf{S}% ^{\star}=\begin{bmatrix}\mathbf{S}^{\star}_{11}&\mathbf{S}^{\star}_{12}\\ \mathbf{S}^{\star}_{21}&\mathbf{S}^{\star}_{22}\end{bmatrix}," class="ltx_Math" display="block" id="S2.E9.m1.3"><semantics id="S2.E9.m1.3a"><mrow id="S2.E9.m1.3.3.1"><mrow id="S2.E9.m1.3.3.1.1.2" xref="S2.E9.m1.3.3.1.1.3.cmml"><mrow id="S2.E9.m1.3.3.1.1.1.1" xref="S2.E9.m1.3.3.1.1.1.1.cmml"><mi id="S2.E9.m1.3.3.1.1.1.1.2" xref="S2.E9.m1.3.3.1.1.1.1.2.cmml">𝚺</mi><mo id="S2.E9.m1.3.3.1.1.1.1.1" xref="S2.E9.m1.3.3.1.1.1.1.1.cmml">=</mo><mrow id="S2.E9.m1.1.1.3" xref="S2.E9.m1.1.1.2.cmml"><mo id="S2.E9.m1.1.1.3.1" xref="S2.E9.m1.1.1.2.1.cmml">[</mo><mtable columnspacing="5pt" displaystyle="true" id="S2.E9.m1.1.1.1.1" rowspacing="0pt" xref="S2.E9.m1.1.1.1.1.cmml"><mtr id="S2.E9.m1.1.1.1.1a" xref="S2.E9.m1.1.1.1.1.cmml"><mtd id="S2.E9.m1.1.1.1.1b" xref="S2.E9.m1.1.1.1.1.cmml"><msub id="S2.E9.m1.1.1.1.1.1.1.1" xref="S2.E9.m1.1.1.1.1.1.1.1.cmml"><mi id="S2.E9.m1.1.1.1.1.1.1.1.2" xref="S2.E9.m1.1.1.1.1.1.1.1.2.cmml">𝚺</mi><mn id="S2.E9.m1.1.1.1.1.1.1.1.3" xref="S2.E9.m1.1.1.1.1.1.1.1.3.cmml">11</mn></msub></mtd><mtd id="S2.E9.m1.1.1.1.1c" xref="S2.E9.m1.1.1.1.1.cmml"><msub id="S2.E9.m1.1.1.1.1.1.2.1" xref="S2.E9.m1.1.1.1.1.1.2.1.cmml"><mi id="S2.E9.m1.1.1.1.1.1.2.1.2" xref="S2.E9.m1.1.1.1.1.1.2.1.2.cmml">𝚺</mi><mn id="S2.E9.m1.1.1.1.1.1.2.1.3" xref="S2.E9.m1.1.1.1.1.1.2.1.3.cmml">12</mn></msub></mtd></mtr><mtr id="S2.E9.m1.1.1.1.1d" xref="S2.E9.m1.1.1.1.1.cmml"><mtd id="S2.E9.m1.1.1.1.1e" xref="S2.E9.m1.1.1.1.1.cmml"><msub id="S2.E9.m1.1.1.1.1.2.1.1" xref="S2.E9.m1.1.1.1.1.2.1.1.cmml"><mi id="S2.E9.m1.1.1.1.1.2.1.1.2" xref="S2.E9.m1.1.1.1.1.2.1.1.2.cmml">𝚺</mi><mn id="S2.E9.m1.1.1.1.1.2.1.1.3" xref="S2.E9.m1.1.1.1.1.2.1.1.3.cmml">21</mn></msub></mtd><mtd id="S2.E9.m1.1.1.1.1f" xref="S2.E9.m1.1.1.1.1.cmml"><msub id="S2.E9.m1.1.1.1.1.2.2.1" xref="S2.E9.m1.1.1.1.1.2.2.1.cmml"><mi id="S2.E9.m1.1.1.1.1.2.2.1.2" xref="S2.E9.m1.1.1.1.1.2.2.1.2.cmml">𝚺</mi><mn id="S2.E9.m1.1.1.1.1.2.2.1.3" xref="S2.E9.m1.1.1.1.1.2.2.1.3.cmml">22</mn></msub></mtd></mtr></mtable><mo id="S2.E9.m1.1.1.3.2" xref="S2.E9.m1.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S2.E9.m1.3.3.1.1.2.3" rspace="1.167em" xref="S2.E9.m1.3.3.1.1.3a.cmml">,</mo><mrow id="S2.E9.m1.3.3.1.1.2.2" xref="S2.E9.m1.3.3.1.1.2.2.cmml"><msup id="S2.E9.m1.3.3.1.1.2.2.2" xref="S2.E9.m1.3.3.1.1.2.2.2.cmml"><mi id="S2.E9.m1.3.3.1.1.2.2.2.2" xref="S2.E9.m1.3.3.1.1.2.2.2.2.cmml">𝐒</mi><mo id="S2.E9.m1.3.3.1.1.2.2.2.3" xref="S2.E9.m1.3.3.1.1.2.2.2.3.cmml">⋆</mo></msup><mo id="S2.E9.m1.3.3.1.1.2.2.1" xref="S2.E9.m1.3.3.1.1.2.2.1.cmml">=</mo><mrow id="S2.E9.m1.2.2.3" xref="S2.E9.m1.2.2.2.cmml"><mo id="S2.E9.m1.2.2.3.1" xref="S2.E9.m1.2.2.2.1.cmml">[</mo><mtable columnspacing="5pt" displaystyle="true" id="S2.E9.m1.2.2.1.1" rowspacing="0pt" xref="S2.E9.m1.2.2.1.1.cmml"><mtr id="S2.E9.m1.2.2.1.1a" xref="S2.E9.m1.2.2.1.1.cmml"><mtd id="S2.E9.m1.2.2.1.1b" xref="S2.E9.m1.2.2.1.1.cmml"><msubsup id="S2.E9.m1.2.2.1.1.1.1.1" xref="S2.E9.m1.2.2.1.1.1.1.1.cmml"><mi id="S2.E9.m1.2.2.1.1.1.1.1.2.2" xref="S2.E9.m1.2.2.1.1.1.1.1.2.2.cmml">𝐒</mi><mn id="S2.E9.m1.2.2.1.1.1.1.1.3" xref="S2.E9.m1.2.2.1.1.1.1.1.3.cmml">11</mn><mo id="S2.E9.m1.2.2.1.1.1.1.1.2.3" xref="S2.E9.m1.2.2.1.1.1.1.1.2.3.cmml">⋆</mo></msubsup></mtd><mtd id="S2.E9.m1.2.2.1.1c" xref="S2.E9.m1.2.2.1.1.cmml"><msubsup id="S2.E9.m1.2.2.1.1.1.2.1" xref="S2.E9.m1.2.2.1.1.1.2.1.cmml"><mi id="S2.E9.m1.2.2.1.1.1.2.1.2.2" xref="S2.E9.m1.2.2.1.1.1.2.1.2.2.cmml">𝐒</mi><mn id="S2.E9.m1.2.2.1.1.1.2.1.3" xref="S2.E9.m1.2.2.1.1.1.2.1.3.cmml">12</mn><mo id="S2.E9.m1.2.2.1.1.1.2.1.2.3" xref="S2.E9.m1.2.2.1.1.1.2.1.2.3.cmml">⋆</mo></msubsup></mtd></mtr><mtr id="S2.E9.m1.2.2.1.1d" xref="S2.E9.m1.2.2.1.1.cmml"><mtd id="S2.E9.m1.2.2.1.1e" 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encoding="application/x-llamapun" id="S2.E9.m1.3d">bold_Σ = [ start_ARG start_ROW start_CELL bold_Σ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL start_CELL bold_Σ start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL bold_Σ start_POSTSUBSCRIPT 21 end_POSTSUBSCRIPT end_CELL start_CELL bold_Σ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] , bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT = [ start_ARG start_ROW start_CELL bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL start_CELL bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 21 end_POSTSUBSCRIPT end_CELL start_CELL bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT end_CELL end_ROW 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">(9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS5.SSS1.p2.11">where both <math alttext="\boldsymbol{\Sigma}_{11}" class="ltx_Math" display="inline" id="S2.SS5.SSS1.p2.9.m1.1"><semantics id="S2.SS5.SSS1.p2.9.m1.1a"><msub id="S2.SS5.SSS1.p2.9.m1.1.1" xref="S2.SS5.SSS1.p2.9.m1.1.1.cmml"><mi id="S2.SS5.SSS1.p2.9.m1.1.1.2" xref="S2.SS5.SSS1.p2.9.m1.1.1.2.cmml">𝚺</mi><mn id="S2.SS5.SSS1.p2.9.m1.1.1.3" xref="S2.SS5.SSS1.p2.9.m1.1.1.3.cmml">11</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS1.p2.9.m1.1b"><apply id="S2.SS5.SSS1.p2.9.m1.1.1.cmml" xref="S2.SS5.SSS1.p2.9.m1.1.1"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p2.9.m1.1.1.1.cmml" xref="S2.SS5.SSS1.p2.9.m1.1.1">subscript</csymbol><ci id="S2.SS5.SSS1.p2.9.m1.1.1.2.cmml" xref="S2.SS5.SSS1.p2.9.m1.1.1.2">𝚺</ci><cn 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xref="S2.SS5.SSS1.p2.11.m3.1.1.1"></times><apply id="S2.SS5.SSS1.p2.11.m3.1.1.2.cmml" xref="S2.SS5.SSS1.p2.11.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p2.11.m3.1.1.2.1.cmml" xref="S2.SS5.SSS1.p2.11.m3.1.1.2">subscript</csymbol><ci id="S2.SS5.SSS1.p2.11.m3.1.1.2.2.cmml" xref="S2.SS5.SSS1.p2.11.m3.1.1.2.2">𝑝</ci><cn id="S2.SS5.SSS1.p2.11.m3.1.1.2.3.cmml" type="integer" xref="S2.SS5.SSS1.p2.11.m3.1.1.2.3">1</cn></apply><apply id="S2.SS5.SSS1.p2.11.m3.1.1.3.cmml" xref="S2.SS5.SSS1.p2.11.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p2.11.m3.1.1.3.1.cmml" xref="S2.SS5.SSS1.p2.11.m3.1.1.3">subscript</csymbol><ci id="S2.SS5.SSS1.p2.11.m3.1.1.3.2.cmml" xref="S2.SS5.SSS1.p2.11.m3.1.1.3.2">𝑝</ci><cn id="S2.SS5.SSS1.p2.11.m3.1.1.3.3.cmml" type="integer" xref="S2.SS5.SSS1.p2.11.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS1.p2.11.m3.1c">p_{1}\times p_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS1.p2.11.m3.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> matrices.</p> </div> <div class="ltx_para" id="S2.SS5.SSS1.p3"> <p class="ltx_p" id="S2.SS5.SSS1.p3.1">To test the hypothesis of independence of the two subsets of variables, we test</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="\mathcal{H}_{0}:\boldsymbol{\Sigma}_{12}=\mathbf{0}~{}~{}\text{vs.}~{}~{}% \mathcal{H}_{1}:\boldsymbol{\Sigma}_{12}\neq\mathbf{0}." class="ltx_Math" display="block" id="S2.E10.m1.1"><semantics id="S2.E10.m1.1a"><mrow id="S2.E10.m1.1.1.1" xref="S2.E10.m1.1.1.1.1.cmml"><mrow id="S2.E10.m1.1.1.1.1" xref="S2.E10.m1.1.1.1.1.cmml"><msub id="S2.E10.m1.1.1.1.1.2" xref="S2.E10.m1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E10.m1.1.1.1.1.2.2" xref="S2.E10.m1.1.1.1.1.2.2.cmml">ℋ</mi><mn id="S2.E10.m1.1.1.1.1.2.3" xref="S2.E10.m1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S2.E10.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S2.E10.m1.1.1.1.1.3.cmml">:</mo><mrow id="S2.E10.m1.1.1.1.1.4" xref="S2.E10.m1.1.1.1.1.4.cmml"><msub id="S2.E10.m1.1.1.1.1.4.2" xref="S2.E10.m1.1.1.1.1.4.2.cmml"><mi id="S2.E10.m1.1.1.1.1.4.2.2" xref="S2.E10.m1.1.1.1.1.4.2.2.cmml">𝚺</mi><mn id="S2.E10.m1.1.1.1.1.4.2.3" xref="S2.E10.m1.1.1.1.1.4.2.3.cmml">12</mn></msub><mo id="S2.E10.m1.1.1.1.1.4.1" xref="S2.E10.m1.1.1.1.1.4.1.cmml">=</mo><mrow id="S2.E10.m1.1.1.1.1.4.3" xref="S2.E10.m1.1.1.1.1.4.3.cmml"><mn id="S2.E10.m1.1.1.1.1.4.3.2" xref="S2.E10.m1.1.1.1.1.4.3.2.cmml">𝟎</mn><mo id="S2.E10.m1.1.1.1.1.4.3.1" lspace="0.660em" xref="S2.E10.m1.1.1.1.1.4.3.1.cmml"></mo><mtext id="S2.E10.m1.1.1.1.1.4.3.3" xref="S2.E10.m1.1.1.1.1.4.3.3a.cmml">vs.</mtext><mo id="S2.E10.m1.1.1.1.1.4.3.1a" lspace="0.660em" xref="S2.E10.m1.1.1.1.1.4.3.1.cmml"></mo><msub id="S2.E10.m1.1.1.1.1.4.3.4" xref="S2.E10.m1.1.1.1.1.4.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E10.m1.1.1.1.1.4.3.4.2" xref="S2.E10.m1.1.1.1.1.4.3.4.2.cmml">ℋ</mi><mn id="S2.E10.m1.1.1.1.1.4.3.4.3" xref="S2.E10.m1.1.1.1.1.4.3.4.3.cmml">1</mn></msub></mrow></mrow><mo id="S2.E10.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S2.E10.m1.1.1.1.1.5.cmml">:</mo><mrow id="S2.E10.m1.1.1.1.1.6" xref="S2.E10.m1.1.1.1.1.6.cmml"><msub id="S2.E10.m1.1.1.1.1.6.2" xref="S2.E10.m1.1.1.1.1.6.2.cmml"><mi id="S2.E10.m1.1.1.1.1.6.2.2" xref="S2.E10.m1.1.1.1.1.6.2.2.cmml">𝚺</mi><mn id="S2.E10.m1.1.1.1.1.6.2.3" xref="S2.E10.m1.1.1.1.1.6.2.3.cmml">12</mn></msub><mo id="S2.E10.m1.1.1.1.1.6.1" xref="S2.E10.m1.1.1.1.1.6.1.cmml">≠</mo><mn id="S2.E10.m1.1.1.1.1.6.3" xref="S2.E10.m1.1.1.1.1.6.3.cmml">𝟎</mn></mrow></mrow><mo id="S2.E10.m1.1.1.1.2" lspace="0em" xref="S2.E10.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml 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xref="S2.E10.m1.1.1.1.1.4.2.2">𝚺</ci><cn id="S2.E10.m1.1.1.1.1.4.2.3.cmml" type="integer" xref="S2.E10.m1.1.1.1.1.4.2.3">12</cn></apply><apply id="S2.E10.m1.1.1.1.1.4.3.cmml" xref="S2.E10.m1.1.1.1.1.4.3"><times id="S2.E10.m1.1.1.1.1.4.3.1.cmml" xref="S2.E10.m1.1.1.1.1.4.3.1"></times><cn id="S2.E10.m1.1.1.1.1.4.3.2.cmml" type="integer" xref="S2.E10.m1.1.1.1.1.4.3.2">0</cn><ci id="S2.E10.m1.1.1.1.1.4.3.3a.cmml" xref="S2.E10.m1.1.1.1.1.4.3.3"><mtext id="S2.E10.m1.1.1.1.1.4.3.3.cmml" xref="S2.E10.m1.1.1.1.1.4.3.3">vs.</mtext></ci><apply id="S2.E10.m1.1.1.1.1.4.3.4.cmml" xref="S2.E10.m1.1.1.1.1.4.3.4"><csymbol cd="ambiguous" id="S2.E10.m1.1.1.1.1.4.3.4.1.cmml" xref="S2.E10.m1.1.1.1.1.4.3.4">subscript</csymbol><ci id="S2.E10.m1.1.1.1.1.4.3.4.2.cmml" xref="S2.E10.m1.1.1.1.1.4.3.4.2">ℋ</ci><cn id="S2.E10.m1.1.1.1.1.4.3.4.3.cmml" type="integer" xref="S2.E10.m1.1.1.1.1.4.3.4.3">1</cn></apply></apply></apply></apply><apply id="S2.E10.m1.1.1.1.1c.cmml" xref="S2.E10.m1.1.1.1"><ci id="S2.E10.m1.1.1.1.1.5.cmml" xref="S2.E10.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2503.14711v1#S2.E10.m1.1.1.1.1.4.cmml" id="S2.E10.m1.1.1.1.1d.cmml" xref="S2.E10.m1.1.1.1"></share><apply id="S2.E10.m1.1.1.1.1.6.cmml" xref="S2.E10.m1.1.1.1.1.6"><neq id="S2.E10.m1.1.1.1.1.6.1.cmml" xref="S2.E10.m1.1.1.1.1.6.1"></neq><apply id="S2.E10.m1.1.1.1.1.6.2.cmml" xref="S2.E10.m1.1.1.1.1.6.2"><csymbol cd="ambiguous" id="S2.E10.m1.1.1.1.1.6.2.1.cmml" xref="S2.E10.m1.1.1.1.1.6.2">subscript</csymbol><ci id="S2.E10.m1.1.1.1.1.6.2.2.cmml" xref="S2.E10.m1.1.1.1.1.6.2.2">𝚺</ci><cn id="S2.E10.m1.1.1.1.1.6.2.3.cmml" type="integer" xref="S2.E10.m1.1.1.1.1.6.2.3">12</cn></apply><cn id="S2.E10.m1.1.1.1.1.6.3.cmml" type="integer" xref="S2.E10.m1.1.1.1.1.6.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E10.m1.1c">\mathcal{H}_{0}:\boldsymbol{\Sigma}_{12}=\mathbf{0}~{}~{}\text{vs.}~{}~{}% \mathcal{H}_{1}:\boldsymbol{\Sigma}_{12}\neq\mathbf{0}.</annotation><annotation encoding="application/x-llamapun" id="S2.E10.m1.1d">caligraphic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT : bold_Σ start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT = bold_0 vs. caligraphic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : bold_Σ start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT ≠ bold_0 .</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> </div> <div class="ltx_para" id="S2.SS5.SSS1.p4"> <p class="ltx_p" id="S2.SS5.SSS1.p4.6"><cite class="ltx_cite ltx_citemacro_citet">Klein, Moura, & Sinha (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib8" title="">2021</a>)</cite> used</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="T_{3}^{\star}=\frac{\left|\mathbf{S}^{\star}\right|}{\left|\mathbf{S}^{\star}_% {11}\right|\left|\mathbf{S}^{\star}_{22}\right|}," class="ltx_Math" display="block" id="S2.E11.m1.4"><semantics id="S2.E11.m1.4a"><mrow id="S2.E11.m1.4.4.1" xref="S2.E11.m1.4.4.1.1.cmml"><mrow id="S2.E11.m1.4.4.1.1" xref="S2.E11.m1.4.4.1.1.cmml"><msubsup id="S2.E11.m1.4.4.1.1.2" xref="S2.E11.m1.4.4.1.1.2.cmml"><mi id="S2.E11.m1.4.4.1.1.2.2.2" xref="S2.E11.m1.4.4.1.1.2.2.2.cmml">T</mi><mn id="S2.E11.m1.4.4.1.1.2.2.3" xref="S2.E11.m1.4.4.1.1.2.2.3.cmml">3</mn><mo id="S2.E11.m1.4.4.1.1.2.3" xref="S2.E11.m1.4.4.1.1.2.3.cmml">⋆</mo></msubsup><mo id="S2.E11.m1.4.4.1.1.1" xref="S2.E11.m1.4.4.1.1.1.cmml">=</mo><mfrac id="S2.E11.m1.3.3" xref="S2.E11.m1.3.3.cmml"><mrow id="S2.E11.m1.1.1.1.1" xref="S2.E11.m1.1.1.1.2.cmml"><mo id="S2.E11.m1.1.1.1.1.2" xref="S2.E11.m1.1.1.1.2.1.cmml">|</mo><msup id="S2.E11.m1.1.1.1.1.1" xref="S2.E11.m1.1.1.1.1.1.cmml"><mi id="S2.E11.m1.1.1.1.1.1.2" xref="S2.E11.m1.1.1.1.1.1.2.cmml">𝐒</mi><mo id="S2.E11.m1.1.1.1.1.1.3" xref="S2.E11.m1.1.1.1.1.1.3.cmml">⋆</mo></msup><mo id="S2.E11.m1.1.1.1.1.3" xref="S2.E11.m1.1.1.1.2.1.cmml">|</mo></mrow><mrow id="S2.E11.m1.3.3.3" xref="S2.E11.m1.3.3.3.cmml"><mrow id="S2.E11.m1.2.2.2.1.1" xref="S2.E11.m1.2.2.2.1.2.cmml"><mo id="S2.E11.m1.2.2.2.1.1.2" xref="S2.E11.m1.2.2.2.1.2.1.cmml">|</mo><msubsup id="S2.E11.m1.2.2.2.1.1.1" xref="S2.E11.m1.2.2.2.1.1.1.cmml"><mi id="S2.E11.m1.2.2.2.1.1.1.2.2" xref="S2.E11.m1.2.2.2.1.1.1.2.2.cmml">𝐒</mi><mn id="S2.E11.m1.2.2.2.1.1.1.3" xref="S2.E11.m1.2.2.2.1.1.1.3.cmml">11</mn><mo id="S2.E11.m1.2.2.2.1.1.1.2.3" xref="S2.E11.m1.2.2.2.1.1.1.2.3.cmml">⋆</mo></msubsup><mo id="S2.E11.m1.2.2.2.1.1.3" xref="S2.E11.m1.2.2.2.1.2.1.cmml">|</mo></mrow><mo id="S2.E11.m1.3.3.3.3" xref="S2.E11.m1.3.3.3.3.cmml"></mo><mrow id="S2.E11.m1.3.3.3.2.1" 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id="S2.E11.m1.3.3.3.2.1.1.cmml" xref="S2.E11.m1.3.3.3.2.1.1"><csymbol cd="ambiguous" id="S2.E11.m1.3.3.3.2.1.1.1.cmml" xref="S2.E11.m1.3.3.3.2.1.1">subscript</csymbol><apply id="S2.E11.m1.3.3.3.2.1.1.2.cmml" xref="S2.E11.m1.3.3.3.2.1.1"><csymbol cd="ambiguous" id="S2.E11.m1.3.3.3.2.1.1.2.1.cmml" xref="S2.E11.m1.3.3.3.2.1.1">superscript</csymbol><ci id="S2.E11.m1.3.3.3.2.1.1.2.2.cmml" xref="S2.E11.m1.3.3.3.2.1.1.2.2">𝐒</ci><ci id="S2.E11.m1.3.3.3.2.1.1.2.3.cmml" xref="S2.E11.m1.3.3.3.2.1.1.2.3">⋆</ci></apply><cn id="S2.E11.m1.3.3.3.2.1.1.3.cmml" type="integer" xref="S2.E11.m1.3.3.3.2.1.1.3">22</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E11.m1.4c">T_{3}^{\star}=\frac{\left|\mathbf{S}^{\star}\right|}{\left|\mathbf{S}^{\star}_% {11}\right|\left|\mathbf{S}^{\star}_{22}\right|},</annotation><annotation encoding="application/x-llamapun" id="S2.E11.m1.4d">italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT = divide start_ARG | bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT | end_ARG start_ARG | bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT | | bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT | 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">(11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS5.SSS1.p4.7">and showed that its distribution is stochastically equivalent to the distribution of</p> <table class="ltx_equation ltx_eqn_table" id="S2.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="\frac{\left|\boldsymbol{\Omega}_{2}\right|}{\left|\boldsymbol{\Omega}_{2,11}% \right|\left|\boldsymbol{\Omega}_{2,22}\right|}" class="ltx_Math" display="block" id="S2.E12.m1.7"><semantics id="S2.E12.m1.7a"><mfrac id="S2.E12.m1.7.7" xref="S2.E12.m1.7.7.cmml"><mrow id="S2.E12.m1.1.1.1.1" xref="S2.E12.m1.1.1.1.2.cmml"><mo id="S2.E12.m1.1.1.1.1.2" xref="S2.E12.m1.1.1.1.2.1.cmml">|</mo><msub id="S2.E12.m1.1.1.1.1.1" xref="S2.E12.m1.1.1.1.1.1.cmml"><mi id="S2.E12.m1.1.1.1.1.1.2" xref="S2.E12.m1.1.1.1.1.1.2.cmml">𝛀</mi><mn id="S2.E12.m1.1.1.1.1.1.3" xref="S2.E12.m1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.E12.m1.1.1.1.1.3" xref="S2.E12.m1.1.1.1.2.1.cmml">|</mo></mrow><mrow id="S2.E12.m1.7.7.7" xref="S2.E12.m1.7.7.7.cmml"><mrow id="S2.E12.m1.6.6.6.5.1" xref="S2.E12.m1.6.6.6.5.2.cmml"><mo id="S2.E12.m1.6.6.6.5.1.2" xref="S2.E12.m1.6.6.6.5.2.1.cmml">|</mo><msub id="S2.E12.m1.6.6.6.5.1.1" xref="S2.E12.m1.6.6.6.5.1.1.cmml"><mi id="S2.E12.m1.6.6.6.5.1.1.2" xref="S2.E12.m1.6.6.6.5.1.1.2.cmml">𝛀</mi><mrow id="S2.E12.m1.3.3.3.2.2.4" xref="S2.E12.m1.3.3.3.2.2.3.cmml"><mn id="S2.E12.m1.2.2.2.1.1.1" xref="S2.E12.m1.2.2.2.1.1.1.cmml">2</mn><mo id="S2.E12.m1.3.3.3.2.2.4.1" xref="S2.E12.m1.3.3.3.2.2.3.cmml">,</mo><mn id="S2.E12.m1.3.3.3.2.2.2" xref="S2.E12.m1.3.3.3.2.2.2.cmml">11</mn></mrow></msub><mo id="S2.E12.m1.6.6.6.5.1.3" xref="S2.E12.m1.6.6.6.5.2.1.cmml">|</mo></mrow><mo id="S2.E12.m1.7.7.7.7" xref="S2.E12.m1.7.7.7.7.cmml"></mo><mrow id="S2.E12.m1.7.7.7.6.1" xref="S2.E12.m1.7.7.7.6.2.cmml"><mo id="S2.E12.m1.7.7.7.6.1.2" xref="S2.E12.m1.7.7.7.6.2.1.cmml">|</mo><msub id="S2.E12.m1.7.7.7.6.1.1" xref="S2.E12.m1.7.7.7.6.1.1.cmml"><mi id="S2.E12.m1.7.7.7.6.1.1.2" xref="S2.E12.m1.7.7.7.6.1.1.2.cmml">𝛀</mi><mrow id="S2.E12.m1.5.5.5.4.2.4" xref="S2.E12.m1.5.5.5.4.2.3.cmml"><mn id="S2.E12.m1.4.4.4.3.1.1" xref="S2.E12.m1.4.4.4.3.1.1.cmml">2</mn><mo id="S2.E12.m1.5.5.5.4.2.4.1" xref="S2.E12.m1.5.5.5.4.2.3.cmml">,</mo><mn id="S2.E12.m1.5.5.5.4.2.2" 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id="S2.E12.m1.6.6.6.5.2.1.cmml" xref="S2.E12.m1.6.6.6.5.1.2"></abs><apply id="S2.E12.m1.6.6.6.5.1.1.cmml" xref="S2.E12.m1.6.6.6.5.1.1"><csymbol cd="ambiguous" id="S2.E12.m1.6.6.6.5.1.1.1.cmml" xref="S2.E12.m1.6.6.6.5.1.1">subscript</csymbol><ci id="S2.E12.m1.6.6.6.5.1.1.2.cmml" xref="S2.E12.m1.6.6.6.5.1.1.2">𝛀</ci><list id="S2.E12.m1.3.3.3.2.2.3.cmml" xref="S2.E12.m1.3.3.3.2.2.4"><cn id="S2.E12.m1.2.2.2.1.1.1.cmml" type="integer" xref="S2.E12.m1.2.2.2.1.1.1">2</cn><cn id="S2.E12.m1.3.3.3.2.2.2.cmml" type="integer" xref="S2.E12.m1.3.3.3.2.2.2">11</cn></list></apply></apply><apply id="S2.E12.m1.7.7.7.6.2.cmml" xref="S2.E12.m1.7.7.7.6.1"><abs id="S2.E12.m1.7.7.7.6.2.1.cmml" xref="S2.E12.m1.7.7.7.6.1.2"></abs><apply id="S2.E12.m1.7.7.7.6.1.1.cmml" xref="S2.E12.m1.7.7.7.6.1.1"><csymbol cd="ambiguous" id="S2.E12.m1.7.7.7.6.1.1.1.cmml" xref="S2.E12.m1.7.7.7.6.1.1">subscript</csymbol><ci id="S2.E12.m1.7.7.7.6.1.1.2.cmml" xref="S2.E12.m1.7.7.7.6.1.1.2">𝛀</ci><list id="S2.E12.m1.5.5.5.4.2.3.cmml" xref="S2.E12.m1.5.5.5.4.2.4"><cn id="S2.E12.m1.4.4.4.3.1.1.cmml" type="integer" xref="S2.E12.m1.4.4.4.3.1.1">2</cn><cn id="S2.E12.m1.5.5.5.4.2.2.cmml" type="integer" xref="S2.E12.m1.5.5.5.4.2.2">22</cn></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E12.m1.7c">\frac{\left|\boldsymbol{\Omega}_{2}\right|}{\left|\boldsymbol{\Omega}_{2,11}% \right|\left|\boldsymbol{\Omega}_{2,22}\right|}</annotation><annotation encoding="application/x-llamapun" id="S2.E12.m1.7d">divide start_ARG | bold_Ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | end_ARG start_ARG | bold_Ω start_POSTSUBSCRIPT 2 , 11 end_POSTSUBSCRIPT | | bold_Ω start_POSTSUBSCRIPT 2 , 22 end_POSTSUBSCRIPT | 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="S2.SS5.SSS1.p4.3">where <math alttext="\boldsymbol{\Omega}_{2}\sim\mathcal{W}_{p}\left(n-1,\frac{1}{n-1}\boldsymbol{% \Omega}_{1}\right)" class="ltx_Math" display="inline" id="S2.SS5.SSS1.p4.1.m1.2"><semantics id="S2.SS5.SSS1.p4.1.m1.2a"><mrow id="S2.SS5.SSS1.p4.1.m1.2.2" xref="S2.SS5.SSS1.p4.1.m1.2.2.cmml"><msub id="S2.SS5.SSS1.p4.1.m1.2.2.4" xref="S2.SS5.SSS1.p4.1.m1.2.2.4.cmml"><mi id="S2.SS5.SSS1.p4.1.m1.2.2.4.2" xref="S2.SS5.SSS1.p4.1.m1.2.2.4.2.cmml">𝛀</mi><mn id="S2.SS5.SSS1.p4.1.m1.2.2.4.3" xref="S2.SS5.SSS1.p4.1.m1.2.2.4.3.cmml">2</mn></msub><mo id="S2.SS5.SSS1.p4.1.m1.2.2.3" xref="S2.SS5.SSS1.p4.1.m1.2.2.3.cmml">∼</mo><mrow id="S2.SS5.SSS1.p4.1.m1.2.2.2" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.cmml"><msub id="S2.SS5.SSS1.p4.1.m1.2.2.2.4" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.SSS1.p4.1.m1.2.2.2.4.2" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.4.2.cmml">𝒲</mi><mi id="S2.SS5.SSS1.p4.1.m1.2.2.2.4.3" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.4.3.cmml">p</mi></msub><mo id="S2.SS5.SSS1.p4.1.m1.2.2.2.3" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.3.cmml"></mo><mrow id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.3.cmml"><mo id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.3" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.3.cmml">(</mo><mrow id="S2.SS5.SSS1.p4.1.m1.1.1.1.1.1.1" xref="S2.SS5.SSS1.p4.1.m1.1.1.1.1.1.1.cmml"><mi id="S2.SS5.SSS1.p4.1.m1.1.1.1.1.1.1.2" xref="S2.SS5.SSS1.p4.1.m1.1.1.1.1.1.1.2.cmml">n</mi><mo id="S2.SS5.SSS1.p4.1.m1.1.1.1.1.1.1.1" xref="S2.SS5.SSS1.p4.1.m1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S2.SS5.SSS1.p4.1.m1.1.1.1.1.1.1.3" xref="S2.SS5.SSS1.p4.1.m1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.4" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.3.cmml">,</mo><mrow id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.cmml"><mfrac id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.cmml"><mn id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.2" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.2.cmml">1</mn><mrow id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.3" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.3.cmml"><mi id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.3.2" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.3.2.cmml">n</mi><mo id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.3.1" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.3.1.cmml">−</mo><mn id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.3.3" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.3.3.cmml">1</mn></mrow></mfrac><mo id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.1" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.1.cmml"></mo><msub id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.3" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.3.cmml"><mi id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.3.2" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.3.2.cmml">𝛀</mi><mn id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.3.3" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.5" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS1.p4.1.m1.2b"><apply id="S2.SS5.SSS1.p4.1.m1.2.2.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2"><csymbol cd="latexml" id="S2.SS5.SSS1.p4.1.m1.2.2.3.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.3">similar-to</csymbol><apply id="S2.SS5.SSS1.p4.1.m1.2.2.4.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.4"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p4.1.m1.2.2.4.1.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.4">subscript</csymbol><ci id="S2.SS5.SSS1.p4.1.m1.2.2.4.2.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.4.2">𝛀</ci><cn id="S2.SS5.SSS1.p4.1.m1.2.2.4.3.cmml" type="integer" xref="S2.SS5.SSS1.p4.1.m1.2.2.4.3">2</cn></apply><apply id="S2.SS5.SSS1.p4.1.m1.2.2.2.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.2"><times id="S2.SS5.SSS1.p4.1.m1.2.2.2.3.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.3"></times><apply id="S2.SS5.SSS1.p4.1.m1.2.2.2.4.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.4"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p4.1.m1.2.2.2.4.1.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.4">subscript</csymbol><ci id="S2.SS5.SSS1.p4.1.m1.2.2.2.4.2.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.4.2">𝒲</ci><ci id="S2.SS5.SSS1.p4.1.m1.2.2.2.4.3.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.4.3">𝑝</ci></apply><interval closure="open" id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.3.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2"><apply id="S2.SS5.SSS1.p4.1.m1.1.1.1.1.1.1.cmml" xref="S2.SS5.SSS1.p4.1.m1.1.1.1.1.1.1"><minus id="S2.SS5.SSS1.p4.1.m1.1.1.1.1.1.1.1.cmml" xref="S2.SS5.SSS1.p4.1.m1.1.1.1.1.1.1.1"></minus><ci id="S2.SS5.SSS1.p4.1.m1.1.1.1.1.1.1.2.cmml" xref="S2.SS5.SSS1.p4.1.m1.1.1.1.1.1.1.2">𝑛</ci><cn id="S2.SS5.SSS1.p4.1.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS5.SSS1.p4.1.m1.1.1.1.1.1.1.3">1</cn></apply><apply id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2"><times id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.1.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.1"></times><apply id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2"><divide id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.1.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2"></divide><cn id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.2.cmml" type="integer" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.2">1</cn><apply id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.3.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.3"><minus id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.3.1.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.3.1"></minus><ci id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.3.2.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.3.2">𝑛</ci><cn id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.3.3.cmml" type="integer" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.2.3.3">1</cn></apply></apply><apply id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.3.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.3.1.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.3">subscript</csymbol><ci id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.3.2.cmml" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.3.2">𝛀</ci><cn id="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.3.3.cmml" type="integer" xref="S2.SS5.SSS1.p4.1.m1.2.2.2.2.2.2.3.3">1</cn></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS1.p4.1.m1.2c">\boldsymbol{\Omega}_{2}\sim\mathcal{W}_{p}\left(n-1,\frac{1}{n-1}\boldsymbol{% \Omega}_{1}\right)</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS1.p4.1.m1.2d">bold_Ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ caligraphic_W start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_n - 1 , divide start_ARG 1 end_ARG start_ARG italic_n - 1 end_ARG bold_Ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, and where <math alttext="\boldsymbol{\Omega}_{1}\sim\mathcal{W}_{p}\left(n-1,\mathbf{I}_{p}\right)" class="ltx_Math" display="inline" id="S2.SS5.SSS1.p4.2.m2.2"><semantics id="S2.SS5.SSS1.p4.2.m2.2a"><mrow id="S2.SS5.SSS1.p4.2.m2.2.2" xref="S2.SS5.SSS1.p4.2.m2.2.2.cmml"><msub id="S2.SS5.SSS1.p4.2.m2.2.2.4" xref="S2.SS5.SSS1.p4.2.m2.2.2.4.cmml"><mi id="S2.SS5.SSS1.p4.2.m2.2.2.4.2" xref="S2.SS5.SSS1.p4.2.m2.2.2.4.2.cmml">𝛀</mi><mn id="S2.SS5.SSS1.p4.2.m2.2.2.4.3" xref="S2.SS5.SSS1.p4.2.m2.2.2.4.3.cmml">1</mn></msub><mo id="S2.SS5.SSS1.p4.2.m2.2.2.3" xref="S2.SS5.SSS1.p4.2.m2.2.2.3.cmml">∼</mo><mrow id="S2.SS5.SSS1.p4.2.m2.2.2.2" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.cmml"><msub id="S2.SS5.SSS1.p4.2.m2.2.2.2.4" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.SSS1.p4.2.m2.2.2.2.4.2" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.4.2.cmml">𝒲</mi><mi id="S2.SS5.SSS1.p4.2.m2.2.2.2.4.3" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.4.3.cmml">p</mi></msub><mo id="S2.SS5.SSS1.p4.2.m2.2.2.2.3" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.3.cmml"></mo><mrow id="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.2.3.cmml"><mo id="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2.3" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.2.3.cmml">(</mo><mrow id="S2.SS5.SSS1.p4.2.m2.1.1.1.1.1.1" xref="S2.SS5.SSS1.p4.2.m2.1.1.1.1.1.1.cmml"><mi id="S2.SS5.SSS1.p4.2.m2.1.1.1.1.1.1.2" xref="S2.SS5.SSS1.p4.2.m2.1.1.1.1.1.1.2.cmml">n</mi><mo id="S2.SS5.SSS1.p4.2.m2.1.1.1.1.1.1.1" xref="S2.SS5.SSS1.p4.2.m2.1.1.1.1.1.1.1.cmml">−</mo><mn id="S2.SS5.SSS1.p4.2.m2.1.1.1.1.1.1.3" xref="S2.SS5.SSS1.p4.2.m2.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2.4" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.2.3.cmml">,</mo><msub id="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2.2" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2.2.cmml"><mi id="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2.2.2" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2.2.2.cmml">𝐈</mi><mi id="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2.2.3" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2.2.3.cmml">p</mi></msub><mo id="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2.5" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS1.p4.2.m2.2b"><apply id="S2.SS5.SSS1.p4.2.m2.2.2.cmml" xref="S2.SS5.SSS1.p4.2.m2.2.2"><csymbol cd="latexml" id="S2.SS5.SSS1.p4.2.m2.2.2.3.cmml" xref="S2.SS5.SSS1.p4.2.m2.2.2.3">similar-to</csymbol><apply id="S2.SS5.SSS1.p4.2.m2.2.2.4.cmml" xref="S2.SS5.SSS1.p4.2.m2.2.2.4"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p4.2.m2.2.2.4.1.cmml" xref="S2.SS5.SSS1.p4.2.m2.2.2.4">subscript</csymbol><ci id="S2.SS5.SSS1.p4.2.m2.2.2.4.2.cmml" xref="S2.SS5.SSS1.p4.2.m2.2.2.4.2">𝛀</ci><cn id="S2.SS5.SSS1.p4.2.m2.2.2.4.3.cmml" type="integer" xref="S2.SS5.SSS1.p4.2.m2.2.2.4.3">1</cn></apply><apply id="S2.SS5.SSS1.p4.2.m2.2.2.2.cmml" xref="S2.SS5.SSS1.p4.2.m2.2.2.2"><times id="S2.SS5.SSS1.p4.2.m2.2.2.2.3.cmml" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.3"></times><apply id="S2.SS5.SSS1.p4.2.m2.2.2.2.4.cmml" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.4"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p4.2.m2.2.2.2.4.1.cmml" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.4">subscript</csymbol><ci id="S2.SS5.SSS1.p4.2.m2.2.2.2.4.2.cmml" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.4.2">𝒲</ci><ci id="S2.SS5.SSS1.p4.2.m2.2.2.2.4.3.cmml" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.4.3">𝑝</ci></apply><interval closure="open" id="S2.SS5.SSS1.p4.2.m2.2.2.2.2.3.cmml" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2"><apply id="S2.SS5.SSS1.p4.2.m2.1.1.1.1.1.1.cmml" xref="S2.SS5.SSS1.p4.2.m2.1.1.1.1.1.1"><minus id="S2.SS5.SSS1.p4.2.m2.1.1.1.1.1.1.1.cmml" xref="S2.SS5.SSS1.p4.2.m2.1.1.1.1.1.1.1"></minus><ci id="S2.SS5.SSS1.p4.2.m2.1.1.1.1.1.1.2.cmml" xref="S2.SS5.SSS1.p4.2.m2.1.1.1.1.1.1.2">𝑛</ci><cn id="S2.SS5.SSS1.p4.2.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS5.SSS1.p4.2.m2.1.1.1.1.1.1.3">1</cn></apply><apply id="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2.2.cmml" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2.2.1.cmml" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2.2">subscript</csymbol><ci id="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2.2.2.cmml" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2.2.2">𝐈</ci><ci id="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2.2.3.cmml" xref="S2.SS5.SSS1.p4.2.m2.2.2.2.2.2.2.3">𝑝</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS1.p4.2.m2.2c">\boldsymbol{\Omega}_{1}\sim\mathcal{W}_{p}\left(n-1,\mathbf{I}_{p}\right)</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS1.p4.2.m2.2d">bold_Ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∼ caligraphic_W start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_n - 1 , bold_I start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT )</annotation></semantics></math>, considering that <math alttext="\boldsymbol{\Omega}_{2}" class="ltx_Math" display="inline" id="S2.SS5.SSS1.p4.3.m3.1"><semantics id="S2.SS5.SSS1.p4.3.m3.1a"><msub id="S2.SS5.SSS1.p4.3.m3.1.1" xref="S2.SS5.SSS1.p4.3.m3.1.1.cmml"><mi id="S2.SS5.SSS1.p4.3.m3.1.1.2" xref="S2.SS5.SSS1.p4.3.m3.1.1.2.cmml">𝛀</mi><mn id="S2.SS5.SSS1.p4.3.m3.1.1.3" xref="S2.SS5.SSS1.p4.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS1.p4.3.m3.1b"><apply id="S2.SS5.SSS1.p4.3.m3.1.1.cmml" xref="S2.SS5.SSS1.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p4.3.m3.1.1.1.cmml" xref="S2.SS5.SSS1.p4.3.m3.1.1">subscript</csymbol><ci id="S2.SS5.SSS1.p4.3.m3.1.1.2.cmml" xref="S2.SS5.SSS1.p4.3.m3.1.1.2">𝛀</ci><cn id="S2.SS5.SSS1.p4.3.m3.1.1.3.cmml" type="integer" xref="S2.SS5.SSS1.p4.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS1.p4.3.m3.1c">\boldsymbol{\Omega}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS1.p4.3.m3.1d">bold_Ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is also partitioned as</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex5"> <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="\begin{bmatrix}\boldsymbol{\Omega}_{2,11}&\boldsymbol{\Omega}_{2,12}\\ \boldsymbol{\Omega}_{2,21}&\boldsymbol{\Omega}_{2,22}\end{bmatrix}\ " class="ltx_Math" display="block" id="S2.Ex5.m1.1"><semantics id="S2.Ex5.m1.1a"><mrow id="S2.Ex5.m1.1.1.3" xref="S2.Ex5.m1.1.1.2.cmml"><mo id="S2.Ex5.m1.1.1.3.1" xref="S2.Ex5.m1.1.1.2.1.cmml">[</mo><mtable columnspacing="5pt" displaystyle="true" id="S2.Ex5.m1.1.1.1.1" rowspacing="0pt" xref="S2.Ex5.m1.1.1.1.1.cmml"><mtr id="S2.Ex5.m1.1.1.1.1a" xref="S2.Ex5.m1.1.1.1.1.cmml"><mtd id="S2.Ex5.m1.1.1.1.1b" xref="S2.Ex5.m1.1.1.1.1.cmml"><msub id="S2.Ex5.m1.1.1.1.1.2.2.2.2" xref="S2.Ex5.m1.1.1.1.1.2.2.2.2.cmml"><mi id="S2.Ex5.m1.1.1.1.1.2.2.2.2.4" xref="S2.Ex5.m1.1.1.1.1.2.2.2.2.4.cmml">𝛀</mi><mrow id="S2.Ex5.m1.1.1.1.1.2.2.2.2.2.2.4" xref="S2.Ex5.m1.1.1.1.1.2.2.2.2.2.2.3.cmml"><mn id="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">2</mn><mo id="S2.Ex5.m1.1.1.1.1.2.2.2.2.2.2.4.1" xref="S2.Ex5.m1.1.1.1.1.2.2.2.2.2.2.3.cmml">,</mo><mn id="S2.Ex5.m1.1.1.1.1.2.2.2.2.2.2.2" xref="S2.Ex5.m1.1.1.1.1.2.2.2.2.2.2.2.cmml">11</mn></mrow></msub></mtd><mtd id="S2.Ex5.m1.1.1.1.1c" xref="S2.Ex5.m1.1.1.1.1.cmml"><msub id="S2.Ex5.m1.1.1.1.1.4.4.4.2" xref="S2.Ex5.m1.1.1.1.1.4.4.4.2.cmml"><mi id="S2.Ex5.m1.1.1.1.1.4.4.4.2.4" xref="S2.Ex5.m1.1.1.1.1.4.4.4.2.4.cmml">𝛀</mi><mrow id="S2.Ex5.m1.1.1.1.1.4.4.4.2.2.2.4" xref="S2.Ex5.m1.1.1.1.1.4.4.4.2.2.2.3.cmml"><mn id="S2.Ex5.m1.1.1.1.1.3.3.3.1.1.1.1" xref="S2.Ex5.m1.1.1.1.1.3.3.3.1.1.1.1.cmml">2</mn><mo id="S2.Ex5.m1.1.1.1.1.4.4.4.2.2.2.4.1" xref="S2.Ex5.m1.1.1.1.1.4.4.4.2.2.2.3.cmml">,</mo><mn id="S2.Ex5.m1.1.1.1.1.4.4.4.2.2.2.2" xref="S2.Ex5.m1.1.1.1.1.4.4.4.2.2.2.2.cmml">12</mn></mrow></msub></mtd></mtr><mtr id="S2.Ex5.m1.1.1.1.1d" xref="S2.Ex5.m1.1.1.1.1.cmml"><mtd id="S2.Ex5.m1.1.1.1.1e" xref="S2.Ex5.m1.1.1.1.1.cmml"><msub id="S2.Ex5.m1.1.1.1.1.6.6.2.2" xref="S2.Ex5.m1.1.1.1.1.6.6.2.2.cmml"><mi id="S2.Ex5.m1.1.1.1.1.6.6.2.2.4" xref="S2.Ex5.m1.1.1.1.1.6.6.2.2.4.cmml">𝛀</mi><mrow id="S2.Ex5.m1.1.1.1.1.6.6.2.2.2.2.4" xref="S2.Ex5.m1.1.1.1.1.6.6.2.2.2.2.3.cmml"><mn id="S2.Ex5.m1.1.1.1.1.5.5.1.1.1.1.1" xref="S2.Ex5.m1.1.1.1.1.5.5.1.1.1.1.1.cmml">2</mn><mo id="S2.Ex5.m1.1.1.1.1.6.6.2.2.2.2.4.1" xref="S2.Ex5.m1.1.1.1.1.6.6.2.2.2.2.3.cmml">,</mo><mn id="S2.Ex5.m1.1.1.1.1.6.6.2.2.2.2.2" xref="S2.Ex5.m1.1.1.1.1.6.6.2.2.2.2.2.cmml">21</mn></mrow></msub></mtd><mtd id="S2.Ex5.m1.1.1.1.1f" xref="S2.Ex5.m1.1.1.1.1.cmml"><msub id="S2.Ex5.m1.1.1.1.1.8.8.4.2" xref="S2.Ex5.m1.1.1.1.1.8.8.4.2.cmml"><mi id="S2.Ex5.m1.1.1.1.1.8.8.4.2.4" xref="S2.Ex5.m1.1.1.1.1.8.8.4.2.4.cmml">𝛀</mi><mrow id="S2.Ex5.m1.1.1.1.1.8.8.4.2.2.2.4" xref="S2.Ex5.m1.1.1.1.1.8.8.4.2.2.2.3.cmml"><mn id="S2.Ex5.m1.1.1.1.1.7.7.3.1.1.1.1" xref="S2.Ex5.m1.1.1.1.1.7.7.3.1.1.1.1.cmml">2</mn><mo id="S2.Ex5.m1.1.1.1.1.8.8.4.2.2.2.4.1" xref="S2.Ex5.m1.1.1.1.1.8.8.4.2.2.2.3.cmml">,</mo><mn id="S2.Ex5.m1.1.1.1.1.8.8.4.2.2.2.2" xref="S2.Ex5.m1.1.1.1.1.8.8.4.2.2.2.2.cmml">22</mn></mrow></msub></mtd></mtr></mtable><mo id="S2.Ex5.m1.1.1.3.2" xref="S2.Ex5.m1.1.1.2.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex5.m1.1b"><apply id="S2.Ex5.m1.1.1.2.cmml" xref="S2.Ex5.m1.1.1.3"><csymbol cd="latexml" id="S2.Ex5.m1.1.1.2.1.cmml" xref="S2.Ex5.m1.1.1.3.1">matrix</csymbol><matrix id="S2.Ex5.m1.1.1.1.1.cmml" xref="S2.Ex5.m1.1.1.1.1"><matrixrow id="S2.Ex5.m1.1.1.1.1a.cmml" xref="S2.Ex5.m1.1.1.1.1"><apply id="S2.Ex5.m1.1.1.1.1.2.2.2.2.cmml" xref="S2.Ex5.m1.1.1.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S2.Ex5.m1.1.1.1.1.2.2.2.2.3.cmml" xref="S2.Ex5.m1.1.1.1.1.2.2.2.2">subscript</csymbol><ci id="S2.Ex5.m1.1.1.1.1.2.2.2.2.4.cmml" xref="S2.Ex5.m1.1.1.1.1.2.2.2.2.4">𝛀</ci><list id="S2.Ex5.m1.1.1.1.1.2.2.2.2.2.2.3.cmml" xref="S2.Ex5.m1.1.1.1.1.2.2.2.2.2.2.4"><cn id="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" type="integer" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.1.1">2</cn><cn id="S2.Ex5.m1.1.1.1.1.2.2.2.2.2.2.2.cmml" type="integer" xref="S2.Ex5.m1.1.1.1.1.2.2.2.2.2.2.2">11</cn></list></apply><apply 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id="S2.Ex5.m1.1c">\begin{bmatrix}\boldsymbol{\Omega}_{2,11}&\boldsymbol{\Omega}_{2,12}\\ \boldsymbol{\Omega}_{2,21}&\boldsymbol{\Omega}_{2,22}\end{bmatrix}\ </annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m1.1d">[ start_ARG start_ROW start_CELL bold_Ω start_POSTSUBSCRIPT 2 , 11 end_POSTSUBSCRIPT end_CELL start_CELL bold_Ω start_POSTSUBSCRIPT 2 , 12 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL bold_Ω start_POSTSUBSCRIPT 2 , 21 end_POSTSUBSCRIPT end_CELL start_CELL bold_Ω start_POSTSUBSCRIPT 2 , 22 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS5.SSS1.p4.5">with <math alttext="\boldsymbol{\Omega}_{2,11}" class="ltx_Math" display="inline" id="S2.SS5.SSS1.p4.4.m1.2"><semantics id="S2.SS5.SSS1.p4.4.m1.2a"><msub id="S2.SS5.SSS1.p4.4.m1.2.3" xref="S2.SS5.SSS1.p4.4.m1.2.3.cmml"><mi id="S2.SS5.SSS1.p4.4.m1.2.3.2" 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xref="S2.SS5.SSS1.p5.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS1.p5.1.m1.1b"><apply id="S2.SS5.SSS1.p5.1.m1.1.1.cmml" xref="S2.SS5.SSS1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p5.1.m1.1.1.1.cmml" xref="S2.SS5.SSS1.p5.1.m1.1.1">subscript</csymbol><ci id="S2.SS5.SSS1.p5.1.m1.1.1.2.cmml" xref="S2.SS5.SSS1.p5.1.m1.1.1.2">ℋ</ci><cn id="S2.SS5.SSS1.p5.1.m1.1.1.3.cmml" type="integer" xref="S2.SS5.SSS1.p5.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS1.p5.1.m1.1c">\mathcal{H}_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS1.p5.1.m1.1d">caligraphic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> should be rejected for a level of significance <math alttext="\alpha" class="ltx_Math" display="inline" id="S2.SS5.SSS1.p5.2.m2.1"><semantics id="S2.SS5.SSS1.p5.2.m2.1a"><mi id="S2.SS5.SSS1.p5.2.m2.1.1" xref="S2.SS5.SSS1.p5.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS1.p5.2.m2.1b"><ci id="S2.SS5.SSS1.p5.2.m2.1.1.cmml" xref="S2.SS5.SSS1.p5.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS1.p5.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS1.p5.2.m2.1d">italic_α</annotation></semantics></math>, if the observed value of <math alttext="T_{3}^{\star}" class="ltx_Math" display="inline" id="S2.SS5.SSS1.p5.3.m3.1"><semantics id="S2.SS5.SSS1.p5.3.m3.1a"><msubsup id="S2.SS5.SSS1.p5.3.m3.1.1" xref="S2.SS5.SSS1.p5.3.m3.1.1.cmml"><mi id="S2.SS5.SSS1.p5.3.m3.1.1.2.2" xref="S2.SS5.SSS1.p5.3.m3.1.1.2.2.cmml">T</mi><mn id="S2.SS5.SSS1.p5.3.m3.1.1.2.3" xref="S2.SS5.SSS1.p5.3.m3.1.1.2.3.cmml">3</mn><mo id="S2.SS5.SSS1.p5.3.m3.1.1.3" xref="S2.SS5.SSS1.p5.3.m3.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS1.p5.3.m3.1b"><apply 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class="ltx_Math" display="inline" id="S2.SS5.SSS1.p5.4.m4.2"><semantics id="S2.SS5.SSS1.p5.4.m4.2a"><msubsup id="S2.SS5.SSS1.p5.4.m4.2.3" xref="S2.SS5.SSS1.p5.4.m4.2.3.cmml"><mi id="S2.SS5.SSS1.p5.4.m4.2.3.2.2" xref="S2.SS5.SSS1.p5.4.m4.2.3.2.2.cmml">t</mi><mrow id="S2.SS5.SSS1.p5.4.m4.2.2.2.4" xref="S2.SS5.SSS1.p5.4.m4.2.2.2.3.cmml"><mn id="S2.SS5.SSS1.p5.4.m4.1.1.1.1" xref="S2.SS5.SSS1.p5.4.m4.1.1.1.1.cmml">3</mn><mo id="S2.SS5.SSS1.p5.4.m4.2.2.2.4.1" xref="S2.SS5.SSS1.p5.4.m4.2.2.2.3.cmml">;</mo><mi id="S2.SS5.SSS1.p5.4.m4.2.2.2.2" xref="S2.SS5.SSS1.p5.4.m4.2.2.2.2.cmml">α</mi></mrow><mo id="S2.SS5.SSS1.p5.4.m4.2.3.3" xref="S2.SS5.SSS1.p5.4.m4.2.3.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS1.p5.4.m4.2b"><apply id="S2.SS5.SSS1.p5.4.m4.2.3.cmml" xref="S2.SS5.SSS1.p5.4.m4.2.3"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p5.4.m4.2.3.1.cmml" xref="S2.SS5.SSS1.p5.4.m4.2.3">superscript</csymbol><apply id="S2.SS5.SSS1.p5.4.m4.2.3.2.cmml" xref="S2.SS5.SSS1.p5.4.m4.2.3"><csymbol cd="ambiguous" id="S2.SS5.SSS1.p5.4.m4.2.3.2.1.cmml" xref="S2.SS5.SSS1.p5.4.m4.2.3">subscript</csymbol><ci id="S2.SS5.SSS1.p5.4.m4.2.3.2.2.cmml" xref="S2.SS5.SSS1.p5.4.m4.2.3.2.2">𝑡</ci><list id="S2.SS5.SSS1.p5.4.m4.2.2.2.3.cmml" xref="S2.SS5.SSS1.p5.4.m4.2.2.2.4"><cn id="S2.SS5.SSS1.p5.4.m4.1.1.1.1.cmml" type="integer" xref="S2.SS5.SSS1.p5.4.m4.1.1.1.1">3</cn><ci id="S2.SS5.SSS1.p5.4.m4.2.2.2.2.cmml" xref="S2.SS5.SSS1.p5.4.m4.2.2.2.2">𝛼</ci></list></apply><ci id="S2.SS5.SSS1.p5.4.m4.2.3.3.cmml" xref="S2.SS5.SSS1.p5.4.m4.2.3.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS1.p5.4.m4.2c">t_{3;\alpha}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS1.p5.4.m4.2d">italic_t start_POSTSUBSCRIPT 3 ; italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>, the <math alttext="\alpha" class="ltx_Math" display="inline" id="S2.SS5.SSS1.p5.5.m5.1"><semantics id="S2.SS5.SSS1.p5.5.m5.1a"><mi id="S2.SS5.SSS1.p5.5.m5.1.1" xref="S2.SS5.SSS1.p5.5.m5.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS1.p5.5.m5.1b"><ci id="S2.SS5.SSS1.p5.5.m5.1.1.cmml" xref="S2.SS5.SSS1.p5.5.m5.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS1.p5.5.m5.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS1.p5.5.m5.1d">italic_α</annotation></semantics></math>th percentile of (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E12" title="In 2.5.1 Independence test ‣ 2.5 Sphericity test ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">12</span></a>).</p> </div> </section> <section class="ltx_subsubsection" id="S2.SS5.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">2.5.2 </span>Regression of One set of Variables on other</h4> <div class="ltx_para" id="S2.SS5.SSS2.p1"> <p class="ltx_p" id="S2.SS5.SSS2.p1.2">To quantify the relationships between two different subsets of variables, one can perform a test for the regression of one set of variables on another. This test aims to determine how changes (variations in one subset of variables can predict/explain the variations in another <cite class="ltx_cite ltx_citemacro_citep">(Moura, Coelho, & Sinha, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib19" title="">2024</a>)</cite>. Specifically, the regression coefficients, represented by the matrix <math alttext="\boldsymbol{\Delta}=\boldsymbol{\Sigma}_{12}\boldsymbol{\Sigma}_{22}^{-1}" class="ltx_Math" display="inline" id="S2.SS5.SSS2.p1.1.m1.1"><semantics id="S2.SS5.SSS2.p1.1.m1.1a"><mrow id="S2.SS5.SSS2.p1.1.m1.1.1" xref="S2.SS5.SSS2.p1.1.m1.1.1.cmml"><mi id="S2.SS5.SSS2.p1.1.m1.1.1.2" xref="S2.SS5.SSS2.p1.1.m1.1.1.2.cmml">𝚫</mi><mo id="S2.SS5.SSS2.p1.1.m1.1.1.1" xref="S2.SS5.SSS2.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S2.SS5.SSS2.p1.1.m1.1.1.3" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.cmml"><msub id="S2.SS5.SSS2.p1.1.m1.1.1.3.2" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.2.cmml"><mi id="S2.SS5.SSS2.p1.1.m1.1.1.3.2.2" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.2.2.cmml">𝚺</mi><mn id="S2.SS5.SSS2.p1.1.m1.1.1.3.2.3" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.2.3.cmml">12</mn></msub><mo id="S2.SS5.SSS2.p1.1.m1.1.1.3.1" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.1.cmml"></mo><msubsup id="S2.SS5.SSS2.p1.1.m1.1.1.3.3" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.3.cmml"><mi id="S2.SS5.SSS2.p1.1.m1.1.1.3.3.2.2" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.3.2.2.cmml">𝚺</mi><mn id="S2.SS5.SSS2.p1.1.m1.1.1.3.3.2.3" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.3.2.3.cmml">22</mn><mrow id="S2.SS5.SSS2.p1.1.m1.1.1.3.3.3" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.3.3.cmml"><mo id="S2.SS5.SSS2.p1.1.m1.1.1.3.3.3a" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.3.3.cmml">−</mo><mn id="S2.SS5.SSS2.p1.1.m1.1.1.3.3.3.2" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.3.3.2.cmml">1</mn></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS2.p1.1.m1.1b"><apply id="S2.SS5.SSS2.p1.1.m1.1.1.cmml" xref="S2.SS5.SSS2.p1.1.m1.1.1"><eq id="S2.SS5.SSS2.p1.1.m1.1.1.1.cmml" xref="S2.SS5.SSS2.p1.1.m1.1.1.1"></eq><ci id="S2.SS5.SSS2.p1.1.m1.1.1.2.cmml" xref="S2.SS5.SSS2.p1.1.m1.1.1.2">𝚫</ci><apply id="S2.SS5.SSS2.p1.1.m1.1.1.3.cmml" xref="S2.SS5.SSS2.p1.1.m1.1.1.3"><times id="S2.SS5.SSS2.p1.1.m1.1.1.3.1.cmml" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.1"></times><apply id="S2.SS5.SSS2.p1.1.m1.1.1.3.2.cmml" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p1.1.m1.1.1.3.2.1.cmml" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.2">subscript</csymbol><ci id="S2.SS5.SSS2.p1.1.m1.1.1.3.2.2.cmml" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.2.2">𝚺</ci><cn id="S2.SS5.SSS2.p1.1.m1.1.1.3.2.3.cmml" type="integer" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.2.3">12</cn></apply><apply id="S2.SS5.SSS2.p1.1.m1.1.1.3.3.cmml" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p1.1.m1.1.1.3.3.1.cmml" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.3">superscript</csymbol><apply id="S2.SS5.SSS2.p1.1.m1.1.1.3.3.2.cmml" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p1.1.m1.1.1.3.3.2.1.cmml" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.3">subscript</csymbol><ci id="S2.SS5.SSS2.p1.1.m1.1.1.3.3.2.2.cmml" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.3.2.2">𝚺</ci><cn id="S2.SS5.SSS2.p1.1.m1.1.1.3.3.2.3.cmml" type="integer" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.3.2.3">22</cn></apply><apply id="S2.SS5.SSS2.p1.1.m1.1.1.3.3.3.cmml" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.3.3"><minus id="S2.SS5.SSS2.p1.1.m1.1.1.3.3.3.1.cmml" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.3.3"></minus><cn id="S2.SS5.SSS2.p1.1.m1.1.1.3.3.3.2.cmml" type="integer" xref="S2.SS5.SSS2.p1.1.m1.1.1.3.3.3.2">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS2.p1.1.m1.1c">\boldsymbol{\Delta}=\boldsymbol{\Sigma}_{12}\boldsymbol{\Sigma}_{22}^{-1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS2.p1.1.m1.1d">bold_Δ = bold_Σ start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT bold_Σ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, consider the partition of <math alttext="\boldsymbol{\Sigma}" class="ltx_Math" display="inline" id="S2.SS5.SSS2.p1.2.m2.1"><semantics id="S2.SS5.SSS2.p1.2.m2.1a"><mi id="S2.SS5.SSS2.p1.2.m2.1.1" xref="S2.SS5.SSS2.p1.2.m2.1.1.cmml">𝚺</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS2.p1.2.m2.1b"><ci id="S2.SS5.SSS2.p1.2.m2.1.1.cmml" xref="S2.SS5.SSS2.p1.2.m2.1.1">𝚺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS2.p1.2.m2.1c">\boldsymbol{\Sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS2.p1.2.m2.1d">bold_Σ</annotation></semantics></math> as in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E9" title="In 2.5.1 Independence test ‣ 2.5 Sphericity test ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">9</span></a>), describe the conditional relationship between the two subsets of variables. These quantify how changes in the independent variables influence the dependent variables, assuming a multivariate normal framework. Moreover, this test will provide insights into the underlying structure and dependencies between the two subsets of variables.</p> </div> <div class="ltx_para" id="S2.SS5.SSS2.p2"> <p class="ltx_p" id="S2.SS5.SSS2.p2.5">The referred test consists of testing</p> <table class="ltx_equation ltx_eqn_table" id="S2.E13"> <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}:\boldsymbol{\Delta}=\boldsymbol{\Delta}_{0}\quad\text{vs.}% \quad\mathcal{H}_{1}:\boldsymbol{\Delta}\neq\boldsymbol{\Delta}_{0}," class="ltx_Math" display="block" id="S2.E13.m1.2"><semantics id="S2.E13.m1.2a"><mrow id="S2.E13.m1.2.2.1" xref="S2.E13.m1.2.2.1.1.cmml"><mrow id="S2.E13.m1.2.2.1.1" xref="S2.E13.m1.2.2.1.1.cmml"><msub id="S2.E13.m1.2.2.1.1.4" xref="S2.E13.m1.2.2.1.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E13.m1.2.2.1.1.4.2" xref="S2.E13.m1.2.2.1.1.4.2.cmml">ℋ</mi><mn id="S2.E13.m1.2.2.1.1.4.3" xref="S2.E13.m1.2.2.1.1.4.3.cmml">0</mn></msub><mo id="S2.E13.m1.2.2.1.1.5" lspace="0.278em" rspace="0.278em" xref="S2.E13.m1.2.2.1.1.5.cmml">:</mo><mrow id="S2.E13.m1.2.2.1.1.2" xref="S2.E13.m1.2.2.1.1.2.cmml"><mi id="S2.E13.m1.2.2.1.1.2.4" xref="S2.E13.m1.2.2.1.1.2.4.cmml">𝚫</mi><mo id="S2.E13.m1.2.2.1.1.2.3" xref="S2.E13.m1.2.2.1.1.2.3.cmml">=</mo><mrow id="S2.E13.m1.2.2.1.1.2.2.2" xref="S2.E13.m1.2.2.1.1.2.2.3.cmml"><msub id="S2.E13.m1.2.2.1.1.1.1.1.1" xref="S2.E13.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S2.E13.m1.2.2.1.1.1.1.1.1.2" xref="S2.E13.m1.2.2.1.1.1.1.1.1.2.cmml">𝚫</mi><mn id="S2.E13.m1.2.2.1.1.1.1.1.1.3" xref="S2.E13.m1.2.2.1.1.1.1.1.1.3.cmml">0</mn></msub><mspace id="S2.E13.m1.2.2.1.1.2.2.2.3" width="1em" xref="S2.E13.m1.2.2.1.1.2.2.3.cmml"></mspace><mtext id="S2.E13.m1.1.1" xref="S2.E13.m1.1.1a.cmml">vs.</mtext><mspace id="S2.E13.m1.2.2.1.1.2.2.2.4" width="1em" 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id="S2.E13.m1.2.2.1.1d.cmml" xref="S2.E13.m1.2.2.1"></share><apply id="S2.E13.m1.2.2.1.1.7.cmml" xref="S2.E13.m1.2.2.1.1.7"><neq id="S2.E13.m1.2.2.1.1.7.1.cmml" xref="S2.E13.m1.2.2.1.1.7.1"></neq><ci id="S2.E13.m1.2.2.1.1.7.2.cmml" xref="S2.E13.m1.2.2.1.1.7.2">𝚫</ci><apply id="S2.E13.m1.2.2.1.1.7.3.cmml" xref="S2.E13.m1.2.2.1.1.7.3"><csymbol cd="ambiguous" id="S2.E13.m1.2.2.1.1.7.3.1.cmml" xref="S2.E13.m1.2.2.1.1.7.3">subscript</csymbol><ci id="S2.E13.m1.2.2.1.1.7.3.2.cmml" xref="S2.E13.m1.2.2.1.1.7.3.2">𝚫</ci><cn id="S2.E13.m1.2.2.1.1.7.3.3.cmml" type="integer" xref="S2.E13.m1.2.2.1.1.7.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E13.m1.2c">\mathcal{H}_{0}:\boldsymbol{\Delta}=\boldsymbol{\Delta}_{0}\quad\text{vs.}% \quad\mathcal{H}_{1}:\boldsymbol{\Delta}\neq\boldsymbol{\Delta}_{0},</annotation><annotation encoding="application/x-llamapun" id="S2.E13.m1.2d">caligraphic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT : bold_Δ = bold_Δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT vs. caligraphic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : bold_Δ ≠ bold_Δ start_POSTSUBSCRIPT 0 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">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS5.SSS2.p2.4">where <math alttext="\boldsymbol{\Delta}=\boldsymbol{\Sigma}_{12}\boldsymbol{\Sigma}_{22}^{-1}" class="ltx_Math" display="inline" id="S2.SS5.SSS2.p2.1.m1.1"><semantics id="S2.SS5.SSS2.p2.1.m1.1a"><mrow id="S2.SS5.SSS2.p2.1.m1.1.1" xref="S2.SS5.SSS2.p2.1.m1.1.1.cmml"><mi id="S2.SS5.SSS2.p2.1.m1.1.1.2" xref="S2.SS5.SSS2.p2.1.m1.1.1.2.cmml">𝚫</mi><mo id="S2.SS5.SSS2.p2.1.m1.1.1.1" xref="S2.SS5.SSS2.p2.1.m1.1.1.1.cmml">=</mo><mrow id="S2.SS5.SSS2.p2.1.m1.1.1.3" xref="S2.SS5.SSS2.p2.1.m1.1.1.3.cmml"><msub 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id="S2.SS5.SSS2.p2.1.m1.1.1.cmml" xref="S2.SS5.SSS2.p2.1.m1.1.1"><eq id="S2.SS5.SSS2.p2.1.m1.1.1.1.cmml" xref="S2.SS5.SSS2.p2.1.m1.1.1.1"></eq><ci id="S2.SS5.SSS2.p2.1.m1.1.1.2.cmml" xref="S2.SS5.SSS2.p2.1.m1.1.1.2">𝚫</ci><apply id="S2.SS5.SSS2.p2.1.m1.1.1.3.cmml" xref="S2.SS5.SSS2.p2.1.m1.1.1.3"><times id="S2.SS5.SSS2.p2.1.m1.1.1.3.1.cmml" xref="S2.SS5.SSS2.p2.1.m1.1.1.3.1"></times><apply id="S2.SS5.SSS2.p2.1.m1.1.1.3.2.cmml" xref="S2.SS5.SSS2.p2.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p2.1.m1.1.1.3.2.1.cmml" xref="S2.SS5.SSS2.p2.1.m1.1.1.3.2">subscript</csymbol><ci id="S2.SS5.SSS2.p2.1.m1.1.1.3.2.2.cmml" xref="S2.SS5.SSS2.p2.1.m1.1.1.3.2.2">𝚺</ci><cn id="S2.SS5.SSS2.p2.1.m1.1.1.3.2.3.cmml" type="integer" xref="S2.SS5.SSS2.p2.1.m1.1.1.3.2.3">12</cn></apply><apply id="S2.SS5.SSS2.p2.1.m1.1.1.3.3.cmml" xref="S2.SS5.SSS2.p2.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p2.1.m1.1.1.3.3.1.cmml" xref="S2.SS5.SSS2.p2.1.m1.1.1.3.3">superscript</csymbol><apply id="S2.SS5.SSS2.p2.1.m1.1.1.3.3.2.cmml" xref="S2.SS5.SSS2.p2.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p2.1.m1.1.1.3.3.2.1.cmml" xref="S2.SS5.SSS2.p2.1.m1.1.1.3.3">subscript</csymbol><ci id="S2.SS5.SSS2.p2.1.m1.1.1.3.3.2.2.cmml" xref="S2.SS5.SSS2.p2.1.m1.1.1.3.3.2.2">𝚺</ci><cn id="S2.SS5.SSS2.p2.1.m1.1.1.3.3.2.3.cmml" type="integer" xref="S2.SS5.SSS2.p2.1.m1.1.1.3.3.2.3">22</cn></apply><apply id="S2.SS5.SSS2.p2.1.m1.1.1.3.3.3.cmml" xref="S2.SS5.SSS2.p2.1.m1.1.1.3.3.3"><minus id="S2.SS5.SSS2.p2.1.m1.1.1.3.3.3.1.cmml" xref="S2.SS5.SSS2.p2.1.m1.1.1.3.3.3"></minus><cn id="S2.SS5.SSS2.p2.1.m1.1.1.3.3.3.2.cmml" type="integer" xref="S2.SS5.SSS2.p2.1.m1.1.1.3.3.3.2">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS2.p2.1.m1.1c">\boldsymbol{\Delta}=\boldsymbol{\Sigma}_{12}\boldsymbol{\Sigma}_{22}^{-1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS2.p2.1.m1.1d">bold_Δ = bold_Σ start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT bold_Σ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, and where <math alttext="\boldsymbol{\Delta}_{0}" class="ltx_Math" display="inline" id="S2.SS5.SSS2.p2.2.m2.1"><semantics id="S2.SS5.SSS2.p2.2.m2.1a"><msub id="S2.SS5.SSS2.p2.2.m2.1.1" xref="S2.SS5.SSS2.p2.2.m2.1.1.cmml"><mi id="S2.SS5.SSS2.p2.2.m2.1.1.2" xref="S2.SS5.SSS2.p2.2.m2.1.1.2.cmml">𝚫</mi><mn id="S2.SS5.SSS2.p2.2.m2.1.1.3" xref="S2.SS5.SSS2.p2.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS2.p2.2.m2.1b"><apply id="S2.SS5.SSS2.p2.2.m2.1.1.cmml" xref="S2.SS5.SSS2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p2.2.m2.1.1.1.cmml" xref="S2.SS5.SSS2.p2.2.m2.1.1">subscript</csymbol><ci id="S2.SS5.SSS2.p2.2.m2.1.1.2.cmml" xref="S2.SS5.SSS2.p2.2.m2.1.1.2">𝚫</ci><cn id="S2.SS5.SSS2.p2.2.m2.1.1.3.cmml" type="integer" xref="S2.SS5.SSS2.p2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS2.p2.2.m2.1c">\boldsymbol{\Delta}_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS2.p2.2.m2.1d">bold_Δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is some specific <math alttext="p_{1}\times(p-p_{1})" class="ltx_Math" display="inline" id="S2.SS5.SSS2.p2.3.m3.1"><semantics id="S2.SS5.SSS2.p2.3.m3.1a"><mrow id="S2.SS5.SSS2.p2.3.m3.1.1" xref="S2.SS5.SSS2.p2.3.m3.1.1.cmml"><msub id="S2.SS5.SSS2.p2.3.m3.1.1.3" xref="S2.SS5.SSS2.p2.3.m3.1.1.3.cmml"><mi id="S2.SS5.SSS2.p2.3.m3.1.1.3.2" xref="S2.SS5.SSS2.p2.3.m3.1.1.3.2.cmml">p</mi><mn id="S2.SS5.SSS2.p2.3.m3.1.1.3.3" xref="S2.SS5.SSS2.p2.3.m3.1.1.3.3.cmml">1</mn></msub><mo id="S2.SS5.SSS2.p2.3.m3.1.1.2" lspace="0.222em" rspace="0.222em" xref="S2.SS5.SSS2.p2.3.m3.1.1.2.cmml">×</mo><mrow id="S2.SS5.SSS2.p2.3.m3.1.1.1.1" xref="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.cmml"><mo id="S2.SS5.SSS2.p2.3.m3.1.1.1.1.2" stretchy="false" xref="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1" xref="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.cmml"><mi id="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.2" xref="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.2.cmml">p</mi><mo id="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.1" xref="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.1.cmml">−</mo><msub id="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.3" xref="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.3.cmml"><mi id="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.3.2" xref="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.3.2.cmml">p</mi><mn id="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.3.3" xref="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S2.SS5.SSS2.p2.3.m3.1.1.1.1.3" stretchy="false" xref="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS2.p2.3.m3.1b"><apply id="S2.SS5.SSS2.p2.3.m3.1.1.cmml" xref="S2.SS5.SSS2.p2.3.m3.1.1"><times id="S2.SS5.SSS2.p2.3.m3.1.1.2.cmml" xref="S2.SS5.SSS2.p2.3.m3.1.1.2"></times><apply id="S2.SS5.SSS2.p2.3.m3.1.1.3.cmml" xref="S2.SS5.SSS2.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p2.3.m3.1.1.3.1.cmml" xref="S2.SS5.SSS2.p2.3.m3.1.1.3">subscript</csymbol><ci id="S2.SS5.SSS2.p2.3.m3.1.1.3.2.cmml" xref="S2.SS5.SSS2.p2.3.m3.1.1.3.2">𝑝</ci><cn id="S2.SS5.SSS2.p2.3.m3.1.1.3.3.cmml" type="integer" xref="S2.SS5.SSS2.p2.3.m3.1.1.3.3">1</cn></apply><apply id="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.cmml" xref="S2.SS5.SSS2.p2.3.m3.1.1.1.1"><minus id="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.1.cmml" xref="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.1"></minus><ci id="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.2.cmml" xref="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.2">𝑝</ci><apply id="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.3.cmml" xref="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.3.1.cmml" xref="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.3">subscript</csymbol><ci id="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.3.2.cmml" xref="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.3.2">𝑝</ci><cn id="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS5.SSS2.p2.3.m3.1.1.1.1.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS2.p2.3.m3.1c">p_{1}\times(p-p_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS2.p2.3.m3.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × ( italic_p - italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> matrix, considering <math alttext="p_{1}\leq p_{2}" class="ltx_Math" display="inline" id="S2.SS5.SSS2.p2.4.m4.1"><semantics id="S2.SS5.SSS2.p2.4.m4.1a"><mrow id="S2.SS5.SSS2.p2.4.m4.1.1" xref="S2.SS5.SSS2.p2.4.m4.1.1.cmml"><msub id="S2.SS5.SSS2.p2.4.m4.1.1.2" xref="S2.SS5.SSS2.p2.4.m4.1.1.2.cmml"><mi id="S2.SS5.SSS2.p2.4.m4.1.1.2.2" xref="S2.SS5.SSS2.p2.4.m4.1.1.2.2.cmml">p</mi><mn id="S2.SS5.SSS2.p2.4.m4.1.1.2.3" xref="S2.SS5.SSS2.p2.4.m4.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS5.SSS2.p2.4.m4.1.1.1" xref="S2.SS5.SSS2.p2.4.m4.1.1.1.cmml">≤</mo><msub id="S2.SS5.SSS2.p2.4.m4.1.1.3" xref="S2.SS5.SSS2.p2.4.m4.1.1.3.cmml"><mi id="S2.SS5.SSS2.p2.4.m4.1.1.3.2" xref="S2.SS5.SSS2.p2.4.m4.1.1.3.2.cmml">p</mi><mn id="S2.SS5.SSS2.p2.4.m4.1.1.3.3" xref="S2.SS5.SSS2.p2.4.m4.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS2.p2.4.m4.1b"><apply id="S2.SS5.SSS2.p2.4.m4.1.1.cmml" xref="S2.SS5.SSS2.p2.4.m4.1.1"><leq id="S2.SS5.SSS2.p2.4.m4.1.1.1.cmml" xref="S2.SS5.SSS2.p2.4.m4.1.1.1"></leq><apply id="S2.SS5.SSS2.p2.4.m4.1.1.2.cmml" xref="S2.SS5.SSS2.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p2.4.m4.1.1.2.1.cmml" xref="S2.SS5.SSS2.p2.4.m4.1.1.2">subscript</csymbol><ci id="S2.SS5.SSS2.p2.4.m4.1.1.2.2.cmml" xref="S2.SS5.SSS2.p2.4.m4.1.1.2.2">𝑝</ci><cn id="S2.SS5.SSS2.p2.4.m4.1.1.2.3.cmml" type="integer" xref="S2.SS5.SSS2.p2.4.m4.1.1.2.3">1</cn></apply><apply id="S2.SS5.SSS2.p2.4.m4.1.1.3.cmml" xref="S2.SS5.SSS2.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p2.4.m4.1.1.3.1.cmml" xref="S2.SS5.SSS2.p2.4.m4.1.1.3">subscript</csymbol><ci id="S2.SS5.SSS2.p2.4.m4.1.1.3.2.cmml" xref="S2.SS5.SSS2.p2.4.m4.1.1.3.2">𝑝</ci><cn id="S2.SS5.SSS2.p2.4.m4.1.1.3.3.cmml" type="integer" xref="S2.SS5.SSS2.p2.4.m4.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS2.p2.4.m4.1c">p_{1}\leq p_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS2.p2.4.m4.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS5.SSS2.p3"> <p class="ltx_p" id="S2.SS5.SSS2.p3.3"><cite class="ltx_cite ltx_citemacro_citet">Klein, Moura, & Sinha (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib8" title="">2021</a>)</cite> considered the test statistic</p> <table class="ltx_equation ltx_eqn_table" id="S2.E14"> <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="T_{4}^{\star}=\frac{\left|\left(\mathbf{S}^{\star}_{12}\mathbf{S}_{22}^{\star-% 1}-\boldsymbol{\Delta}_{0}\right)\mathbf{S}^{\star}_{22}\left(\mathbf{S}^{% \star}_{12}\mathbf{S}_{22}^{\star-1}-\boldsymbol{\Delta}_{0}\right)^{\top}% \right|}{\left|\mathbf{S}^{\star}_{11}-\mathbf{S}^{\star}_{12}\mathbf{S}_{22}^% {\star-1}\mathbf{S}^{\star}_{21}\right|}," class="ltx_Math" display="block" id="S2.E14.m1.3"><semantics id="S2.E14.m1.3a"><mrow id="S2.E14.m1.3.3.1" xref="S2.E14.m1.3.3.1.1.cmml"><mrow id="S2.E14.m1.3.3.1.1" xref="S2.E14.m1.3.3.1.1.cmml"><msubsup id="S2.E14.m1.3.3.1.1.2" xref="S2.E14.m1.3.3.1.1.2.cmml"><mi id="S2.E14.m1.3.3.1.1.2.2.2" xref="S2.E14.m1.3.3.1.1.2.2.2.cmml">T</mi><mn id="S2.E14.m1.3.3.1.1.2.2.3" xref="S2.E14.m1.3.3.1.1.2.2.3.cmml">4</mn><mo id="S2.E14.m1.3.3.1.1.2.3" xref="S2.E14.m1.3.3.1.1.2.3.cmml">⋆</mo></msubsup><mo id="S2.E14.m1.3.3.1.1.1" xref="S2.E14.m1.3.3.1.1.1.cmml">=</mo><mfrac id="S2.E14.m1.2.2" xref="S2.E14.m1.2.2.cmml"><mrow id="S2.E14.m1.1.1.1.1" xref="S2.E14.m1.1.1.1.2.cmml"><mo id="S2.E14.m1.1.1.1.1.2" xref="S2.E14.m1.1.1.1.2.1.cmml">|</mo><mrow id="S2.E14.m1.1.1.1.1.1" xref="S2.E14.m1.1.1.1.1.1.cmml"><mrow id="S2.E14.m1.1.1.1.1.1.1.1" xref="S2.E14.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E14.m1.1.1.1.1.1.1.1.2" xref="S2.E14.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E14.m1.1.1.1.1.1.1.1.1" xref="S2.E14.m1.1.1.1.1.1.1.1.1.cmml"><mrow id="S2.E14.m1.1.1.1.1.1.1.1.1.2" xref="S2.E14.m1.1.1.1.1.1.1.1.1.2.cmml"><msubsup id="S2.E14.m1.1.1.1.1.1.1.1.1.2.2" xref="S2.E14.m1.1.1.1.1.1.1.1.1.2.2.cmml"><mi id="S2.E14.m1.1.1.1.1.1.1.1.1.2.2.2.2" xref="S2.E14.m1.1.1.1.1.1.1.1.1.2.2.2.2.cmml">𝐒</mi><mn id="S2.E14.m1.1.1.1.1.1.1.1.1.2.2.3" xref="S2.E14.m1.1.1.1.1.1.1.1.1.2.2.3.cmml">12</mn><mo id="S2.E14.m1.1.1.1.1.1.1.1.1.2.2.2.3" xref="S2.E14.m1.1.1.1.1.1.1.1.1.2.2.2.3.cmml">⋆</mo></msubsup><mo 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start_POSTSUPERSCRIPT ⋆ - 1 end_POSTSUPERSCRIPT - bold_Δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT ( bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT bold_S start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ - 1 end_POSTSUPERSCRIPT - bold_Δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT | end_ARG start_ARG | bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT - bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT bold_S start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ - 1 end_POSTSUPERSCRIPT bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 21 end_POSTSUBSCRIPT | 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">(14)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS5.SSS2.p3.4">proving that (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E14" title="In 2.5.2 Regression of One set of Variables on other ‣ 2.5 Sphericity test ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">14</span></a>) is stochastically equivalent to</p> <table class="ltx_equation ltx_eqn_table" id="S2.E15"> <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="\frac{\left|\boldsymbol{\Omega}_{2,12}\boldsymbol{\Omega}_{2,22}^{-1}% \boldsymbol{\Omega}_{2,21}\right|}{\left|\boldsymbol{\Omega}_{2,11}-% 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xref="S2.E15.m1.15.15.15.8.2.2">21</cn></list></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E15.m1.16c">\frac{\left|\boldsymbol{\Omega}_{2,12}\boldsymbol{\Omega}_{2,22}^{-1}% \boldsymbol{\Omega}_{2,21}\right|}{\left|\boldsymbol{\Omega}_{2,11}-% \boldsymbol{\Omega}_{2,12}\boldsymbol{\Omega}_{2,22}^{-1}\boldsymbol{\Omega}_{% 2,21}\right|}.</annotation><annotation encoding="application/x-llamapun" id="S2.E15.m1.16d">divide start_ARG | bold_Ω start_POSTSUBSCRIPT 2 , 12 end_POSTSUBSCRIPT bold_Ω start_POSTSUBSCRIPT 2 , 22 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT bold_Ω start_POSTSUBSCRIPT 2 , 21 end_POSTSUBSCRIPT | end_ARG start_ARG | bold_Ω start_POSTSUBSCRIPT 2 , 11 end_POSTSUBSCRIPT - bold_Ω start_POSTSUBSCRIPT 2 , 12 end_POSTSUBSCRIPT bold_Ω start_POSTSUBSCRIPT 2 , 22 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT bold_Ω start_POSTSUBSCRIPT 2 , 21 end_POSTSUBSCRIPT | 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">(15)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS5.SSS2.p3.2">where <math alttext="\boldsymbol{\Omega}_{2}\sim W_{p}\left(n-1,\frac{1}{n-1}\boldsymbol{\Omega}_{1% }\right)" class="ltx_Math" display="inline" id="S2.SS5.SSS2.p3.1.m1.2"><semantics id="S2.SS5.SSS2.p3.1.m1.2a"><mrow id="S2.SS5.SSS2.p3.1.m1.2.2" xref="S2.SS5.SSS2.p3.1.m1.2.2.cmml"><msub id="S2.SS5.SSS2.p3.1.m1.2.2.4" xref="S2.SS5.SSS2.p3.1.m1.2.2.4.cmml"><mi id="S2.SS5.SSS2.p3.1.m1.2.2.4.2" xref="S2.SS5.SSS2.p3.1.m1.2.2.4.2.cmml">𝛀</mi><mn id="S2.SS5.SSS2.p3.1.m1.2.2.4.3" xref="S2.SS5.SSS2.p3.1.m1.2.2.4.3.cmml">2</mn></msub><mo id="S2.SS5.SSS2.p3.1.m1.2.2.3" xref="S2.SS5.SSS2.p3.1.m1.2.2.3.cmml">∼</mo><mrow id="S2.SS5.SSS2.p3.1.m1.2.2.2" xref="S2.SS5.SSS2.p3.1.m1.2.2.2.cmml"><msub 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xref="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2"><times id="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.1.cmml" xref="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.1"></times><apply id="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.2.cmml" xref="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.2"><divide id="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.2.1.cmml" xref="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.2"></divide><cn id="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.2.2.cmml" type="integer" xref="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.2.2">1</cn><apply id="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.2.3.cmml" xref="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.2.3"><minus id="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.2.3.1.cmml" xref="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.2.3.1"></minus><ci id="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.2.3.2.cmml" xref="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.2.3.2">𝑛</ci><cn id="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.2.3.3.cmml" type="integer" xref="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.2.3.3">1</cn></apply></apply><apply id="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.3.cmml" xref="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.3.1.cmml" xref="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.3">subscript</csymbol><ci id="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.3.2.cmml" xref="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.3.2">𝛀</ci><cn id="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.3.3.cmml" type="integer" xref="S2.SS5.SSS2.p3.1.m1.2.2.2.2.2.2.3.3">1</cn></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS2.p3.1.m1.2c">\boldsymbol{\Omega}_{2}\sim W_{p}\left(n-1,\frac{1}{n-1}\boldsymbol{\Omega}_{1% }\right)</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS2.p3.1.m1.2d">bold_Ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ italic_W start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_n - 1 , divide start_ARG 1 end_ARG start_ARG italic_n - 1 end_ARG bold_Ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, and where <math alttext="\boldsymbol{\Omega}_{1}\sim W_{p}\left(n-1,\mathbf{I}_{p}\right)" class="ltx_Math" display="inline" id="S2.SS5.SSS2.p3.2.m2.2"><semantics id="S2.SS5.SSS2.p3.2.m2.2a"><mrow id="S2.SS5.SSS2.p3.2.m2.2.2" xref="S2.SS5.SSS2.p3.2.m2.2.2.cmml"><msub id="S2.SS5.SSS2.p3.2.m2.2.2.4" xref="S2.SS5.SSS2.p3.2.m2.2.2.4.cmml"><mi id="S2.SS5.SSS2.p3.2.m2.2.2.4.2" xref="S2.SS5.SSS2.p3.2.m2.2.2.4.2.cmml">𝛀</mi><mn id="S2.SS5.SSS2.p3.2.m2.2.2.4.3" xref="S2.SS5.SSS2.p3.2.m2.2.2.4.3.cmml">1</mn></msub><mo id="S2.SS5.SSS2.p3.2.m2.2.2.3" xref="S2.SS5.SSS2.p3.2.m2.2.2.3.cmml">∼</mo><mrow id="S2.SS5.SSS2.p3.2.m2.2.2.2" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.cmml"><msub id="S2.SS5.SSS2.p3.2.m2.2.2.2.4" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.4.cmml"><mi id="S2.SS5.SSS2.p3.2.m2.2.2.2.4.2" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.4.2.cmml">W</mi><mi id="S2.SS5.SSS2.p3.2.m2.2.2.2.4.3" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.4.3.cmml">p</mi></msub><mo id="S2.SS5.SSS2.p3.2.m2.2.2.2.3" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.3.cmml"></mo><mrow id="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.2.3.cmml"><mo id="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2.3" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.2.3.cmml">(</mo><mrow id="S2.SS5.SSS2.p3.2.m2.1.1.1.1.1.1" xref="S2.SS5.SSS2.p3.2.m2.1.1.1.1.1.1.cmml"><mi id="S2.SS5.SSS2.p3.2.m2.1.1.1.1.1.1.2" xref="S2.SS5.SSS2.p3.2.m2.1.1.1.1.1.1.2.cmml">n</mi><mo id="S2.SS5.SSS2.p3.2.m2.1.1.1.1.1.1.1" xref="S2.SS5.SSS2.p3.2.m2.1.1.1.1.1.1.1.cmml">−</mo><mn id="S2.SS5.SSS2.p3.2.m2.1.1.1.1.1.1.3" xref="S2.SS5.SSS2.p3.2.m2.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2.4" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.2.3.cmml">,</mo><msub id="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2.2" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2.2.cmml"><mi id="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2.2.2" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2.2.2.cmml">𝐈</mi><mi id="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2.2.3" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2.2.3.cmml">p</mi></msub><mo id="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2.5" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS2.p3.2.m2.2b"><apply id="S2.SS5.SSS2.p3.2.m2.2.2.cmml" xref="S2.SS5.SSS2.p3.2.m2.2.2"><csymbol cd="latexml" id="S2.SS5.SSS2.p3.2.m2.2.2.3.cmml" xref="S2.SS5.SSS2.p3.2.m2.2.2.3">similar-to</csymbol><apply id="S2.SS5.SSS2.p3.2.m2.2.2.4.cmml" xref="S2.SS5.SSS2.p3.2.m2.2.2.4"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p3.2.m2.2.2.4.1.cmml" xref="S2.SS5.SSS2.p3.2.m2.2.2.4">subscript</csymbol><ci id="S2.SS5.SSS2.p3.2.m2.2.2.4.2.cmml" xref="S2.SS5.SSS2.p3.2.m2.2.2.4.2">𝛀</ci><cn id="S2.SS5.SSS2.p3.2.m2.2.2.4.3.cmml" type="integer" xref="S2.SS5.SSS2.p3.2.m2.2.2.4.3">1</cn></apply><apply id="S2.SS5.SSS2.p3.2.m2.2.2.2.cmml" xref="S2.SS5.SSS2.p3.2.m2.2.2.2"><times id="S2.SS5.SSS2.p3.2.m2.2.2.2.3.cmml" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.3"></times><apply id="S2.SS5.SSS2.p3.2.m2.2.2.2.4.cmml" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.4"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p3.2.m2.2.2.2.4.1.cmml" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.4">subscript</csymbol><ci id="S2.SS5.SSS2.p3.2.m2.2.2.2.4.2.cmml" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.4.2">𝑊</ci><ci id="S2.SS5.SSS2.p3.2.m2.2.2.2.4.3.cmml" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.4.3">𝑝</ci></apply><interval closure="open" id="S2.SS5.SSS2.p3.2.m2.2.2.2.2.3.cmml" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2"><apply id="S2.SS5.SSS2.p3.2.m2.1.1.1.1.1.1.cmml" xref="S2.SS5.SSS2.p3.2.m2.1.1.1.1.1.1"><minus id="S2.SS5.SSS2.p3.2.m2.1.1.1.1.1.1.1.cmml" xref="S2.SS5.SSS2.p3.2.m2.1.1.1.1.1.1.1"></minus><ci id="S2.SS5.SSS2.p3.2.m2.1.1.1.1.1.1.2.cmml" xref="S2.SS5.SSS2.p3.2.m2.1.1.1.1.1.1.2">𝑛</ci><cn id="S2.SS5.SSS2.p3.2.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS5.SSS2.p3.2.m2.1.1.1.1.1.1.3">1</cn></apply><apply id="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2.2.cmml" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2.2.1.cmml" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2.2">subscript</csymbol><ci id="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2.2.2.cmml" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2.2.2">𝐈</ci><ci id="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2.2.3.cmml" xref="S2.SS5.SSS2.p3.2.m2.2.2.2.2.2.2.3">𝑝</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS2.p3.2.m2.2c">\boldsymbol{\Omega}_{1}\sim W_{p}\left(n-1,\mathbf{I}_{p}\right)</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS2.p3.2.m2.2d">bold_Ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∼ italic_W start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_n - 1 , bold_I start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS5.SSS2.p4"> <p class="ltx_p" id="S2.SS5.SSS2.p4.5">For the regression test, <math alttext="\mathcal{H}_{0}" class="ltx_Math" display="inline" id="S2.SS5.SSS2.p4.1.m1.1"><semantics id="S2.SS5.SSS2.p4.1.m1.1a"><msub id="S2.SS5.SSS2.p4.1.m1.1.1" xref="S2.SS5.SSS2.p4.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.SSS2.p4.1.m1.1.1.2" xref="S2.SS5.SSS2.p4.1.m1.1.1.2.cmml">ℋ</mi><mn id="S2.SS5.SSS2.p4.1.m1.1.1.3" xref="S2.SS5.SSS2.p4.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS2.p4.1.m1.1b"><apply id="S2.SS5.SSS2.p4.1.m1.1.1.cmml" xref="S2.SS5.SSS2.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p4.1.m1.1.1.1.cmml" xref="S2.SS5.SSS2.p4.1.m1.1.1">subscript</csymbol><ci id="S2.SS5.SSS2.p4.1.m1.1.1.2.cmml" xref="S2.SS5.SSS2.p4.1.m1.1.1.2">ℋ</ci><cn id="S2.SS5.SSS2.p4.1.m1.1.1.3.cmml" type="integer" xref="S2.SS5.SSS2.p4.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS2.p4.1.m1.1c">\mathcal{H}_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS2.p4.1.m1.1d">caligraphic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> should be rejected at a level of significance <math alttext="\alpha" class="ltx_Math" display="inline" id="S2.SS5.SSS2.p4.2.m2.1"><semantics id="S2.SS5.SSS2.p4.2.m2.1a"><mi id="S2.SS5.SSS2.p4.2.m2.1.1" xref="S2.SS5.SSS2.p4.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS2.p4.2.m2.1b"><ci id="S2.SS5.SSS2.p4.2.m2.1.1.cmml" xref="S2.SS5.SSS2.p4.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS2.p4.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS2.p4.2.m2.1d">italic_α</annotation></semantics></math> if the observed value of <math alttext="T_{4}^{\star}" class="ltx_Math" display="inline" id="S2.SS5.SSS2.p4.3.m3.1"><semantics id="S2.SS5.SSS2.p4.3.m3.1a"><msubsup id="S2.SS5.SSS2.p4.3.m3.1.1" xref="S2.SS5.SSS2.p4.3.m3.1.1.cmml"><mi id="S2.SS5.SSS2.p4.3.m3.1.1.2.2" xref="S2.SS5.SSS2.p4.3.m3.1.1.2.2.cmml">T</mi><mn id="S2.SS5.SSS2.p4.3.m3.1.1.2.3" xref="S2.SS5.SSS2.p4.3.m3.1.1.2.3.cmml">4</mn><mo id="S2.SS5.SSS2.p4.3.m3.1.1.3" xref="S2.SS5.SSS2.p4.3.m3.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS2.p4.3.m3.1b"><apply id="S2.SS5.SSS2.p4.3.m3.1.1.cmml" xref="S2.SS5.SSS2.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p4.3.m3.1.1.1.cmml" xref="S2.SS5.SSS2.p4.3.m3.1.1">superscript</csymbol><apply id="S2.SS5.SSS2.p4.3.m3.1.1.2.cmml" xref="S2.SS5.SSS2.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p4.3.m3.1.1.2.1.cmml" xref="S2.SS5.SSS2.p4.3.m3.1.1">subscript</csymbol><ci id="S2.SS5.SSS2.p4.3.m3.1.1.2.2.cmml" xref="S2.SS5.SSS2.p4.3.m3.1.1.2.2">𝑇</ci><cn id="S2.SS5.SSS2.p4.3.m3.1.1.2.3.cmml" type="integer" xref="S2.SS5.SSS2.p4.3.m3.1.1.2.3">4</cn></apply><ci id="S2.SS5.SSS2.p4.3.m3.1.1.3.cmml" xref="S2.SS5.SSS2.p4.3.m3.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS2.p4.3.m3.1c">T_{4}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS2.p4.3.m3.1d">italic_T start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> is greater than <math alttext="t_{4;\alpha}^{\star}" class="ltx_Math" display="inline" id="S2.SS5.SSS2.p4.4.m4.2"><semantics id="S2.SS5.SSS2.p4.4.m4.2a"><msubsup id="S2.SS5.SSS2.p4.4.m4.2.3" xref="S2.SS5.SSS2.p4.4.m4.2.3.cmml"><mi id="S2.SS5.SSS2.p4.4.m4.2.3.2.2" xref="S2.SS5.SSS2.p4.4.m4.2.3.2.2.cmml">t</mi><mrow id="S2.SS5.SSS2.p4.4.m4.2.2.2.4" xref="S2.SS5.SSS2.p4.4.m4.2.2.2.3.cmml"><mn id="S2.SS5.SSS2.p4.4.m4.1.1.1.1" xref="S2.SS5.SSS2.p4.4.m4.1.1.1.1.cmml">4</mn><mo id="S2.SS5.SSS2.p4.4.m4.2.2.2.4.1" xref="S2.SS5.SSS2.p4.4.m4.2.2.2.3.cmml">;</mo><mi id="S2.SS5.SSS2.p4.4.m4.2.2.2.2" xref="S2.SS5.SSS2.p4.4.m4.2.2.2.2.cmml">α</mi></mrow><mo id="S2.SS5.SSS2.p4.4.m4.2.3.3" xref="S2.SS5.SSS2.p4.4.m4.2.3.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS2.p4.4.m4.2b"><apply id="S2.SS5.SSS2.p4.4.m4.2.3.cmml" xref="S2.SS5.SSS2.p4.4.m4.2.3"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p4.4.m4.2.3.1.cmml" xref="S2.SS5.SSS2.p4.4.m4.2.3">superscript</csymbol><apply id="S2.SS5.SSS2.p4.4.m4.2.3.2.cmml" xref="S2.SS5.SSS2.p4.4.m4.2.3"><csymbol cd="ambiguous" id="S2.SS5.SSS2.p4.4.m4.2.3.2.1.cmml" xref="S2.SS5.SSS2.p4.4.m4.2.3">subscript</csymbol><ci id="S2.SS5.SSS2.p4.4.m4.2.3.2.2.cmml" xref="S2.SS5.SSS2.p4.4.m4.2.3.2.2">𝑡</ci><list id="S2.SS5.SSS2.p4.4.m4.2.2.2.3.cmml" xref="S2.SS5.SSS2.p4.4.m4.2.2.2.4"><cn id="S2.SS5.SSS2.p4.4.m4.1.1.1.1.cmml" type="integer" xref="S2.SS5.SSS2.p4.4.m4.1.1.1.1">4</cn><ci id="S2.SS5.SSS2.p4.4.m4.2.2.2.2.cmml" xref="S2.SS5.SSS2.p4.4.m4.2.2.2.2">𝛼</ci></list></apply><ci id="S2.SS5.SSS2.p4.4.m4.2.3.3.cmml" xref="S2.SS5.SSS2.p4.4.m4.2.3.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS2.p4.4.m4.2c">t_{4;\alpha}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS2.p4.4.m4.2d">italic_t start_POSTSUBSCRIPT 4 ; italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>, the <math alttext="\alpha" class="ltx_Math" display="inline" id="S2.SS5.SSS2.p4.5.m5.1"><semantics id="S2.SS5.SSS2.p4.5.m5.1a"><mi id="S2.SS5.SSS2.p4.5.m5.1.1" xref="S2.SS5.SSS2.p4.5.m5.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.SSS2.p4.5.m5.1b"><ci id="S2.SS5.SSS2.p4.5.m5.1.1.cmml" xref="S2.SS5.SSS2.p4.5.m5.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.SSS2.p4.5.m5.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.SSS2.p4.5.m5.1d">italic_α</annotation></semantics></math>th percentile of (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E15" title="In 2.5.2 Regression of One set of Variables on other ‣ 2.5 Sphericity test ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">15</span></a>).</p> </div> </section> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Description and Implementation of PSInference</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">The <a class="ltx_ref ltx_href" href="https://cran.r-project.org/web/packages/PSinference/index.html" title="">PSinference</a> package is an R package designed to generate singly imputed fully synthetic data using PS under the assumption that the original data follows a multivariate normal distribution. The package provides tools for performing various inferential procedures on the synthetic data, including tests for generalized variance, sphericity, independence between two subsets of variables, and regression of one set of variables on another. These procedures are implemented according to the methodologies described in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2" title="2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">2</span></a>, ensuring robust and accurate statistical inference for synthetic datasets intended for release. It is freely available at the Comprehensive R Archive Network (CRAN) at <a class="ltx_ref ltx_url ltx_font_typewriter" href="https://cran.r-project.org/web/packages/PSinference/index.html" title="">https://cran.r-project.org/web/packages/PSinference/index.html</a>.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.1">The R package <a class="ltx_ref ltx_href" href="https://cran.r-project.org/web/packages/PSinference/index.html" title="">PSInference</a> contains the following functions: <span class="ltx_text ltx_font_typewriter" id="S3.p2.1.1">Canodist</span> (described in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E2" title="In 2.3 Generating fully synthetic data ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">2</span></a>)), <span class="ltx_text ltx_font_typewriter" id="S3.p2.1.2">GVdist</span> (described in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E5" title="In 2.4 Generalized variance ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">5</span></a>) using (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E6" title="In 2.4 Generalized variance ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">6</span></a>)), <span class="ltx_text ltx_font_typewriter" id="S3.p2.1.3">Inddist</span> (described in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E11" title="In 2.5.1 Independence test ‣ 2.5 Sphericity test ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">11</span></a>) using (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E12" title="In 2.5.1 Independence test ‣ 2.5 Sphericity test ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">12</span></a>)), <span class="ltx_text ltx_font_typewriter" id="S3.p2.1.4">Sphdist</span> (described in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E7" title="In 2.5 Sphericity test ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">7</span></a>) using (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E8" title="In 2.5 Sphericity test ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">8</span></a>)), <span class="ltx_text ltx_font_typewriter" id="S3.p2.1.5">partition</span>, and <span class="ltx_text ltx_font_typewriter" id="S3.p2.1.6">SimSynthData</span>, described summarily in Table <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:tab:table_1</span>.</p> </div> <figure class="ltx_table" id="S3.T1"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S3.T1.4.1.1" style="font-size:90%;">Table 1</span>: </span><span class="ltx_text" id="S3.T1.5.2" style="font-size:90%;">A description of functions in the <a class="ltx_ref ltx_href" href="https://cran.r-project.org/web/packages/PSinference/index.html" title="">PSInference</a> package, their arguments, and return values</span></figcaption> <table class="ltx_tabular" id="S3.T1.2"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S3.T1.2.3.1"> <th class="ltx_td ltx_align_justify ltx_align_top ltx_th ltx_th_column ltx_border_t" id="S3.T1.2.3.1.1"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.3.1.1.1"> <span class="ltx_p" id="S3.T1.2.3.1.1.1.1" style="width:85.4pt;"><span class="ltx_text ltx_font_bold" id="S3.T1.2.3.1.1.1.1.1">Function</span></span> </span> </th> <th class="ltx_td ltx_align_justify ltx_align_top ltx_th ltx_th_column ltx_border_t" id="S3.T1.2.3.1.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.3.1.2.1"> <span class="ltx_p" id="S3.T1.2.3.1.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_bold" id="S3.T1.2.3.1.2.1.1.1">Arguments</span></span> </span> </th> <th class="ltx_td ltx_align_justify ltx_align_top ltx_th ltx_th_column ltx_border_t" id="S3.T1.2.3.1.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.3.1.3.1"> <span class="ltx_p" id="S3.T1.2.3.1.3.1.1" style="width:130.9pt;"><span class="ltx_text ltx_font_bold" id="S3.T1.2.3.1.3.1.1.1">Description</span></span> </span> </th> <th class="ltx_td ltx_align_justify ltx_align_top ltx_th ltx_th_column ltx_border_t" id="S3.T1.2.3.1.4"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.3.1.4.1"> <span class="ltx_p" id="S3.T1.2.3.1.4.1.1" style="width:99.6pt;"><span class="ltx_text ltx_font_bold" id="S3.T1.2.3.1.4.1.1.1">Return</span></span> </span> </th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.T1.2.4.1"> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.2.4.1.1"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.4.1.1.1"> <span class="ltx_p" id="S3.T1.2.4.1.1.1.1" style="width:85.4pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.4.1.1.1.1.1">simSynthData()</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.2.4.1.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.4.1.2.1"> <span class="ltx_p" id="S3.T1.2.4.1.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.4.1.2.1.1.1">X</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.2.4.1.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.4.1.3.1"> <span class="ltx_p" id="S3.T1.2.4.1.3.1.1" style="width:130.9pt;">Matrix or dataframe</span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.2.4.1.4"></td> </tr> <tr class="ltx_tr" id="S3.T1.2.5.2"> <td class="ltx_td ltx_align_top" id="S3.T1.2.5.2.1"></td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.5.2.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.5.2.2.1"> <span class="ltx_p" id="S3.T1.2.5.2.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.5.2.2.1.1.1">n_imp</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.5.2.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.5.2.3.1"> <span class="ltx_p" id="S3.T1.2.5.2.3.1.1" style="width:130.9pt;">Sample size</span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.5.2.4"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.5.2.4.1"> <span class="ltx_p" id="S3.T1.2.5.2.4.1.1" style="width:99.6pt;">Matrix of generated synthetic dataset</span> </span> </td> </tr> <tr class="ltx_tr" id="S3.T1.2.6.3"> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.2.6.3.1"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.6.3.1.1"> <span class="ltx_p" id="S3.T1.2.6.3.1.1.1" style="width:85.4pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.6.3.1.1.1.1">GVdist()</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.2.6.3.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.6.3.2.1"> <span class="ltx_p" id="S3.T1.2.6.3.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.6.3.2.1.1.1">nsample</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.2.6.3.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.6.3.3.1"> <span class="ltx_p" id="S3.T1.2.6.3.3.1.1" style="width:130.9pt;">Sample size</span> </span> </td> <td class="ltx_td ltx_align_top ltx_border_t" id="S3.T1.2.6.3.4"></td> </tr> <tr class="ltx_tr" id="S3.T1.2.7.4"> <td class="ltx_td ltx_align_top" id="S3.T1.2.7.4.1"></td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.7.4.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.7.4.2.1"> <span class="ltx_p" id="S3.T1.2.7.4.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.7.4.2.1.1.1">pvariates</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.7.4.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.7.4.3.1"> <span class="ltx_p" id="S3.T1.2.7.4.3.1.1" style="width:130.9pt;">Number of variables</span> </span> </td> <td class="ltx_td ltx_align_top" id="S3.T1.2.7.4.4"></td> </tr> <tr class="ltx_tr" id="S3.T1.2.8.5"> <td class="ltx_td ltx_align_top" id="S3.T1.2.8.5.1"></td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.8.5.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.8.5.2.1"> <span class="ltx_p" id="S3.T1.2.8.5.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.8.5.2.1.1.1">iterations</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.8.5.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.8.5.3.1"> <span class="ltx_p" id="S3.T1.2.8.5.3.1.1" style="width:130.9pt;">Number of iterations for simulating values from the distribution and finding the quantiles. Default is 10000</span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.8.5.4"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.8.5.4.1"> <span class="ltx_p" id="S3.T1.2.8.5.4.1.1" style="width:99.6pt;">Vector of simulated distribution values for generalized variance quantiles</span> </span> </td> </tr> <tr class="ltx_tr" id="S3.T1.2.9.6"> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.2.9.6.1"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.9.6.1.1"> <span class="ltx_p" id="S3.T1.2.9.6.1.1.1" style="width:85.4pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.9.6.1.1.1.1">Sphdist()</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.2.9.6.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.9.6.2.1"> <span class="ltx_p" id="S3.T1.2.9.6.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.9.6.2.1.1.1">nsample</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.2.9.6.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.9.6.3.1"> <span class="ltx_p" id="S3.T1.2.9.6.3.1.1" style="width:130.9pt;">Sample size</span> </span> </td> <td class="ltx_td ltx_align_top ltx_border_t" id="S3.T1.2.9.6.4"></td> </tr> <tr class="ltx_tr" id="S3.T1.2.10.7"> <td class="ltx_td ltx_align_top" id="S3.T1.2.10.7.1"></td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.10.7.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.10.7.2.1"> <span class="ltx_p" id="S3.T1.2.10.7.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.10.7.2.1.1.1">pvariates</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.10.7.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.10.7.3.1"> <span class="ltx_p" id="S3.T1.2.10.7.3.1.1" style="width:130.9pt;">Number of variables</span> </span> </td> <td class="ltx_td ltx_align_top" id="S3.T1.2.10.7.4"></td> </tr> <tr class="ltx_tr" id="S3.T1.2.11.8"> <td class="ltx_td ltx_align_top" id="S3.T1.2.11.8.1"></td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.11.8.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.11.8.2.1"> <span class="ltx_p" id="S3.T1.2.11.8.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.11.8.2.1.1.1">iterations</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.11.8.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.11.8.3.1"> <span class="ltx_p" id="S3.T1.2.11.8.3.1.1" style="width:130.9pt;">Number of iterations for simulating values from the distribution and finding the quantiles. Default is 10000</span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.11.8.4"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.11.8.4.1"> <span class="ltx_p" id="S3.T1.2.11.8.4.1.1" style="width:99.6pt;">Vector of simulated distribution values for sphericity quantiles</span> </span> </td> </tr> <tr class="ltx_tr" id="S3.T1.1.1"> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.1.1.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.1.1.2.1"> <span class="ltx_p" id="S3.T1.1.1.2.1.1" style="width:85.4pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.1.1.2.1.1.1">Inddist()</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.1.1.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.1.1.3.1"> <span class="ltx_p" id="S3.T1.1.1.3.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.1.1.3.1.1.1">part</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.1.1.1"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.1.1.1.1"> <span class="ltx_p" id="S3.T1.1.1.1.1.1" style="width:130.9pt;">Length of the first subset of variables (<math alttext="p_{1}" class="ltx_Math" display="inline" id="S3.T1.1.1.1.1.1.m1.1"><semantics id="S3.T1.1.1.1.1.1.m1.1a"><msub id="S3.T1.1.1.1.1.1.m1.1.1" xref="S3.T1.1.1.1.1.1.m1.1.1.cmml"><mi id="S3.T1.1.1.1.1.1.m1.1.1.2" xref="S3.T1.1.1.1.1.1.m1.1.1.2.cmml">p</mi><mn id="S3.T1.1.1.1.1.1.m1.1.1.3" xref="S3.T1.1.1.1.1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.T1.1.1.1.1.1.m1.1b"><apply id="S3.T1.1.1.1.1.1.m1.1.1.cmml" xref="S3.T1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T1.1.1.1.1.1.m1.1.1.1.cmml" xref="S3.T1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S3.T1.1.1.1.1.1.m1.1.1.2.cmml" xref="S3.T1.1.1.1.1.1.m1.1.1.2">𝑝</ci><cn id="S3.T1.1.1.1.1.1.m1.1.1.3.cmml" type="integer" xref="S3.T1.1.1.1.1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.1.1.1.1.1.m1.1c">p_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.1.1.1.1.1.m1.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>)</span> </span> </td> <td class="ltx_td ltx_align_top ltx_border_t" id="S3.T1.1.1.4"></td> </tr> <tr class="ltx_tr" id="S3.T1.2.12.9"> <td class="ltx_td ltx_align_top" id="S3.T1.2.12.9.1"></td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.12.9.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.12.9.2.1"> <span class="ltx_p" id="S3.T1.2.12.9.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.12.9.2.1.1.1">nsample</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.12.9.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.12.9.3.1"> <span class="ltx_p" id="S3.T1.2.12.9.3.1.1" style="width:130.9pt;">Sample size</span> </span> </td> <td class="ltx_td ltx_align_top" id="S3.T1.2.12.9.4"></td> </tr> <tr class="ltx_tr" id="S3.T1.2.13.10"> <td class="ltx_td ltx_align_top" id="S3.T1.2.13.10.1"></td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.13.10.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.13.10.2.1"> <span class="ltx_p" id="S3.T1.2.13.10.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.13.10.2.1.1.1">pvariates</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.13.10.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.13.10.3.1"> <span class="ltx_p" id="S3.T1.2.13.10.3.1.1" style="width:130.9pt;">Number of variables</span> </span> </td> <td class="ltx_td ltx_align_top" id="S3.T1.2.13.10.4"></td> </tr> <tr class="ltx_tr" id="S3.T1.2.14.11"> <td class="ltx_td ltx_align_top" id="S3.T1.2.14.11.1"></td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.14.11.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.14.11.2.1"> <span class="ltx_p" id="S3.T1.2.14.11.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.14.11.2.1.1.1">iterations</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.14.11.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.14.11.3.1"> <span class="ltx_p" id="S3.T1.2.14.11.3.1.1" style="width:130.9pt;">Number of iterations for simulating values from the distribution and finding the quantiles. Default is 10000</span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.14.11.4"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.14.11.4.1"> <span class="ltx_p" id="S3.T1.2.14.11.4.1.1" style="width:99.6pt;">Vector of simulated distribution values for independence test quantiles</span> </span> </td> </tr> <tr class="ltx_tr" id="S3.T1.2.2"> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.2.2.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.2.2.1"> <span class="ltx_p" id="S3.T1.2.2.2.1.1" style="width:85.4pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.2.2.1.1.1">canodist()</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.2.2.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.2.3.1"> <span class="ltx_p" id="S3.T1.2.2.3.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.2.3.1.1.1">part</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.2.2.1"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.2.1.1"> <span class="ltx_p" id="S3.T1.2.2.1.1.1" style="width:130.9pt;">Length of the first subset of variables (<math alttext="p_{1}" class="ltx_Math" display="inline" id="S3.T1.2.2.1.1.1.m1.1"><semantics id="S3.T1.2.2.1.1.1.m1.1a"><msub id="S3.T1.2.2.1.1.1.m1.1.1" xref="S3.T1.2.2.1.1.1.m1.1.1.cmml"><mi id="S3.T1.2.2.1.1.1.m1.1.1.2" xref="S3.T1.2.2.1.1.1.m1.1.1.2.cmml">p</mi><mn id="S3.T1.2.2.1.1.1.m1.1.1.3" xref="S3.T1.2.2.1.1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.T1.2.2.1.1.1.m1.1b"><apply id="S3.T1.2.2.1.1.1.m1.1.1.cmml" xref="S3.T1.2.2.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T1.2.2.1.1.1.m1.1.1.1.cmml" xref="S3.T1.2.2.1.1.1.m1.1.1">subscript</csymbol><ci id="S3.T1.2.2.1.1.1.m1.1.1.2.cmml" xref="S3.T1.2.2.1.1.1.m1.1.1.2">𝑝</ci><cn id="S3.T1.2.2.1.1.1.m1.1.1.3.cmml" type="integer" xref="S3.T1.2.2.1.1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.2.2.1.1.1.m1.1c">p_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.2.2.1.1.1.m1.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>)</span> </span> </td> <td class="ltx_td ltx_align_top ltx_border_t" id="S3.T1.2.2.4"></td> </tr> <tr class="ltx_tr" id="S3.T1.2.15.12"> <td class="ltx_td ltx_align_top" id="S3.T1.2.15.12.1"></td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.15.12.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.15.12.2.1"> <span class="ltx_p" id="S3.T1.2.15.12.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.15.12.2.1.1.1">nsample</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.15.12.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.15.12.3.1"> <span class="ltx_p" id="S3.T1.2.15.12.3.1.1" style="width:130.9pt;">Sample size</span> </span> </td> <td class="ltx_td ltx_align_top" id="S3.T1.2.15.12.4"></td> </tr> <tr class="ltx_tr" id="S3.T1.2.16.13"> <td class="ltx_td ltx_align_top" id="S3.T1.2.16.13.1"></td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.16.13.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.16.13.2.1"> <span class="ltx_p" id="S3.T1.2.16.13.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.16.13.2.1.1.1">pvariates</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.16.13.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.16.13.3.1"> <span class="ltx_p" id="S3.T1.2.16.13.3.1.1" style="width:130.9pt;">Number of variables</span> </span> </td> <td class="ltx_td ltx_align_top" id="S3.T1.2.16.13.4"></td> </tr> <tr class="ltx_tr" id="S3.T1.2.17.14"> <td class="ltx_td ltx_align_top" id="S3.T1.2.17.14.1"></td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.17.14.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.17.14.2.1"> <span class="ltx_p" id="S3.T1.2.17.14.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.17.14.2.1.1.1">iterations</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.17.14.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.17.14.3.1"> <span class="ltx_p" id="S3.T1.2.17.14.3.1.1" style="width:130.9pt;">Number of iterations for simulating values from the distribution and finding the quantiles. Default is 10000</span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.17.14.4"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.17.14.4.1"> <span class="ltx_p" id="S3.T1.2.17.14.4.1.1" style="width:99.6pt;">Vector of simulated distribution values for regression test quantiles</span> </span> </td> </tr> <tr class="ltx_tr" id="S3.T1.2.18.15"> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.2.18.15.1"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.18.15.1.1"> <span class="ltx_p" id="S3.T1.2.18.15.1.1.1" style="width:85.4pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.18.15.1.1.1.1">partition()</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.2.18.15.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.18.15.2.1"> <span class="ltx_p" id="S3.T1.2.18.15.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.18.15.2.1.1.1">Matrix</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" id="S3.T1.2.18.15.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.18.15.3.1"> <span class="ltx_p" id="S3.T1.2.18.15.3.1.1" style="width:130.9pt;">A matrix to split</span> </span> </td> <td class="ltx_td ltx_align_top ltx_border_t" id="S3.T1.2.18.15.4"></td> </tr> <tr class="ltx_tr" id="S3.T1.2.19.16"> <td class="ltx_td ltx_align_top" id="S3.T1.2.19.16.1"></td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.19.16.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.19.16.2.1"> <span class="ltx_p" id="S3.T1.2.19.16.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.19.16.2.1.1.1">nrows</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top" id="S3.T1.2.19.16.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.19.16.3.1"> <span class="ltx_p" id="S3.T1.2.19.16.3.1.1" style="width:130.9pt;">Positive integer indicating the number of row blocks</span> </span> </td> <td class="ltx_td ltx_align_top" id="S3.T1.2.19.16.4"></td> </tr> <tr class="ltx_tr" id="S3.T1.2.20.17"> <td class="ltx_td ltx_align_top ltx_border_bb" id="S3.T1.2.20.17.1"></td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_bb" id="S3.T1.2.20.17.2"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.20.17.2.1"> <span class="ltx_p" id="S3.T1.2.20.17.2.1.1" style="width:71.1pt;"><span class="ltx_text ltx_font_typewriter" id="S3.T1.2.20.17.2.1.1.1">ncols</span></span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_bb" id="S3.T1.2.20.17.3"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.20.17.3.1"> <span class="ltx_p" id="S3.T1.2.20.17.3.1.1" style="width:130.9pt;">Positive integer indicating the number of column blocks</span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_bb" id="S3.T1.2.20.17.4"> <span class="ltx_inline-block ltx_align_top" id="S3.T1.2.20.17.4.1"> <span class="ltx_p" id="S3.T1.2.20.17.4.1.1" style="width:99.6pt;">List of partitioned sub-matrices</span> </span> </td> </tr> </tbody> </table> </figure> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.1">We must consider that the functions <span class="ltx_text ltx_font_typewriter" id="S3.p3.1.1">Inddist</span> and <span class="ltx_text ltx_font_typewriter" id="S3.p3.1.2">Sphdist</span> require the number of variables belonging to the first subset to be specified. For this, the user needs to understand that the variables will be selected in the order they appear in the dataset.</p> </div> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Numerical Studies and Demonstration of Programming Code</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">Even though the entire theoretical background can be found in <cite class="ltx_cite ltx_citemacro_citet">Klein, Moura, & Sinha (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib8" title="">2021</a>)</cite>, where all the proofs are analytical, there remains a need for practical verification of simulated data. We will use the practical verification of <cite class="ltx_cite ltx_citemacro_citet">Klein, Moura, & Sinha (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#bib.bib8" title="">2021</a>)</cite> theory to illustrate the use of the <a class="ltx_ref ltx_href" href="https://cran.r-project.org/web/packages/PSinference/index.html" title="">PSinference</a> package for generating synthetic data and conducting the inferential tests discussed in this paper.</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.2">For that purpose, under the multivariate normal model we will consider <math alttext="p=4" class="ltx_Math" display="inline" id="S4.p2.1.m1.1"><semantics id="S4.p2.1.m1.1a"><mrow id="S4.p2.1.m1.1.1" xref="S4.p2.1.m1.1.1.cmml"><mi id="S4.p2.1.m1.1.1.2" xref="S4.p2.1.m1.1.1.2.cmml">p</mi><mo id="S4.p2.1.m1.1.1.1" xref="S4.p2.1.m1.1.1.1.cmml">=</mo><mn id="S4.p2.1.m1.1.1.3" xref="S4.p2.1.m1.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.1.m1.1b"><apply id="S4.p2.1.m1.1.1.cmml" xref="S4.p2.1.m1.1.1"><eq id="S4.p2.1.m1.1.1.1.cmml" xref="S4.p2.1.m1.1.1.1"></eq><ci id="S4.p2.1.m1.1.1.2.cmml" xref="S4.p2.1.m1.1.1.2">𝑝</ci><cn id="S4.p2.1.m1.1.1.3.cmml" type="integer" xref="S4.p2.1.m1.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.1.m1.1c">p=4</annotation><annotation encoding="application/x-llamapun" id="S4.p2.1.m1.1d">italic_p = 4</annotation></semantics></math>, the number of variables, and <math alttext="\boldsymbol{\mu}=(1,2,3,4)^{\top}" class="ltx_Math" display="inline" id="S4.p2.2.m2.4"><semantics id="S4.p2.2.m2.4a"><mrow id="S4.p2.2.m2.4.5" xref="S4.p2.2.m2.4.5.cmml"><mi id="S4.p2.2.m2.4.5.2" xref="S4.p2.2.m2.4.5.2.cmml">𝝁</mi><mo id="S4.p2.2.m2.4.5.1" xref="S4.p2.2.m2.4.5.1.cmml">=</mo><msup id="S4.p2.2.m2.4.5.3" xref="S4.p2.2.m2.4.5.3.cmml"><mrow id="S4.p2.2.m2.4.5.3.2.2" xref="S4.p2.2.m2.4.5.3.2.1.cmml"><mo id="S4.p2.2.m2.4.5.3.2.2.1" stretchy="false" xref="S4.p2.2.m2.4.5.3.2.1.cmml">(</mo><mn id="S4.p2.2.m2.1.1" xref="S4.p2.2.m2.1.1.cmml">1</mn><mo id="S4.p2.2.m2.4.5.3.2.2.2" xref="S4.p2.2.m2.4.5.3.2.1.cmml">,</mo><mn id="S4.p2.2.m2.2.2" xref="S4.p2.2.m2.2.2.cmml">2</mn><mo id="S4.p2.2.m2.4.5.3.2.2.3" xref="S4.p2.2.m2.4.5.3.2.1.cmml">,</mo><mn id="S4.p2.2.m2.3.3" xref="S4.p2.2.m2.3.3.cmml">3</mn><mo id="S4.p2.2.m2.4.5.3.2.2.4" xref="S4.p2.2.m2.4.5.3.2.1.cmml">,</mo><mn id="S4.p2.2.m2.4.4" xref="S4.p2.2.m2.4.4.cmml">4</mn><mo id="S4.p2.2.m2.4.5.3.2.2.5" stretchy="false" xref="S4.p2.2.m2.4.5.3.2.1.cmml">)</mo></mrow><mo id="S4.p2.2.m2.4.5.3.3" xref="S4.p2.2.m2.4.5.3.3.cmml">⊤</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.2.m2.4b"><apply id="S4.p2.2.m2.4.5.cmml" xref="S4.p2.2.m2.4.5"><eq id="S4.p2.2.m2.4.5.1.cmml" xref="S4.p2.2.m2.4.5.1"></eq><ci id="S4.p2.2.m2.4.5.2.cmml" xref="S4.p2.2.m2.4.5.2">𝝁</ci><apply id="S4.p2.2.m2.4.5.3.cmml" xref="S4.p2.2.m2.4.5.3"><csymbol cd="ambiguous" id="S4.p2.2.m2.4.5.3.1.cmml" xref="S4.p2.2.m2.4.5.3">superscript</csymbol><vector id="S4.p2.2.m2.4.5.3.2.1.cmml" xref="S4.p2.2.m2.4.5.3.2.2"><cn id="S4.p2.2.m2.1.1.cmml" type="integer" xref="S4.p2.2.m2.1.1">1</cn><cn id="S4.p2.2.m2.2.2.cmml" type="integer" xref="S4.p2.2.m2.2.2">2</cn><cn id="S4.p2.2.m2.3.3.cmml" type="integer" xref="S4.p2.2.m2.3.3">3</cn><cn id="S4.p2.2.m2.4.4.cmml" type="integer" xref="S4.p2.2.m2.4.4">4</cn></vector><csymbol cd="latexml" id="S4.p2.2.m2.4.5.3.3.cmml" xref="S4.p2.2.m2.4.5.3.3">top</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.2.m2.4c">\boldsymbol{\mu}=(1,2,3,4)^{\top}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m2.4d">bold_italic_μ = ( 1 , 2 , 3 , 4 ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>, the population mean. 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1 end_CELL start_CELL 0.5 end_CELL start_CELL 0.5 end_CELL end_ROW start_ROW start_CELL 0.5 end_CELL start_CELL 0.5 end_CELL start_CELL 1 end_CELL start_CELL 0.5 end_CELL end_ROW start_ROW start_CELL 0.5 end_CELL start_CELL 0.5 end_CELL start_CELL 0.5 end_CELL start_CELL 1 end_CELL end_ROW end_ARRAY ) , bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL 0.5 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0.5 end_CELL start_CELL 2 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 3 end_CELL start_CELL 0.2 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0.2 end_CELL start_CELL 4 end_CELL end_ROW end_ARRAY )</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">(16)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.6">chosen to illustrate the performance for all tests presented in this work. We used Monte Carlo simulations with <math alttext="10^{5}" class="ltx_Math" display="inline" id="S4.p2.3.m1.1"><semantics id="S4.p2.3.m1.1a"><msup id="S4.p2.3.m1.1.1" xref="S4.p2.3.m1.1.1.cmml"><mn id="S4.p2.3.m1.1.1.2" xref="S4.p2.3.m1.1.1.2.cmml">10</mn><mn id="S4.p2.3.m1.1.1.3" xref="S4.p2.3.m1.1.1.3.cmml">5</mn></msup><annotation-xml encoding="MathML-Content" id="S4.p2.3.m1.1b"><apply id="S4.p2.3.m1.1.1.cmml" xref="S4.p2.3.m1.1.1"><csymbol cd="ambiguous" id="S4.p2.3.m1.1.1.1.cmml" xref="S4.p2.3.m1.1.1">superscript</csymbol><cn id="S4.p2.3.m1.1.1.2.cmml" type="integer" xref="S4.p2.3.m1.1.1.2">10</cn><cn id="S4.p2.3.m1.1.1.3.cmml" type="integer" xref="S4.p2.3.m1.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.3.m1.1c">10^{5}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.3.m1.1d">10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT</annotation></semantics></math> iterations and estimated the coverage probability (<math alttext="cov" class="ltx_Math" display="inline" id="S4.p2.4.m2.1"><semantics id="S4.p2.4.m2.1a"><mrow id="S4.p2.4.m2.1.1" xref="S4.p2.4.m2.1.1.cmml"><mi id="S4.p2.4.m2.1.1.2" xref="S4.p2.4.m2.1.1.2.cmml">c</mi><mo id="S4.p2.4.m2.1.1.1" xref="S4.p2.4.m2.1.1.1.cmml"></mo><mi id="S4.p2.4.m2.1.1.3" xref="S4.p2.4.m2.1.1.3.cmml">o</mi><mo id="S4.p2.4.m2.1.1.1a" xref="S4.p2.4.m2.1.1.1.cmml"></mo><mi id="S4.p2.4.m2.1.1.4" xref="S4.p2.4.m2.1.1.4.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.4.m2.1b"><apply id="S4.p2.4.m2.1.1.cmml" xref="S4.p2.4.m2.1.1"><times id="S4.p2.4.m2.1.1.1.cmml" xref="S4.p2.4.m2.1.1.1"></times><ci id="S4.p2.4.m2.1.1.2.cmml" xref="S4.p2.4.m2.1.1.2">𝑐</ci><ci id="S4.p2.4.m2.1.1.3.cmml" xref="S4.p2.4.m2.1.1.3">𝑜</ci><ci id="S4.p2.4.m2.1.1.4.cmml" xref="S4.p2.4.m2.1.1.4">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.4.m2.1c">cov</annotation><annotation encoding="application/x-llamapun" id="S4.p2.4.m2.1d">italic_c italic_o italic_v</annotation></semantics></math>) for <math alttext="\alpha=0.05" class="ltx_Math" display="inline" id="S4.p2.5.m3.1"><semantics id="S4.p2.5.m3.1a"><mrow id="S4.p2.5.m3.1.1" xref="S4.p2.5.m3.1.1.cmml"><mi id="S4.p2.5.m3.1.1.2" xref="S4.p2.5.m3.1.1.2.cmml">α</mi><mo id="S4.p2.5.m3.1.1.1" xref="S4.p2.5.m3.1.1.1.cmml">=</mo><mn id="S4.p2.5.m3.1.1.3" xref="S4.p2.5.m3.1.1.3.cmml">0.05</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.5.m3.1b"><apply id="S4.p2.5.m3.1.1.cmml" xref="S4.p2.5.m3.1.1"><eq id="S4.p2.5.m3.1.1.1.cmml" xref="S4.p2.5.m3.1.1.1"></eq><ci id="S4.p2.5.m3.1.1.2.cmml" xref="S4.p2.5.m3.1.1.2">𝛼</ci><cn id="S4.p2.5.m3.1.1.3.cmml" type="float" xref="S4.p2.5.m3.1.1.3">0.05</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.5.m3.1c">\alpha=0.05</annotation><annotation encoding="application/x-llamapun" id="S4.p2.5.m3.1d">italic_α = 0.05</annotation></semantics></math> significance level. The estimated coverage probability is the proportion of iterations where the observed values of the test statistics fall inside the non-rejection region. For every iteration the PS single imputed dataset is created using the models described in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2" title="2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">2</span></a> and the sample sizes considered were <math alttext="n=10,20,100,500" class="ltx_Math" display="inline" id="S4.p2.6.m4.4"><semantics id="S4.p2.6.m4.4a"><mrow id="S4.p2.6.m4.4.5" xref="S4.p2.6.m4.4.5.cmml"><mi id="S4.p2.6.m4.4.5.2" xref="S4.p2.6.m4.4.5.2.cmml">n</mi><mo id="S4.p2.6.m4.4.5.1" xref="S4.p2.6.m4.4.5.1.cmml">=</mo><mrow id="S4.p2.6.m4.4.5.3.2" xref="S4.p2.6.m4.4.5.3.1.cmml"><mn id="S4.p2.6.m4.1.1" xref="S4.p2.6.m4.1.1.cmml">10</mn><mo id="S4.p2.6.m4.4.5.3.2.1" xref="S4.p2.6.m4.4.5.3.1.cmml">,</mo><mn id="S4.p2.6.m4.2.2" xref="S4.p2.6.m4.2.2.cmml">20</mn><mo id="S4.p2.6.m4.4.5.3.2.2" xref="S4.p2.6.m4.4.5.3.1.cmml">,</mo><mn id="S4.p2.6.m4.3.3" xref="S4.p2.6.m4.3.3.cmml">100</mn><mo id="S4.p2.6.m4.4.5.3.2.3" xref="S4.p2.6.m4.4.5.3.1.cmml">,</mo><mn id="S4.p2.6.m4.4.4" xref="S4.p2.6.m4.4.4.cmml">500</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.6.m4.4b"><apply id="S4.p2.6.m4.4.5.cmml" xref="S4.p2.6.m4.4.5"><eq id="S4.p2.6.m4.4.5.1.cmml" xref="S4.p2.6.m4.4.5.1"></eq><ci id="S4.p2.6.m4.4.5.2.cmml" xref="S4.p2.6.m4.4.5.2">𝑛</ci><list id="S4.p2.6.m4.4.5.3.1.cmml" xref="S4.p2.6.m4.4.5.3.2"><cn id="S4.p2.6.m4.1.1.cmml" type="integer" xref="S4.p2.6.m4.1.1">10</cn><cn id="S4.p2.6.m4.2.2.cmml" type="integer" xref="S4.p2.6.m4.2.2">20</cn><cn id="S4.p2.6.m4.3.3.cmml" type="integer" xref="S4.p2.6.m4.3.3">100</cn><cn id="S4.p2.6.m4.4.4.cmml" type="integer" xref="S4.p2.6.m4.4.4">500</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.6.m4.4c">n=10,20,100,500</annotation><annotation encoding="application/x-llamapun" id="S4.p2.6.m4.4d">italic_n = 10 , 20 , 100 , 500</annotation></semantics></math>.</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>Average coverage probability of the Generalized Variance</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.7">For the generalized variance, the illustrative example is applied to the covariance matrices <math alttext="\boldsymbol{\Sigma}_{3}" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.1"><semantics id="S4.SS1.p1.1.m1.1a"><msub id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml"><mi id="S4.SS1.p1.1.m1.1.1.2" xref="S4.SS1.p1.1.m1.1.1.2.cmml">𝚺</mi><mn id="S4.SS1.p1.1.m1.1.1.3" xref="S4.SS1.p1.1.m1.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><apply id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.1.m1.1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1">subscript</csymbol><ci id="S4.SS1.p1.1.m1.1.1.2.cmml" xref="S4.SS1.p1.1.m1.1.1.2">𝚺</ci><cn id="S4.SS1.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.SS1.p1.1.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">\boldsymbol{\Sigma}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.1d">bold_Σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.SS1.p1.2.m2.1"><semantics id="S4.SS1.p1.2.m2.1a"><msub id="S4.SS1.p1.2.m2.1.1" xref="S4.SS1.p1.2.m2.1.1.cmml"><mi id="S4.SS1.p1.2.m2.1.1.2" xref="S4.SS1.p1.2.m2.1.1.2.cmml">𝚺</mi><mn id="S4.SS1.p1.2.m2.1.1.3" xref="S4.SS1.p1.2.m2.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.2.m2.1b"><apply id="S4.SS1.p1.2.m2.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.2.m2.1.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS1.p1.2.m2.1.1.2.cmml" xref="S4.SS1.p1.2.m2.1.1.2">𝚺</ci><cn id="S4.SS1.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.SS1.p1.2.m2.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.2.m2.1c">\boldsymbol{\Sigma}_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.2.m2.1d">bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math> defined in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4.E16" title="In 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">16</span></a>). These matrices, along with the defined mean vector <math alttext="\boldsymbol{\mu}" class="ltx_Math" display="inline" id="S4.SS1.p1.3.m3.1"><semantics id="S4.SS1.p1.3.m3.1a"><mi id="S4.SS1.p1.3.m3.1.1" xref="S4.SS1.p1.3.m3.1.1.cmml">𝝁</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.3.m3.1b"><ci id="S4.SS1.p1.3.m3.1.1.cmml" xref="S4.SS1.p1.3.m3.1.1">𝝁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.3.m3.1c">\boldsymbol{\mu}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.3.m3.1d">bold_italic_μ</annotation></semantics></math>, are used in each iteration to generate an original dataset, which is then used to create a synthetic dataset via the <span class="ltx_text ltx_font_typewriter" id="S4.SS1.p1.7.1">simSynthData</span> function. In each iteration, the observed values of <math alttext="T_{1}^{\star}" class="ltx_Math" display="inline" id="S4.SS1.p1.4.m4.1"><semantics id="S4.SS1.p1.4.m4.1a"><msubsup id="S4.SS1.p1.4.m4.1.1" xref="S4.SS1.p1.4.m4.1.1.cmml"><mi id="S4.SS1.p1.4.m4.1.1.2.2" xref="S4.SS1.p1.4.m4.1.1.2.2.cmml">T</mi><mn id="S4.SS1.p1.4.m4.1.1.2.3" xref="S4.SS1.p1.4.m4.1.1.2.3.cmml">1</mn><mo id="S4.SS1.p1.4.m4.1.1.3" xref="S4.SS1.p1.4.m4.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.4.m4.1b"><apply id="S4.SS1.p1.4.m4.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.4.m4.1.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1">superscript</csymbol><apply id="S4.SS1.p1.4.m4.1.1.2.cmml" xref="S4.SS1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.4.m4.1.1.2.1.cmml" xref="S4.SS1.p1.4.m4.1.1">subscript</csymbol><ci id="S4.SS1.p1.4.m4.1.1.2.2.cmml" xref="S4.SS1.p1.4.m4.1.1.2.2">𝑇</ci><cn id="S4.SS1.p1.4.m4.1.1.2.3.cmml" type="integer" xref="S4.SS1.p1.4.m4.1.1.2.3">1</cn></apply><ci id="S4.SS1.p1.4.m4.1.1.3.cmml" xref="S4.SS1.p1.4.m4.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.4.m4.1c">T_{1}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.4.m4.1d">italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> are calculated for both scenarios and stored to compute the coverage probabilities (<math alttext="cov" class="ltx_Math" display="inline" id="S4.SS1.p1.5.m5.1"><semantics id="S4.SS1.p1.5.m5.1a"><mrow id="S4.SS1.p1.5.m5.1.1" xref="S4.SS1.p1.5.m5.1.1.cmml"><mi id="S4.SS1.p1.5.m5.1.1.2" xref="S4.SS1.p1.5.m5.1.1.2.cmml">c</mi><mo id="S4.SS1.p1.5.m5.1.1.1" xref="S4.SS1.p1.5.m5.1.1.1.cmml"></mo><mi id="S4.SS1.p1.5.m5.1.1.3" xref="S4.SS1.p1.5.m5.1.1.3.cmml">o</mi><mo id="S4.SS1.p1.5.m5.1.1.1a" xref="S4.SS1.p1.5.m5.1.1.1.cmml"></mo><mi id="S4.SS1.p1.5.m5.1.1.4" xref="S4.SS1.p1.5.m5.1.1.4.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.5.m5.1b"><apply id="S4.SS1.p1.5.m5.1.1.cmml" xref="S4.SS1.p1.5.m5.1.1"><times id="S4.SS1.p1.5.m5.1.1.1.cmml" xref="S4.SS1.p1.5.m5.1.1.1"></times><ci id="S4.SS1.p1.5.m5.1.1.2.cmml" xref="S4.SS1.p1.5.m5.1.1.2">𝑐</ci><ci id="S4.SS1.p1.5.m5.1.1.3.cmml" xref="S4.SS1.p1.5.m5.1.1.3">𝑜</ci><ci id="S4.SS1.p1.5.m5.1.1.4.cmml" xref="S4.SS1.p1.5.m5.1.1.4">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.5.m5.1c">cov</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.5.m5.1d">italic_c italic_o italic_v</annotation></semantics></math>) for the two cases considered. Finally, the two <math alttext="cov" class="ltx_Math" display="inline" id="S4.SS1.p1.6.m6.1"><semantics id="S4.SS1.p1.6.m6.1a"><mrow id="S4.SS1.p1.6.m6.1.1" xref="S4.SS1.p1.6.m6.1.1.cmml"><mi id="S4.SS1.p1.6.m6.1.1.2" xref="S4.SS1.p1.6.m6.1.1.2.cmml">c</mi><mo id="S4.SS1.p1.6.m6.1.1.1" xref="S4.SS1.p1.6.m6.1.1.1.cmml"></mo><mi id="S4.SS1.p1.6.m6.1.1.3" xref="S4.SS1.p1.6.m6.1.1.3.cmml">o</mi><mo id="S4.SS1.p1.6.m6.1.1.1a" xref="S4.SS1.p1.6.m6.1.1.1.cmml"></mo><mi id="S4.SS1.p1.6.m6.1.1.4" xref="S4.SS1.p1.6.m6.1.1.4.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.6.m6.1b"><apply id="S4.SS1.p1.6.m6.1.1.cmml" xref="S4.SS1.p1.6.m6.1.1"><times id="S4.SS1.p1.6.m6.1.1.1.cmml" xref="S4.SS1.p1.6.m6.1.1.1"></times><ci id="S4.SS1.p1.6.m6.1.1.2.cmml" xref="S4.SS1.p1.6.m6.1.1.2">𝑐</ci><ci id="S4.SS1.p1.6.m6.1.1.3.cmml" xref="S4.SS1.p1.6.m6.1.1.3">𝑜</ci><ci id="S4.SS1.p1.6.m6.1.1.4.cmml" xref="S4.SS1.p1.6.m6.1.1.4">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.6.m6.1c">cov</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.6.m6.1d">italic_c italic_o italic_v</annotation></semantics></math> values are computed after calculating the quantiles for <math alttext="T_{1}^{\star}" class="ltx_Math" display="inline" id="S4.SS1.p1.7.m7.1"><semantics id="S4.SS1.p1.7.m7.1a"><msubsup id="S4.SS1.p1.7.m7.1.1" xref="S4.SS1.p1.7.m7.1.1.cmml"><mi id="S4.SS1.p1.7.m7.1.1.2.2" xref="S4.SS1.p1.7.m7.1.1.2.2.cmml">T</mi><mn id="S4.SS1.p1.7.m7.1.1.2.3" xref="S4.SS1.p1.7.m7.1.1.2.3.cmml">1</mn><mo id="S4.SS1.p1.7.m7.1.1.3" xref="S4.SS1.p1.7.m7.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.7.m7.1b"><apply id="S4.SS1.p1.7.m7.1.1.cmml" xref="S4.SS1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.7.m7.1.1.1.cmml" xref="S4.SS1.p1.7.m7.1.1">superscript</csymbol><apply id="S4.SS1.p1.7.m7.1.1.2.cmml" xref="S4.SS1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.7.m7.1.1.2.1.cmml" xref="S4.SS1.p1.7.m7.1.1">subscript</csymbol><ci id="S4.SS1.p1.7.m7.1.1.2.2.cmml" xref="S4.SS1.p1.7.m7.1.1.2.2">𝑇</ci><cn id="S4.SS1.p1.7.m7.1.1.2.3.cmml" type="integer" xref="S4.SS1.p1.7.m7.1.1.2.3">1</cn></apply><ci id="S4.SS1.p1.7.m7.1.1.3.cmml" xref="S4.SS1.p1.7.m7.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.7.m7.1c">T_{1}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.7.m7.1d">italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> using the distribution obtained with <span class="ltx_text ltx_font_typewriter" id="S4.SS1.p1.7.2">GVdist</span>, and the results are presented.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <pre class="ltx_verbatim ltx_font_typewriter" id="S4.SS1.p2.1"> library(MASS) library(PSinference) # Set random seed for reproducibility set.seed(123) # Total number of Monte Carlo simulations N <- 100000 # Sample size n <- 10 #replace by 20, 100, 500 for all other cases # Population mean vector mu <- c(1, 2, 3, 4) # Number of covariates (dimensionality) p <- length(mu) # Covariance matrix 1 (Diagonal matrix with variance 1) sigma_1 <- diag(c(1, 1, 1, 1), 4, 4) # Covariance matrix 2 (Diagonal matrix with variance 5) sigma_2 <- diag(c(5, 5, 5, 5), 4, 4) # Covariance matrix 3 (homogeneous) sigma_3 <- matrix(c(1, .5, .5, .5, .5, 1, .5, .5, .5, .5, 1, .5, .5, .5, .5, 1), 4, 4) # Covariance matrix 4 (heterogeneous) sigma_4 <- matrix(c(1, .5, 0, 0, .5, 2, 0, 0, 0, 0, 3, .2, 0, 0, .2, 4), 4, 4) # Get the generalized variance distribution from PSinference Ts <- GVdist(nsample = n, pvariates = p, iterations = N) # Quantiles for the simulated distribution at alpha = 0.05 q975 <- quantile(Ts, probs = c(.975)) q025 <- quantile(Ts, probs = c(.025)) # Initialize result vectors for two test datasets T1 <- c() T2 <- c() # Monte Carlo loop for generating synthetic data and perform simulation for (i in 1:N) { # Generate original data samples from multivariate normal distribution # for mu and sigma_3 and sigma_4 x1 <- mvrnorm(n, mu, sigma_3) x2 <- mvrnorm(n, mu, sigma_4) # Generate PS synthetic data for both datasets using simSynthData # from PSinference v1 <- simSynthData(x1) v2 <- simSynthData(x2) # PS estimates of mu (mean vector) for both datasets mean_v1 <- apply(v1, 2, mean) mean_v2 <- apply(v2, 2, mean) # Create matrices of mean values for vectorized subtraction MEANv1 <- matrix(mean_v1, n, 4, byrow = TRUE) MEANv2 <- matrix(mean_v2, n, 4, byrow = TRUE) # Compute the covariance matrices of the synthetic data # (could be done just using var function) s_star1 <- t(v1 - MEANv1) \%*\% (v1 - MEANv1) s_star2 <- t(v2 - MEANv2) \%*\% (v2 - MEANv2) # Calculate observed T*_1 statistics for both datasets T1temp <- ((n - 1)^p) * det(s_star1) / det(sigma_3) T2temp <- ((n - 1)^p) * det(s_star2) / det(sigma_4) T1[i] <- T1temp T2[i] <- T2temp } # Print the quantiles of the simulated T1 values (observed and distribution) print(quantile(T1, probs = c(0, 0.1, .5, .9, 1))) print(quantile(T2, probs = c(0, 0.1, .5, .9, 1))) print(quantile(Ts, probs = c(0, 0.1, .5, .9, 1))) # Compute average coverage probabilities # Rejection rate for 1st observed T*_1 rej1 <- mean(T1 < q025) + mean(T1 > q975) # Rejection rate for 2nd observed T*_1 rej2 <- mean(T2 < q025) + mean(T2 > q975) cov1 <- 1 - rej1 # Coverage probability for 1st case cov2 <- 1 - rej2 # Coverage probability for 2nd case # Print the coverage probabilities print(c(cov1, cov2)) </pre> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2 </span>Average coverage probability for the Sphericity Test</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.3">Similar to the previous example/numerical studies, we illustrate the Sphericity Scenario, but now using matrices <math alttext="\boldsymbol{\Sigma}_{1}" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><msub id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml"><mi id="S4.SS2.p1.1.m1.1.1.2" xref="S4.SS2.p1.1.m1.1.1.2.cmml">𝚺</mi><mn id="S4.SS2.p1.1.m1.1.1.3" xref="S4.SS2.p1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><apply id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.1.m1.1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p1.1.m1.1.1.2.cmml" xref="S4.SS2.p1.1.m1.1.1.2">𝚺</ci><cn id="S4.SS2.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">\boldsymbol{\Sigma}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{\Sigma}_{2}" class="ltx_Math" display="inline" id="S4.SS2.p1.2.m2.1"><semantics id="S4.SS2.p1.2.m2.1a"><msub id="S4.SS2.p1.2.m2.1.1" xref="S4.SS2.p1.2.m2.1.1.cmml"><mi id="S4.SS2.p1.2.m2.1.1.2" xref="S4.SS2.p1.2.m2.1.1.2.cmml">𝚺</mi><mn id="S4.SS2.p1.2.m2.1.1.3" xref="S4.SS2.p1.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.2.m2.1b"><apply id="S4.SS2.p1.2.m2.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.2.m2.1.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.p1.2.m2.1.1.2.cmml" xref="S4.SS2.p1.2.m2.1.1.2">𝚺</ci><cn id="S4.SS2.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.m2.1c">\boldsymbol{\Sigma}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.2.m2.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> defined in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4.E16" title="In 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">16</span></a>), which are spherical matrices. Obviously, now we base our simulations on <math alttext="T_{2}^{\star}" class="ltx_Math" display="inline" id="S4.SS2.p1.3.m3.1"><semantics id="S4.SS2.p1.3.m3.1a"><msubsup id="S4.SS2.p1.3.m3.1.1" xref="S4.SS2.p1.3.m3.1.1.cmml"><mi id="S4.SS2.p1.3.m3.1.1.2.2" xref="S4.SS2.p1.3.m3.1.1.2.2.cmml">T</mi><mn id="S4.SS2.p1.3.m3.1.1.2.3" xref="S4.SS2.p1.3.m3.1.1.2.3.cmml">2</mn><mo id="S4.SS2.p1.3.m3.1.1.3" xref="S4.SS2.p1.3.m3.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.3.m3.1b"><apply id="S4.SS2.p1.3.m3.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.3.m3.1.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1">superscript</csymbol><apply id="S4.SS2.p1.3.m3.1.1.2.cmml" xref="S4.SS2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.3.m3.1.1.2.1.cmml" xref="S4.SS2.p1.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.p1.3.m3.1.1.2.2.cmml" xref="S4.SS2.p1.3.m3.1.1.2.2">𝑇</ci><cn id="S4.SS2.p1.3.m3.1.1.2.3.cmml" type="integer" xref="S4.SS2.p1.3.m3.1.1.2.3">2</cn></apply><ci id="S4.SS2.p1.3.m3.1.1.3.cmml" xref="S4.SS2.p1.3.m3.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.3.m3.1c">T_{2}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.3.m3.1d">italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> and the function <span class="ltx_text ltx_font_typewriter" id="S4.SS2.p1.3.1">Sphdist</span>.</p> </div> <div class="ltx_para" id="S4.SS2.p2"> <pre class="ltx_verbatim ltx_font_typewriter" id="S4.SS2.p2.1"> # Using the same given parameters, vectors and matrices set.seed(123) # Set random seed for reproducibility # Monte Carlo simulation to generate distribution for T*_(2, alpha) Ts <- Sphdist(n, p, N) # Quantile alpha = 0.05 of the simulated values q05 <- quantile(Ts, probs = c(.05)) # Initialize result vectors for the test statistics T1 <- c() T2 <- c() # Monte Carlo loop to calculate T*_2 statistics for # two different covariance matrices for (i in 1:N) { # Generate original data samples from multivariate normal distribution # for mu and sigma_1 and sigma_2 x1 <- mvrnorm(n, mu, sigma_1) x2 <- mvrnorm(n, mu, sigma_2) # Generate PS synthetic data for both datasets using simSynthData # from PSinference v1 <- simSynthData(x1) v2 <- simSynthData(x2) # PS estimates of mu for both datasets mean_v1 <- apply(v1, 2, mean) mean_v2 <- apply(v2, 2, mean) # Create matrices of mean values for vectorized subtraction MEANv1 <- matrix(mean_v1, n, 4, byrow = TRUE) MEANv2 <- matrix(mean_v2, n, 4, byrow = TRUE) # Covariance matrices of the synthetic data s_star1 <- t(v1 - MEANv1) \%*\% (v1 - MEANv1) s_star2 <- t(v2 - MEANv2) \%*\% (v2 - MEANv2) # Calculate T*_2 statistics for both datasets T1temp <- (det(s_star1)^(1/p)) / sum(diag(s_star1)) T2temp <- (det(s_star2)^(1/p)) / sum(diag(s_star2)) T1[i] <- T1temp # Store the statistic for dataset 1 T2[i] <- T2temp # Store the statistic for dataset 2 } # Print quantiles of the T1 and T2 distributions print(quantile(T1, probs = c(0, 0.1, 0.5, 0.9, 1))) print(quantile(T2, probs = c(0, 0.1, 0.5, 0.9, 1))) print(quantile(Ts, probs = c(0, 0.1, 0.5, 0.9, 1))) # Calculate and print coverage probabilities cov1 <- mean(T1 > q05) cov2 <- mean(T2 > q05) print(c(cov1, cov2)) </pre> </div> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.3 </span>Average coverage probability for the Independence Test</h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.8">The simulation process here follows the same structure of the previous two scenarios, applied to matrices <math alttext="\boldsymbol{\Sigma}_{1}" class="ltx_Math" display="inline" id="S4.SS3.p1.1.m1.1"><semantics id="S4.SS3.p1.1.m1.1a"><msub id="S4.SS3.p1.1.m1.1.1" xref="S4.SS3.p1.1.m1.1.1.cmml"><mi id="S4.SS3.p1.1.m1.1.1.2" xref="S4.SS3.p1.1.m1.1.1.2.cmml">𝚺</mi><mn id="S4.SS3.p1.1.m1.1.1.3" xref="S4.SS3.p1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.1.m1.1b"><apply id="S4.SS3.p1.1.m1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.1.m1.1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1">subscript</csymbol><ci id="S4.SS3.p1.1.m1.1.1.2.cmml" xref="S4.SS3.p1.1.m1.1.1.2">𝚺</ci><cn id="S4.SS3.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.SS3.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.1.m1.1c">\boldsymbol{\Sigma}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.1.m1.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.SS3.p1.2.m2.1"><semantics id="S4.SS3.p1.2.m2.1a"><msub id="S4.SS3.p1.2.m2.1.1" xref="S4.SS3.p1.2.m2.1.1.cmml"><mi id="S4.SS3.p1.2.m2.1.1.2" xref="S4.SS3.p1.2.m2.1.1.2.cmml">𝚺</mi><mn id="S4.SS3.p1.2.m2.1.1.3" xref="S4.SS3.p1.2.m2.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.2.m2.1b"><apply id="S4.SS3.p1.2.m2.1.1.cmml" xref="S4.SS3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.2.m2.1.1.1.cmml" xref="S4.SS3.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS3.p1.2.m2.1.1.2.cmml" xref="S4.SS3.p1.2.m2.1.1.2">𝚺</ci><cn id="S4.SS3.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.SS3.p1.2.m2.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.2.m2.1c">\boldsymbol{\Sigma}_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.2.m2.1d">bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math> defined in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4.E16" title="In 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">16</span></a>). However, since the partition of <math alttext="\mathbf{S}^{\star}" class="ltx_Math" display="inline" id="S4.SS3.p1.3.m3.1"><semantics id="S4.SS3.p1.3.m3.1a"><msup id="S4.SS3.p1.3.m3.1.1" xref="S4.SS3.p1.3.m3.1.1.cmml"><mi id="S4.SS3.p1.3.m3.1.1.2" xref="S4.SS3.p1.3.m3.1.1.2.cmml">𝐒</mi><mo id="S4.SS3.p1.3.m3.1.1.3" xref="S4.SS3.p1.3.m3.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.3.m3.1b"><apply id="S4.SS3.p1.3.m3.1.1.cmml" xref="S4.SS3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.3.m3.1.1.1.cmml" xref="S4.SS3.p1.3.m3.1.1">superscript</csymbol><ci id="S4.SS3.p1.3.m3.1.1.2.cmml" xref="S4.SS3.p1.3.m3.1.1.2">𝐒</ci><ci id="S4.SS3.p1.3.m3.1.1.3.cmml" xref="S4.SS3.p1.3.m3.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.3.m3.1c">\mathbf{S}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.3.m3.1d">bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> is needed, the <span class="ltx_text ltx_font_typewriter" id="S4.SS3.p1.8.1">partition</span> function is used, as described in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2.E9" title="In 2.5.1 Independence test ‣ 2.5 Sphericity test ‣ 2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">9</span></a>), making <math alttext="p_{1}=1" class="ltx_Math" display="inline" id="S4.SS3.p1.4.m4.1"><semantics id="S4.SS3.p1.4.m4.1a"><mrow id="S4.SS3.p1.4.m4.1.1" xref="S4.SS3.p1.4.m4.1.1.cmml"><msub id="S4.SS3.p1.4.m4.1.1.2" xref="S4.SS3.p1.4.m4.1.1.2.cmml"><mi id="S4.SS3.p1.4.m4.1.1.2.2" xref="S4.SS3.p1.4.m4.1.1.2.2.cmml">p</mi><mn id="S4.SS3.p1.4.m4.1.1.2.3" xref="S4.SS3.p1.4.m4.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS3.p1.4.m4.1.1.1" xref="S4.SS3.p1.4.m4.1.1.1.cmml">=</mo><mn id="S4.SS3.p1.4.m4.1.1.3" xref="S4.SS3.p1.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.4.m4.1b"><apply id="S4.SS3.p1.4.m4.1.1.cmml" xref="S4.SS3.p1.4.m4.1.1"><eq id="S4.SS3.p1.4.m4.1.1.1.cmml" xref="S4.SS3.p1.4.m4.1.1.1"></eq><apply id="S4.SS3.p1.4.m4.1.1.2.cmml" xref="S4.SS3.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.4.m4.1.1.2.1.cmml" xref="S4.SS3.p1.4.m4.1.1.2">subscript</csymbol><ci id="S4.SS3.p1.4.m4.1.1.2.2.cmml" xref="S4.SS3.p1.4.m4.1.1.2.2">𝑝</ci><cn id="S4.SS3.p1.4.m4.1.1.2.3.cmml" type="integer" xref="S4.SS3.p1.4.m4.1.1.2.3">1</cn></apply><cn id="S4.SS3.p1.4.m4.1.1.3.cmml" type="integer" xref="S4.SS3.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.4.m4.1c">p_{1}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.4.m4.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math> when using <math alttext="\boldsymbol{\Sigma}_{1}" class="ltx_Math" display="inline" id="S4.SS3.p1.5.m5.1"><semantics id="S4.SS3.p1.5.m5.1a"><msub id="S4.SS3.p1.5.m5.1.1" xref="S4.SS3.p1.5.m5.1.1.cmml"><mi id="S4.SS3.p1.5.m5.1.1.2" xref="S4.SS3.p1.5.m5.1.1.2.cmml">𝚺</mi><mn id="S4.SS3.p1.5.m5.1.1.3" xref="S4.SS3.p1.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.5.m5.1b"><apply id="S4.SS3.p1.5.m5.1.1.cmml" xref="S4.SS3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.5.m5.1.1.1.cmml" xref="S4.SS3.p1.5.m5.1.1">subscript</csymbol><ci id="S4.SS3.p1.5.m5.1.1.2.cmml" xref="S4.SS3.p1.5.m5.1.1.2">𝚺</ci><cn id="S4.SS3.p1.5.m5.1.1.3.cmml" type="integer" xref="S4.SS3.p1.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.5.m5.1c">\boldsymbol{\Sigma}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.5.m5.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="p_{1}=2" class="ltx_Math" display="inline" id="S4.SS3.p1.6.m6.1"><semantics id="S4.SS3.p1.6.m6.1a"><mrow id="S4.SS3.p1.6.m6.1.1" xref="S4.SS3.p1.6.m6.1.1.cmml"><msub id="S4.SS3.p1.6.m6.1.1.2" xref="S4.SS3.p1.6.m6.1.1.2.cmml"><mi id="S4.SS3.p1.6.m6.1.1.2.2" xref="S4.SS3.p1.6.m6.1.1.2.2.cmml">p</mi><mn id="S4.SS3.p1.6.m6.1.1.2.3" xref="S4.SS3.p1.6.m6.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS3.p1.6.m6.1.1.1" xref="S4.SS3.p1.6.m6.1.1.1.cmml">=</mo><mn id="S4.SS3.p1.6.m6.1.1.3" xref="S4.SS3.p1.6.m6.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.6.m6.1b"><apply id="S4.SS3.p1.6.m6.1.1.cmml" xref="S4.SS3.p1.6.m6.1.1"><eq id="S4.SS3.p1.6.m6.1.1.1.cmml" xref="S4.SS3.p1.6.m6.1.1.1"></eq><apply id="S4.SS3.p1.6.m6.1.1.2.cmml" xref="S4.SS3.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.6.m6.1.1.2.1.cmml" xref="S4.SS3.p1.6.m6.1.1.2">subscript</csymbol><ci id="S4.SS3.p1.6.m6.1.1.2.2.cmml" xref="S4.SS3.p1.6.m6.1.1.2.2">𝑝</ci><cn id="S4.SS3.p1.6.m6.1.1.2.3.cmml" type="integer" xref="S4.SS3.p1.6.m6.1.1.2.3">1</cn></apply><cn id="S4.SS3.p1.6.m6.1.1.3.cmml" type="integer" xref="S4.SS3.p1.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.6.m6.1c">p_{1}=2</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.6.m6.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2</annotation></semantics></math> when using <math alttext="\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.SS3.p1.7.m7.1"><semantics id="S4.SS3.p1.7.m7.1a"><msub id="S4.SS3.p1.7.m7.1.1" xref="S4.SS3.p1.7.m7.1.1.cmml"><mi id="S4.SS3.p1.7.m7.1.1.2" xref="S4.SS3.p1.7.m7.1.1.2.cmml">𝚺</mi><mn id="S4.SS3.p1.7.m7.1.1.3" xref="S4.SS3.p1.7.m7.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.7.m7.1b"><apply id="S4.SS3.p1.7.m7.1.1.cmml" xref="S4.SS3.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.7.m7.1.1.1.cmml" xref="S4.SS3.p1.7.m7.1.1">subscript</csymbol><ci id="S4.SS3.p1.7.m7.1.1.2.cmml" xref="S4.SS3.p1.7.m7.1.1.2">𝚺</ci><cn id="S4.SS3.p1.7.m7.1.1.3.cmml" type="integer" xref="S4.SS3.p1.7.m7.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.7.m7.1c">\boldsymbol{\Sigma}_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.7.m7.1d">bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math>. In this case, the numerical study uses the <span class="ltx_text ltx_font_typewriter" id="S4.SS3.p1.8.2">Inddist</span> function to obtain the distribution of <math alttext="T_{3}^{\star}" class="ltx_Math" display="inline" id="S4.SS3.p1.8.m8.1"><semantics id="S4.SS3.p1.8.m8.1a"><msubsup id="S4.SS3.p1.8.m8.1.1" xref="S4.SS3.p1.8.m8.1.1.cmml"><mi id="S4.SS3.p1.8.m8.1.1.2.2" xref="S4.SS3.p1.8.m8.1.1.2.2.cmml">T</mi><mn id="S4.SS3.p1.8.m8.1.1.2.3" xref="S4.SS3.p1.8.m8.1.1.2.3.cmml">3</mn><mo id="S4.SS3.p1.8.m8.1.1.3" xref="S4.SS3.p1.8.m8.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.8.m8.1b"><apply id="S4.SS3.p1.8.m8.1.1.cmml" xref="S4.SS3.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.8.m8.1.1.1.cmml" xref="S4.SS3.p1.8.m8.1.1">superscript</csymbol><apply id="S4.SS3.p1.8.m8.1.1.2.cmml" xref="S4.SS3.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.8.m8.1.1.2.1.cmml" xref="S4.SS3.p1.8.m8.1.1">subscript</csymbol><ci id="S4.SS3.p1.8.m8.1.1.2.2.cmml" xref="S4.SS3.p1.8.m8.1.1.2.2">𝑇</ci><cn id="S4.SS3.p1.8.m8.1.1.2.3.cmml" type="integer" xref="S4.SS3.p1.8.m8.1.1.2.3">3</cn></apply><ci id="S4.SS3.p1.8.m8.1.1.3.cmml" xref="S4.SS3.p1.8.m8.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.8.m8.1c">T_{3}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.8.m8.1d">italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS3.p2"> <pre class="ltx_verbatim ltx_font_typewriter" id="S4.SS3.p2.1"> # Using the same given parameters, vectors and matrices set.seed(123) # Set random seed for reproducibility # partition sizes p1 <- 1 # Partition size 1 p2 <- 2 # Partition size 2 # Perform Monte Carlo simulations to generate the distribution # for the test statistics T1 <- Inddist(p1, n, p, N) # Simulate for the first partition T2 <- Inddist(p2, n, p, N) # Simulate for the second partition # Quantile alpha = 0.05 for the simulated values q05_1 <- quantile(T1, probs = c(.05)) q05_2 <- quantile(T2, probs = c(.05)) # Initialize result vectors for the second set of test statistics T1_1 <- c() T2_1 <- c() # Monte Carlo loop to calculate the T*_3 statistics for # different covariance matrices for (i in 1:N) { # Generate original data samples from multivariate normal distribution # with given mu and sigma_1 and sigma_4 x1 <- mvrnorm(n, mu, sigma_1) # Sample using sigma_1 x2 <- mvrnorm(n, mu, sigma_4) # Sample using sigma_4 # Generate PS synthetic single data for both datasets v1 <- simSynthData(x1) v2 <- simSynthData(x2) # PLS estimates of mu for both datasets mean_v1 <- apply(v1, 2, mean) mean_v2 <- apply(v2, 2, mean) # Create matrices of mean values for vectorized subtraction MEANv1 <- matrix(mean_v1, n, 4, byrow = TRUE) MEANv2 <- matrix(mean_v2, n, 4, byrow = TRUE) # Compute the covariance matrices of the synthetic data s_star1 <- t(v1 - MEANv1) \%*\% (v1 - MEANv1) s_star2 <- t(v2 - MEANv2) \%*\% (v2 - MEANv2) # Partition the covariance matrices into subsets s_star1_11 <- as.matrix(partition(s_star1, p1, p1)[[1]]) s_star1_22 <- as.matrix(partition(s_star1, p1, p1)[[4]]) s_star2_11 <- partition(s_star2, p2, p2)[[1]] s_star2_22 <- partition(s_star2, p2, p2)[[4]] # Calculate the T*_3 statistics for both datasets # T*_3 for first dataset T1temp <- det(s_star1) / (det(s_star1_11) * det(s_star1_22)) # T*_3 for second dataset T2temp <- det(s_star2) / (det(s_star2_11) * det(s_star2_22)) # Store the results in T1_1 and T2_1 T1_1[i] <- T1temp T2_1[i] <- T2temp } # Print quantiles of the distributions for T1, T1_1, T2, and T2_1 print(quantile(T1, probs = c(0, 0.1, .5, .9, 1))) print(quantile(T1_1, probs = c(0, 0.1, .5, .9, 1))) print(quantile(T2, probs = c(0, 0.1, .5, .9, 1))) print(quantile(T2_1, probs = c(0, 0.1, .5, .9, 1))) # Average coverage calculations cov1 <- mean(T1_1 > q05_1) # Coverage for T1_1 cov2 <- mean(T2_1 > q05_2) # Coverage for T2_1 # Print coverage probabilities print(c(cov1, cov2)) </pre> </div> </section> <section class="ltx_subsection" id="S4.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.4 </span>Average coverage probability for the Regression Test</h3> <div class="ltx_para" id="S4.SS4.p1"> <p class="ltx_p" id="S4.SS4.p1.14">For the regression test <math alttext="cov" class="ltx_Math" display="inline" id="S4.SS4.p1.1.m1.1"><semantics id="S4.SS4.p1.1.m1.1a"><mrow id="S4.SS4.p1.1.m1.1.1" xref="S4.SS4.p1.1.m1.1.1.cmml"><mi id="S4.SS4.p1.1.m1.1.1.2" xref="S4.SS4.p1.1.m1.1.1.2.cmml">c</mi><mo id="S4.SS4.p1.1.m1.1.1.1" xref="S4.SS4.p1.1.m1.1.1.1.cmml"></mo><mi id="S4.SS4.p1.1.m1.1.1.3" xref="S4.SS4.p1.1.m1.1.1.3.cmml">o</mi><mo id="S4.SS4.p1.1.m1.1.1.1a" xref="S4.SS4.p1.1.m1.1.1.1.cmml"></mo><mi id="S4.SS4.p1.1.m1.1.1.4" xref="S4.SS4.p1.1.m1.1.1.4.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.1.m1.1b"><apply id="S4.SS4.p1.1.m1.1.1.cmml" xref="S4.SS4.p1.1.m1.1.1"><times id="S4.SS4.p1.1.m1.1.1.1.cmml" xref="S4.SS4.p1.1.m1.1.1.1"></times><ci id="S4.SS4.p1.1.m1.1.1.2.cmml" xref="S4.SS4.p1.1.m1.1.1.2">𝑐</ci><ci id="S4.SS4.p1.1.m1.1.1.3.cmml" xref="S4.SS4.p1.1.m1.1.1.3">𝑜</ci><ci id="S4.SS4.p1.1.m1.1.1.4.cmml" xref="S4.SS4.p1.1.m1.1.1.4">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.1.m1.1c">cov</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.1.m1.1d">italic_c italic_o italic_v</annotation></semantics></math> computation, we use the covariance matrices <math alttext="\boldsymbol{\Sigma}_{3}" class="ltx_Math" display="inline" id="S4.SS4.p1.2.m2.1"><semantics id="S4.SS4.p1.2.m2.1a"><msub id="S4.SS4.p1.2.m2.1.1" xref="S4.SS4.p1.2.m2.1.1.cmml"><mi id="S4.SS4.p1.2.m2.1.1.2" xref="S4.SS4.p1.2.m2.1.1.2.cmml">𝚺</mi><mn id="S4.SS4.p1.2.m2.1.1.3" xref="S4.SS4.p1.2.m2.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.2.m2.1b"><apply id="S4.SS4.p1.2.m2.1.1.cmml" xref="S4.SS4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS4.p1.2.m2.1.1.1.cmml" xref="S4.SS4.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS4.p1.2.m2.1.1.2.cmml" xref="S4.SS4.p1.2.m2.1.1.2">𝚺</ci><cn id="S4.SS4.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.SS4.p1.2.m2.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.2.m2.1c">\boldsymbol{\Sigma}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.2.m2.1d">bold_Σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.SS4.p1.3.m3.1"><semantics id="S4.SS4.p1.3.m3.1a"><msub id="S4.SS4.p1.3.m3.1.1" xref="S4.SS4.p1.3.m3.1.1.cmml"><mi id="S4.SS4.p1.3.m3.1.1.2" xref="S4.SS4.p1.3.m3.1.1.2.cmml">𝚺</mi><mn id="S4.SS4.p1.3.m3.1.1.3" xref="S4.SS4.p1.3.m3.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.3.m3.1b"><apply id="S4.SS4.p1.3.m3.1.1.cmml" xref="S4.SS4.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS4.p1.3.m3.1.1.1.cmml" xref="S4.SS4.p1.3.m3.1.1">subscript</csymbol><ci id="S4.SS4.p1.3.m3.1.1.2.cmml" xref="S4.SS4.p1.3.m3.1.1.2">𝚺</ci><cn id="S4.SS4.p1.3.m3.1.1.3.cmml" type="integer" xref="S4.SS4.p1.3.m3.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.3.m3.1c">\boldsymbol{\Sigma}_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.3.m3.1d">bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math>, as defined in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4.E16" title="In 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">16</span></a>). At each iteration, the observed values of the test statistic <math alttext="T_{4}^{\star}" class="ltx_Math" display="inline" id="S4.SS4.p1.4.m4.1"><semantics id="S4.SS4.p1.4.m4.1a"><msubsup id="S4.SS4.p1.4.m4.1.1" xref="S4.SS4.p1.4.m4.1.1.cmml"><mi id="S4.SS4.p1.4.m4.1.1.2.2" xref="S4.SS4.p1.4.m4.1.1.2.2.cmml">T</mi><mn id="S4.SS4.p1.4.m4.1.1.2.3" xref="S4.SS4.p1.4.m4.1.1.2.3.cmml">4</mn><mo id="S4.SS4.p1.4.m4.1.1.3" xref="S4.SS4.p1.4.m4.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.4.m4.1b"><apply id="S4.SS4.p1.4.m4.1.1.cmml" xref="S4.SS4.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS4.p1.4.m4.1.1.1.cmml" xref="S4.SS4.p1.4.m4.1.1">superscript</csymbol><apply id="S4.SS4.p1.4.m4.1.1.2.cmml" xref="S4.SS4.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS4.p1.4.m4.1.1.2.1.cmml" xref="S4.SS4.p1.4.m4.1.1">subscript</csymbol><ci id="S4.SS4.p1.4.m4.1.1.2.2.cmml" xref="S4.SS4.p1.4.m4.1.1.2.2">𝑇</ci><cn id="S4.SS4.p1.4.m4.1.1.2.3.cmml" type="integer" xref="S4.SS4.p1.4.m4.1.1.2.3">4</cn></apply><ci id="S4.SS4.p1.4.m4.1.1.3.cmml" xref="S4.SS4.p1.4.m4.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.4.m4.1c">T_{4}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.4.m4.1d">italic_T start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> are computed for both scenarios, making <math alttext="p_{1}=2" class="ltx_Math" display="inline" id="S4.SS4.p1.5.m5.1"><semantics id="S4.SS4.p1.5.m5.1a"><mrow id="S4.SS4.p1.5.m5.1.1" xref="S4.SS4.p1.5.m5.1.1.cmml"><msub id="S4.SS4.p1.5.m5.1.1.2" xref="S4.SS4.p1.5.m5.1.1.2.cmml"><mi id="S4.SS4.p1.5.m5.1.1.2.2" xref="S4.SS4.p1.5.m5.1.1.2.2.cmml">p</mi><mn id="S4.SS4.p1.5.m5.1.1.2.3" xref="S4.SS4.p1.5.m5.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS4.p1.5.m5.1.1.1" xref="S4.SS4.p1.5.m5.1.1.1.cmml">=</mo><mn id="S4.SS4.p1.5.m5.1.1.3" xref="S4.SS4.p1.5.m5.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.5.m5.1b"><apply id="S4.SS4.p1.5.m5.1.1.cmml" xref="S4.SS4.p1.5.m5.1.1"><eq id="S4.SS4.p1.5.m5.1.1.1.cmml" xref="S4.SS4.p1.5.m5.1.1.1"></eq><apply id="S4.SS4.p1.5.m5.1.1.2.cmml" xref="S4.SS4.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p1.5.m5.1.1.2.1.cmml" xref="S4.SS4.p1.5.m5.1.1.2">subscript</csymbol><ci id="S4.SS4.p1.5.m5.1.1.2.2.cmml" xref="S4.SS4.p1.5.m5.1.1.2.2">𝑝</ci><cn id="S4.SS4.p1.5.m5.1.1.2.3.cmml" type="integer" xref="S4.SS4.p1.5.m5.1.1.2.3">1</cn></apply><cn id="S4.SS4.p1.5.m5.1.1.3.cmml" type="integer" xref="S4.SS4.p1.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.5.m5.1c">p_{1}=2</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.5.m5.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2</annotation></semantics></math> when using <math alttext="\boldsymbol{\Sigma}_{3}" class="ltx_Math" display="inline" id="S4.SS4.p1.6.m6.1"><semantics id="S4.SS4.p1.6.m6.1a"><msub id="S4.SS4.p1.6.m6.1.1" xref="S4.SS4.p1.6.m6.1.1.cmml"><mi id="S4.SS4.p1.6.m6.1.1.2" xref="S4.SS4.p1.6.m6.1.1.2.cmml">𝚺</mi><mn id="S4.SS4.p1.6.m6.1.1.3" xref="S4.SS4.p1.6.m6.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.6.m6.1b"><apply id="S4.SS4.p1.6.m6.1.1.cmml" xref="S4.SS4.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS4.p1.6.m6.1.1.1.cmml" xref="S4.SS4.p1.6.m6.1.1">subscript</csymbol><ci id="S4.SS4.p1.6.m6.1.1.2.cmml" xref="S4.SS4.p1.6.m6.1.1.2">𝚺</ci><cn id="S4.SS4.p1.6.m6.1.1.3.cmml" type="integer" xref="S4.SS4.p1.6.m6.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.6.m6.1c">\boldsymbol{\Sigma}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.6.m6.1d">bold_Σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="p_{1}=1" class="ltx_Math" display="inline" id="S4.SS4.p1.7.m7.1"><semantics id="S4.SS4.p1.7.m7.1a"><mrow id="S4.SS4.p1.7.m7.1.1" xref="S4.SS4.p1.7.m7.1.1.cmml"><msub id="S4.SS4.p1.7.m7.1.1.2" xref="S4.SS4.p1.7.m7.1.1.2.cmml"><mi id="S4.SS4.p1.7.m7.1.1.2.2" xref="S4.SS4.p1.7.m7.1.1.2.2.cmml">p</mi><mn id="S4.SS4.p1.7.m7.1.1.2.3" xref="S4.SS4.p1.7.m7.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS4.p1.7.m7.1.1.1" xref="S4.SS4.p1.7.m7.1.1.1.cmml">=</mo><mn id="S4.SS4.p1.7.m7.1.1.3" xref="S4.SS4.p1.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.7.m7.1b"><apply id="S4.SS4.p1.7.m7.1.1.cmml" xref="S4.SS4.p1.7.m7.1.1"><eq id="S4.SS4.p1.7.m7.1.1.1.cmml" xref="S4.SS4.p1.7.m7.1.1.1"></eq><apply id="S4.SS4.p1.7.m7.1.1.2.cmml" xref="S4.SS4.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS4.p1.7.m7.1.1.2.1.cmml" xref="S4.SS4.p1.7.m7.1.1.2">subscript</csymbol><ci id="S4.SS4.p1.7.m7.1.1.2.2.cmml" xref="S4.SS4.p1.7.m7.1.1.2.2">𝑝</ci><cn id="S4.SS4.p1.7.m7.1.1.2.3.cmml" type="integer" xref="S4.SS4.p1.7.m7.1.1.2.3">1</cn></apply><cn id="S4.SS4.p1.7.m7.1.1.3.cmml" type="integer" xref="S4.SS4.p1.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.7.m7.1c">p_{1}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.7.m7.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math> when using <math alttext="\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.SS4.p1.8.m8.1"><semantics id="S4.SS4.p1.8.m8.1a"><msub id="S4.SS4.p1.8.m8.1.1" xref="S4.SS4.p1.8.m8.1.1.cmml"><mi id="S4.SS4.p1.8.m8.1.1.2" xref="S4.SS4.p1.8.m8.1.1.2.cmml">𝚺</mi><mn id="S4.SS4.p1.8.m8.1.1.3" xref="S4.SS4.p1.8.m8.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.8.m8.1b"><apply id="S4.SS4.p1.8.m8.1.1.cmml" xref="S4.SS4.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS4.p1.8.m8.1.1.1.cmml" xref="S4.SS4.p1.8.m8.1.1">subscript</csymbol><ci id="S4.SS4.p1.8.m8.1.1.2.cmml" xref="S4.SS4.p1.8.m8.1.1.2">𝚺</ci><cn id="S4.SS4.p1.8.m8.1.1.3.cmml" type="integer" xref="S4.SS4.p1.8.m8.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.8.m8.1c">\boldsymbol{\Sigma}_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.8.m8.1d">bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math>. Here, the test statistic <math alttext="T_{4}^{\star}" class="ltx_Math" display="inline" id="S4.SS4.p1.9.m9.1"><semantics id="S4.SS4.p1.9.m9.1a"><msubsup id="S4.SS4.p1.9.m9.1.1" xref="S4.SS4.p1.9.m9.1.1.cmml"><mi id="S4.SS4.p1.9.m9.1.1.2.2" xref="S4.SS4.p1.9.m9.1.1.2.2.cmml">T</mi><mn id="S4.SS4.p1.9.m9.1.1.2.3" xref="S4.SS4.p1.9.m9.1.1.2.3.cmml">4</mn><mo id="S4.SS4.p1.9.m9.1.1.3" xref="S4.SS4.p1.9.m9.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.9.m9.1b"><apply id="S4.SS4.p1.9.m9.1.1.cmml" xref="S4.SS4.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS4.p1.9.m9.1.1.1.cmml" xref="S4.SS4.p1.9.m9.1.1">superscript</csymbol><apply id="S4.SS4.p1.9.m9.1.1.2.cmml" xref="S4.SS4.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS4.p1.9.m9.1.1.2.1.cmml" xref="S4.SS4.p1.9.m9.1.1">subscript</csymbol><ci id="S4.SS4.p1.9.m9.1.1.2.2.cmml" xref="S4.SS4.p1.9.m9.1.1.2.2">𝑇</ci><cn id="S4.SS4.p1.9.m9.1.1.2.3.cmml" type="integer" xref="S4.SS4.p1.9.m9.1.1.2.3">4</cn></apply><ci id="S4.SS4.p1.9.m9.1.1.3.cmml" xref="S4.SS4.p1.9.m9.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.9.m9.1c">T_{4}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.9.m9.1d">italic_T start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> is based on the matrix of regression coefficients <math alttext="\boldsymbol{\Delta}=\boldsymbol{\Sigma}_{12}\boldsymbol{\Sigma}_{22}^{-1}" class="ltx_Math" display="inline" id="S4.SS4.p1.10.m10.1"><semantics id="S4.SS4.p1.10.m10.1a"><mrow id="S4.SS4.p1.10.m10.1.1" xref="S4.SS4.p1.10.m10.1.1.cmml"><mi id="S4.SS4.p1.10.m10.1.1.2" xref="S4.SS4.p1.10.m10.1.1.2.cmml">𝚫</mi><mo id="S4.SS4.p1.10.m10.1.1.1" xref="S4.SS4.p1.10.m10.1.1.1.cmml">=</mo><mrow id="S4.SS4.p1.10.m10.1.1.3" xref="S4.SS4.p1.10.m10.1.1.3.cmml"><msub id="S4.SS4.p1.10.m10.1.1.3.2" xref="S4.SS4.p1.10.m10.1.1.3.2.cmml"><mi id="S4.SS4.p1.10.m10.1.1.3.2.2" xref="S4.SS4.p1.10.m10.1.1.3.2.2.cmml">𝚺</mi><mn id="S4.SS4.p1.10.m10.1.1.3.2.3" xref="S4.SS4.p1.10.m10.1.1.3.2.3.cmml">12</mn></msub><mo id="S4.SS4.p1.10.m10.1.1.3.1" xref="S4.SS4.p1.10.m10.1.1.3.1.cmml"></mo><msubsup id="S4.SS4.p1.10.m10.1.1.3.3" xref="S4.SS4.p1.10.m10.1.1.3.3.cmml"><mi id="S4.SS4.p1.10.m10.1.1.3.3.2.2" xref="S4.SS4.p1.10.m10.1.1.3.3.2.2.cmml">𝚺</mi><mn id="S4.SS4.p1.10.m10.1.1.3.3.2.3" xref="S4.SS4.p1.10.m10.1.1.3.3.2.3.cmml">22</mn><mrow id="S4.SS4.p1.10.m10.1.1.3.3.3" xref="S4.SS4.p1.10.m10.1.1.3.3.3.cmml"><mo id="S4.SS4.p1.10.m10.1.1.3.3.3a" xref="S4.SS4.p1.10.m10.1.1.3.3.3.cmml">−</mo><mn id="S4.SS4.p1.10.m10.1.1.3.3.3.2" xref="S4.SS4.p1.10.m10.1.1.3.3.3.2.cmml">1</mn></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.10.m10.1b"><apply id="S4.SS4.p1.10.m10.1.1.cmml" xref="S4.SS4.p1.10.m10.1.1"><eq id="S4.SS4.p1.10.m10.1.1.1.cmml" xref="S4.SS4.p1.10.m10.1.1.1"></eq><ci id="S4.SS4.p1.10.m10.1.1.2.cmml" xref="S4.SS4.p1.10.m10.1.1.2">𝚫</ci><apply id="S4.SS4.p1.10.m10.1.1.3.cmml" xref="S4.SS4.p1.10.m10.1.1.3"><times id="S4.SS4.p1.10.m10.1.1.3.1.cmml" xref="S4.SS4.p1.10.m10.1.1.3.1"></times><apply id="S4.SS4.p1.10.m10.1.1.3.2.cmml" xref="S4.SS4.p1.10.m10.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS4.p1.10.m10.1.1.3.2.1.cmml" xref="S4.SS4.p1.10.m10.1.1.3.2">subscript</csymbol><ci id="S4.SS4.p1.10.m10.1.1.3.2.2.cmml" xref="S4.SS4.p1.10.m10.1.1.3.2.2">𝚺</ci><cn id="S4.SS4.p1.10.m10.1.1.3.2.3.cmml" type="integer" xref="S4.SS4.p1.10.m10.1.1.3.2.3">12</cn></apply><apply id="S4.SS4.p1.10.m10.1.1.3.3.cmml" xref="S4.SS4.p1.10.m10.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS4.p1.10.m10.1.1.3.3.1.cmml" xref="S4.SS4.p1.10.m10.1.1.3.3">superscript</csymbol><apply id="S4.SS4.p1.10.m10.1.1.3.3.2.cmml" xref="S4.SS4.p1.10.m10.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS4.p1.10.m10.1.1.3.3.2.1.cmml" xref="S4.SS4.p1.10.m10.1.1.3.3">subscript</csymbol><ci id="S4.SS4.p1.10.m10.1.1.3.3.2.2.cmml" xref="S4.SS4.p1.10.m10.1.1.3.3.2.2">𝚺</ci><cn id="S4.SS4.p1.10.m10.1.1.3.3.2.3.cmml" type="integer" xref="S4.SS4.p1.10.m10.1.1.3.3.2.3">22</cn></apply><apply id="S4.SS4.p1.10.m10.1.1.3.3.3.cmml" xref="S4.SS4.p1.10.m10.1.1.3.3.3"><minus id="S4.SS4.p1.10.m10.1.1.3.3.3.1.cmml" xref="S4.SS4.p1.10.m10.1.1.3.3.3"></minus><cn id="S4.SS4.p1.10.m10.1.1.3.3.3.2.cmml" type="integer" xref="S4.SS4.p1.10.m10.1.1.3.3.3.2">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.10.m10.1c">\boldsymbol{\Delta}=\boldsymbol{\Sigma}_{12}\boldsymbol{\Sigma}_{22}^{-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.10.m10.1d">bold_Δ = bold_Σ start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT bold_Σ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, with corresponding values of <math alttext="p_{1}" class="ltx_Math" display="inline" id="S4.SS4.p1.11.m11.1"><semantics id="S4.SS4.p1.11.m11.1a"><msub id="S4.SS4.p1.11.m11.1.1" xref="S4.SS4.p1.11.m11.1.1.cmml"><mi id="S4.SS4.p1.11.m11.1.1.2" xref="S4.SS4.p1.11.m11.1.1.2.cmml">p</mi><mn id="S4.SS4.p1.11.m11.1.1.3" xref="S4.SS4.p1.11.m11.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.11.m11.1b"><apply id="S4.SS4.p1.11.m11.1.1.cmml" xref="S4.SS4.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS4.p1.11.m11.1.1.1.cmml" xref="S4.SS4.p1.11.m11.1.1">subscript</csymbol><ci id="S4.SS4.p1.11.m11.1.1.2.cmml" xref="S4.SS4.p1.11.m11.1.1.2">𝑝</ci><cn id="S4.SS4.p1.11.m11.1.1.3.cmml" type="integer" xref="S4.SS4.p1.11.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.11.m11.1c">p_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.11.m11.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. The observed values of <math alttext="T_{4}^{\star}" class="ltx_Math" display="inline" id="S4.SS4.p1.12.m12.1"><semantics id="S4.SS4.p1.12.m12.1a"><msubsup id="S4.SS4.p1.12.m12.1.1" xref="S4.SS4.p1.12.m12.1.1.cmml"><mi id="S4.SS4.p1.12.m12.1.1.2.2" xref="S4.SS4.p1.12.m12.1.1.2.2.cmml">T</mi><mn id="S4.SS4.p1.12.m12.1.1.2.3" xref="S4.SS4.p1.12.m12.1.1.2.3.cmml">4</mn><mo id="S4.SS4.p1.12.m12.1.1.3" xref="S4.SS4.p1.12.m12.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.12.m12.1b"><apply id="S4.SS4.p1.12.m12.1.1.cmml" xref="S4.SS4.p1.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS4.p1.12.m12.1.1.1.cmml" xref="S4.SS4.p1.12.m12.1.1">superscript</csymbol><apply id="S4.SS4.p1.12.m12.1.1.2.cmml" xref="S4.SS4.p1.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS4.p1.12.m12.1.1.2.1.cmml" xref="S4.SS4.p1.12.m12.1.1">subscript</csymbol><ci id="S4.SS4.p1.12.m12.1.1.2.2.cmml" xref="S4.SS4.p1.12.m12.1.1.2.2">𝑇</ci><cn id="S4.SS4.p1.12.m12.1.1.2.3.cmml" type="integer" xref="S4.SS4.p1.12.m12.1.1.2.3">4</cn></apply><ci id="S4.SS4.p1.12.m12.1.1.3.cmml" xref="S4.SS4.p1.12.m12.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.12.m12.1c">T_{4}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.12.m12.1d">italic_T start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> are stored to compute the coverage probabilities (<math alttext="cov" class="ltx_Math" display="inline" id="S4.SS4.p1.13.m13.1"><semantics id="S4.SS4.p1.13.m13.1a"><mrow id="S4.SS4.p1.13.m13.1.1" xref="S4.SS4.p1.13.m13.1.1.cmml"><mi id="S4.SS4.p1.13.m13.1.1.2" xref="S4.SS4.p1.13.m13.1.1.2.cmml">c</mi><mo id="S4.SS4.p1.13.m13.1.1.1" xref="S4.SS4.p1.13.m13.1.1.1.cmml"></mo><mi id="S4.SS4.p1.13.m13.1.1.3" xref="S4.SS4.p1.13.m13.1.1.3.cmml">o</mi><mo id="S4.SS4.p1.13.m13.1.1.1a" xref="S4.SS4.p1.13.m13.1.1.1.cmml"></mo><mi id="S4.SS4.p1.13.m13.1.1.4" xref="S4.SS4.p1.13.m13.1.1.4.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.13.m13.1b"><apply id="S4.SS4.p1.13.m13.1.1.cmml" xref="S4.SS4.p1.13.m13.1.1"><times id="S4.SS4.p1.13.m13.1.1.1.cmml" xref="S4.SS4.p1.13.m13.1.1.1"></times><ci id="S4.SS4.p1.13.m13.1.1.2.cmml" xref="S4.SS4.p1.13.m13.1.1.2">𝑐</ci><ci id="S4.SS4.p1.13.m13.1.1.3.cmml" xref="S4.SS4.p1.13.m13.1.1.3">𝑜</ci><ci id="S4.SS4.p1.13.m13.1.1.4.cmml" xref="S4.SS4.p1.13.m13.1.1.4">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.13.m13.1c">cov</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.13.m13.1d">italic_c italic_o italic_v</annotation></semantics></math>) for the regression test and using the distribution obtained via the <span class="ltx_text ltx_font_typewriter" id="S4.SS4.p1.14.1">Regdist</span> function, the corresponding <math alttext="cov" class="ltx_Math" display="inline" id="S4.SS4.p1.14.m14.1"><semantics id="S4.SS4.p1.14.m14.1a"><mrow id="S4.SS4.p1.14.m14.1.1" xref="S4.SS4.p1.14.m14.1.1.cmml"><mi id="S4.SS4.p1.14.m14.1.1.2" xref="S4.SS4.p1.14.m14.1.1.2.cmml">c</mi><mo id="S4.SS4.p1.14.m14.1.1.1" xref="S4.SS4.p1.14.m14.1.1.1.cmml"></mo><mi id="S4.SS4.p1.14.m14.1.1.3" xref="S4.SS4.p1.14.m14.1.1.3.cmml">o</mi><mo id="S4.SS4.p1.14.m14.1.1.1a" xref="S4.SS4.p1.14.m14.1.1.1.cmml"></mo><mi id="S4.SS4.p1.14.m14.1.1.4" xref="S4.SS4.p1.14.m14.1.1.4.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.14.m14.1b"><apply id="S4.SS4.p1.14.m14.1.1.cmml" xref="S4.SS4.p1.14.m14.1.1"><times id="S4.SS4.p1.14.m14.1.1.1.cmml" xref="S4.SS4.p1.14.m14.1.1.1"></times><ci id="S4.SS4.p1.14.m14.1.1.2.cmml" xref="S4.SS4.p1.14.m14.1.1.2">𝑐</ci><ci id="S4.SS4.p1.14.m14.1.1.3.cmml" xref="S4.SS4.p1.14.m14.1.1.3">𝑜</ci><ci id="S4.SS4.p1.14.m14.1.1.4.cmml" xref="S4.SS4.p1.14.m14.1.1.4">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.14.m14.1c">cov</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.14.m14.1d">italic_c italic_o italic_v</annotation></semantics></math> values are computed and the results are presented.</p> </div> <div class="ltx_para" id="S4.SS4.p2"> <pre class="ltx_verbatim ltx_font_typewriter" id="S4.SS4.p2.1"> # Using the same given parameters, vectors and matrices set.seed(123) # Set random seed for reproducibility N <- 100000 # Number of Monte Carlo simulations # Sample size and partition sizes n <- 10 # Total sample size p1 <- 2 # Partition size for the first set of variables p2 <- 1 # Partition size for the second set of variables # Partition Sigma_3 into submatrices sigma3_11 <- as.matrix(partition(sigma_3, p1, p1)[[1]]) # Submatrix Sigma_11 sigma3_22 <- as.matrix(partition(sigma_3, p1, p1)[[4]]) # Submatrix Sigma_22 sigma3_12 <- as.matrix(partition(sigma_3, p1, p1)[[2]]) # Submatrix Sigma_12 sigma3_21 <- as.matrix(partition(sigma_3, p1, p1)[[3]]) # Submatrix Sigma_21 Delta1 <- sigma3_12 \%*\% solve(sigma3_22) # Compute Delta_1 for later use # Partition Sigma_4 into submatrices sigma4_11 <- as.matrix(partition(sigma_4, p2, p2)[[1]]) # Submatrix Sigma_11 sigma4_22 <- as.matrix(partition(sigma_4, p2, p2)[[4]]) # Submatrix Sigma_22 sigma4_12 <- as.matrix(partition(sigma_4, p2, p2)[[2]]) # Submatrix Sigma_12 sigma4_21 <- as.matrix(partition(sigma_4, p2, p2)[[3]]) # Submatrix Sigma_21 Delta2 <- t(sigma4_12) \%*\% solve(sigma4_22) # Compute Delta_2 for later use # Simulate canonical distances (T1 and T2) with the initial partition values T1 <- canodist(p1, n, p, N) T2 <- canodist(p2, n, p, N) # Quantiles for alpha = 0.05 of the simulated values (for coverage analysis) q95_1 <- quantile(T1, probs = c(.95)) q95_2 <- quantile(T2, probs = c(.95)) # Initialize result vectors for storing the T*_4 statistics T1_1 <- c() T2_1 <- c() # Monte Carlo loop to generate synthetic data and calculate T*_4 statistics for (i in 1:N) { # Generate original data samples from multivariate normal distributions # with mu and Sigma x1 <- mvrnorm(n, mu, sigma_3) # Data generated with Sigma_3 x2 <- mvrnorm(n, mu, sigma_4) # Data generated with Sigma_4 # Generate PLS synthetic data from the original samples v1 <- simSynthData(x1) v2 <- simSynthData(x2) # Compute the means for the synthetic datasets mean_v1 <- apply(v1, 2, mean) mean_v2 <- apply(v2, 2, mean) # Create matrices of mean values for later calculations MEANv1 <- matrix(mean_v1, n, 4, byrow = TRUE) MEANv2 <- matrix(mean_v2, n, 4, byrow = TRUE) # Covariance matrices of the synthetic data s_star1 <- t(v1 - MEANv1) \%*\% (v1 - MEANv1) s_star2 <- t(v2 - MEANv2) \%*\% (v2 - MEANv2) # Partition the synthetic covariance matrices s_star1_11 <- as.matrix(partition(s_star1, p1, p1)[[1]]) s_star1_22 <- partition(s_star1, p1, p1)[[4]] s_star1_12 <- partition(s_star1, p1, p1)[[2]] s_star1_21 <- partition(s_star1, p1, p1)[[3]] s_star2_11 <- partition(s_star2, p2, p2)[[1]] s_star2_22 <- partition(s_star2, p2, p2)[[4]] s_star2_12 <- partition(s_star2, p2, p2)[[2]] s_star2_21 <- partition(s_star2, p2, p2)[[3]] # Calculate adjusted submatrices and Delta_star for T*_4 statistics s_star1_112 <- s_star1_11 - s_star1_12 \%*\% (solve(s_star1_22) \%*\% s_star1_21) Delta_star1 <- s_star1_12 \%*\% solve(s_star1_22) s_star2_112 <- s_star2_11 - s_star2_12 \%*\% (solve(s_star2_22) \%*\% s_star2_21) Delta_star2 <- s_star2_12 \%*\% solve(s_star2_22) # Calculate T*_4 statistics for both datasets T1temp <- det((Delta_star1 - Delta1) \%*\% (s_star1_22 \%*\% t(Delta_star1 - Delta1))) / det(s_star1_112) T2temp <- det((Delta_star2 - Delta2) \%*\% (s_star2_22 \%*\% t(Delta_star2 - Delta2))) / det(s_star2_112) # Store the results in T1_1 and T2_1 T1_1[i] <- T1temp T2_1[i] <- T2temp } # Print the quantiles for T1, T1_1, T2, and T2_1 distributions print(quantile(T1, probs = c(0, 0.1, .5, .9, 1))) print(quantile(T1_1, probs = c(0, 0.1, .5, .9, 1))) print(quantile(T2, probs = c(0, 0.1, .5, .9, 1))) print(quantile(T2_1, probs = c(0, 0.1, .5, .9, 1))) # Calculate average coverage for T1_1 and T2_1 compared to the 95th # percentile of T1 and T2 cov1 <- mean(T1_1 < q95_1) cov2 <- mean(T2_1 < q95_2) # Print the coverage probabilities print(c(cov1, cov2)) </pre> </div> <div class="ltx_para" id="S4.SS4.p3"> <p class="ltx_p" id="S4.SS4.p3.1">To summarize the results found in the simulations, Table <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4.T2" title="Table 2 ‣ 4.4 Average coverage probability for the Regression Test ‣ 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">2</span></a> shows, for values of <math alttext="n" class="ltx_Math" display="inline" id="S4.SS4.p3.1.m1.1"><semantics id="S4.SS4.p3.1.m1.1a"><mi id="S4.SS4.p3.1.m1.1.1" xref="S4.SS4.p3.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p3.1.m1.1b"><ci id="S4.SS4.p3.1.m1.1.1.cmml" xref="S4.SS4.p3.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p3.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p3.1.m1.1d">italic_n</annotation></semantics></math> and selected covariance matrices defined in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4.E16" title="In 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">16</span></a>), the estimated coverage values for:</p> <ul class="ltx_itemize" id="S4.I1"> <li class="ltx_item" id="S4.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i1.p1"> <p class="ltx_p" id="S4.I1.i1.p1.2">the confidence interval for the Generalized Variance under the column <span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.2.1">Gener. Variance</span> and selected <math alttext="\boldsymbol{\Sigma}_{3}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.1.m1.1"><semantics id="S4.I1.i1.p1.1.m1.1a"><msub id="S4.I1.i1.p1.1.m1.1.1" xref="S4.I1.i1.p1.1.m1.1.1.cmml"><mi id="S4.I1.i1.p1.1.m1.1.1.2" xref="S4.I1.i1.p1.1.m1.1.1.2.cmml">𝚺</mi><mn id="S4.I1.i1.p1.1.m1.1.1.3" xref="S4.I1.i1.p1.1.m1.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.1.m1.1b"><apply id="S4.I1.i1.p1.1.m1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.1.m1.1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1">subscript</csymbol><ci id="S4.I1.i1.p1.1.m1.1.1.2.cmml" xref="S4.I1.i1.p1.1.m1.1.1.2">𝚺</ci><cn id="S4.I1.i1.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.I1.i1.p1.1.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.1.m1.1c">\boldsymbol{\Sigma}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.1.m1.1d">bold_Σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.2.m2.1"><semantics id="S4.I1.i1.p1.2.m2.1a"><msub id="S4.I1.i1.p1.2.m2.1.1" xref="S4.I1.i1.p1.2.m2.1.1.cmml"><mi id="S4.I1.i1.p1.2.m2.1.1.2" xref="S4.I1.i1.p1.2.m2.1.1.2.cmml">𝚺</mi><mn id="S4.I1.i1.p1.2.m2.1.1.3" xref="S4.I1.i1.p1.2.m2.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.2.m2.1b"><apply id="S4.I1.i1.p1.2.m2.1.1.cmml" xref="S4.I1.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.2.m2.1.1.1.cmml" xref="S4.I1.i1.p1.2.m2.1.1">subscript</csymbol><ci id="S4.I1.i1.p1.2.m2.1.1.2.cmml" xref="S4.I1.i1.p1.2.m2.1.1.2">𝚺</ci><cn id="S4.I1.i1.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.I1.i1.p1.2.m2.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.2.m2.1c">\boldsymbol{\Sigma}_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.2.m2.1d">bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math>;</p> </div> </li> <li class="ltx_item" id="S4.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i2.p1"> <p class="ltx_p" id="S4.I1.i2.p1.2">the Sphericity test under the column <span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.2.1">Sphericity</span> and selected <math alttext="\boldsymbol{\Sigma}_{1}" class="ltx_Math" display="inline" id="S4.I1.i2.p1.1.m1.1"><semantics id="S4.I1.i2.p1.1.m1.1a"><msub id="S4.I1.i2.p1.1.m1.1.1" xref="S4.I1.i2.p1.1.m1.1.1.cmml"><mi id="S4.I1.i2.p1.1.m1.1.1.2" xref="S4.I1.i2.p1.1.m1.1.1.2.cmml">𝚺</mi><mn id="S4.I1.i2.p1.1.m1.1.1.3" xref="S4.I1.i2.p1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.1.m1.1b"><apply id="S4.I1.i2.p1.1.m1.1.1.cmml" xref="S4.I1.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.I1.i2.p1.1.m1.1.1.1.cmml" xref="S4.I1.i2.p1.1.m1.1.1">subscript</csymbol><ci id="S4.I1.i2.p1.1.m1.1.1.2.cmml" xref="S4.I1.i2.p1.1.m1.1.1.2">𝚺</ci><cn id="S4.I1.i2.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.I1.i2.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.1.m1.1c">\boldsymbol{\Sigma}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.1.m1.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{\Sigma}_{2}" class="ltx_Math" display="inline" id="S4.I1.i2.p1.2.m2.1"><semantics id="S4.I1.i2.p1.2.m2.1a"><msub id="S4.I1.i2.p1.2.m2.1.1" xref="S4.I1.i2.p1.2.m2.1.1.cmml"><mi id="S4.I1.i2.p1.2.m2.1.1.2" xref="S4.I1.i2.p1.2.m2.1.1.2.cmml">𝚺</mi><mn id="S4.I1.i2.p1.2.m2.1.1.3" xref="S4.I1.i2.p1.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.2.m2.1b"><apply id="S4.I1.i2.p1.2.m2.1.1.cmml" xref="S4.I1.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.I1.i2.p1.2.m2.1.1.1.cmml" xref="S4.I1.i2.p1.2.m2.1.1">subscript</csymbol><ci id="S4.I1.i2.p1.2.m2.1.1.2.cmml" xref="S4.I1.i2.p1.2.m2.1.1.2">𝚺</ci><cn id="S4.I1.i2.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.I1.i2.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.2.m2.1c">\boldsymbol{\Sigma}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.2.m2.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>;</p> </div> </li> <li class="ltx_item" id="S4.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i3.p1"> <p class="ltx_p" id="S4.I1.i3.p1.4">the Independence test under the column <span class="ltx_text ltx_font_italic" id="S4.I1.i3.p1.4.1">Independence</span> and selected <math alttext="\boldsymbol{\Sigma}_{1}" class="ltx_Math" display="inline" id="S4.I1.i3.p1.1.m1.1"><semantics id="S4.I1.i3.p1.1.m1.1a"><msub id="S4.I1.i3.p1.1.m1.1.1" xref="S4.I1.i3.p1.1.m1.1.1.cmml"><mi id="S4.I1.i3.p1.1.m1.1.1.2" xref="S4.I1.i3.p1.1.m1.1.1.2.cmml">𝚺</mi><mn id="S4.I1.i3.p1.1.m1.1.1.3" xref="S4.I1.i3.p1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i3.p1.1.m1.1b"><apply id="S4.I1.i3.p1.1.m1.1.1.cmml" xref="S4.I1.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.I1.i3.p1.1.m1.1.1.1.cmml" xref="S4.I1.i3.p1.1.m1.1.1">subscript</csymbol><ci id="S4.I1.i3.p1.1.m1.1.1.2.cmml" xref="S4.I1.i3.p1.1.m1.1.1.2">𝚺</ci><cn id="S4.I1.i3.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.I1.i3.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i3.p1.1.m1.1c">\boldsymbol{\Sigma}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i3.p1.1.m1.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.I1.i3.p1.2.m2.1"><semantics id="S4.I1.i3.p1.2.m2.1a"><msub id="S4.I1.i3.p1.2.m2.1.1" xref="S4.I1.i3.p1.2.m2.1.1.cmml"><mi id="S4.I1.i3.p1.2.m2.1.1.2" xref="S4.I1.i3.p1.2.m2.1.1.2.cmml">𝚺</mi><mn id="S4.I1.i3.p1.2.m2.1.1.3" xref="S4.I1.i3.p1.2.m2.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i3.p1.2.m2.1b"><apply id="S4.I1.i3.p1.2.m2.1.1.cmml" xref="S4.I1.i3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.I1.i3.p1.2.m2.1.1.1.cmml" xref="S4.I1.i3.p1.2.m2.1.1">subscript</csymbol><ci id="S4.I1.i3.p1.2.m2.1.1.2.cmml" xref="S4.I1.i3.p1.2.m2.1.1.2">𝚺</ci><cn id="S4.I1.i3.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.I1.i3.p1.2.m2.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i3.p1.2.m2.1c">\boldsymbol{\Sigma}_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i3.p1.2.m2.1d">bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math>, for <math alttext="p_{1}=1" class="ltx_Math" display="inline" id="S4.I1.i3.p1.3.m3.1"><semantics id="S4.I1.i3.p1.3.m3.1a"><mrow id="S4.I1.i3.p1.3.m3.1.1" xref="S4.I1.i3.p1.3.m3.1.1.cmml"><msub id="S4.I1.i3.p1.3.m3.1.1.2" xref="S4.I1.i3.p1.3.m3.1.1.2.cmml"><mi id="S4.I1.i3.p1.3.m3.1.1.2.2" xref="S4.I1.i3.p1.3.m3.1.1.2.2.cmml">p</mi><mn id="S4.I1.i3.p1.3.m3.1.1.2.3" xref="S4.I1.i3.p1.3.m3.1.1.2.3.cmml">1</mn></msub><mo id="S4.I1.i3.p1.3.m3.1.1.1" xref="S4.I1.i3.p1.3.m3.1.1.1.cmml">=</mo><mn id="S4.I1.i3.p1.3.m3.1.1.3" xref="S4.I1.i3.p1.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i3.p1.3.m3.1b"><apply id="S4.I1.i3.p1.3.m3.1.1.cmml" xref="S4.I1.i3.p1.3.m3.1.1"><eq id="S4.I1.i3.p1.3.m3.1.1.1.cmml" xref="S4.I1.i3.p1.3.m3.1.1.1"></eq><apply id="S4.I1.i3.p1.3.m3.1.1.2.cmml" xref="S4.I1.i3.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.I1.i3.p1.3.m3.1.1.2.1.cmml" xref="S4.I1.i3.p1.3.m3.1.1.2">subscript</csymbol><ci id="S4.I1.i3.p1.3.m3.1.1.2.2.cmml" xref="S4.I1.i3.p1.3.m3.1.1.2.2">𝑝</ci><cn id="S4.I1.i3.p1.3.m3.1.1.2.3.cmml" type="integer" xref="S4.I1.i3.p1.3.m3.1.1.2.3">1</cn></apply><cn id="S4.I1.i3.p1.3.m3.1.1.3.cmml" type="integer" xref="S4.I1.i3.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i3.p1.3.m3.1c">p_{1}=1</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i3.p1.3.m3.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math> and <math alttext="p_{1}=2" class="ltx_Math" display="inline" id="S4.I1.i3.p1.4.m4.1"><semantics id="S4.I1.i3.p1.4.m4.1a"><mrow id="S4.I1.i3.p1.4.m4.1.1" xref="S4.I1.i3.p1.4.m4.1.1.cmml"><msub id="S4.I1.i3.p1.4.m4.1.1.2" xref="S4.I1.i3.p1.4.m4.1.1.2.cmml"><mi id="S4.I1.i3.p1.4.m4.1.1.2.2" xref="S4.I1.i3.p1.4.m4.1.1.2.2.cmml">p</mi><mn id="S4.I1.i3.p1.4.m4.1.1.2.3" xref="S4.I1.i3.p1.4.m4.1.1.2.3.cmml">1</mn></msub><mo id="S4.I1.i3.p1.4.m4.1.1.1" xref="S4.I1.i3.p1.4.m4.1.1.1.cmml">=</mo><mn id="S4.I1.i3.p1.4.m4.1.1.3" xref="S4.I1.i3.p1.4.m4.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i3.p1.4.m4.1b"><apply id="S4.I1.i3.p1.4.m4.1.1.cmml" xref="S4.I1.i3.p1.4.m4.1.1"><eq id="S4.I1.i3.p1.4.m4.1.1.1.cmml" xref="S4.I1.i3.p1.4.m4.1.1.1"></eq><apply id="S4.I1.i3.p1.4.m4.1.1.2.cmml" xref="S4.I1.i3.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.I1.i3.p1.4.m4.1.1.2.1.cmml" xref="S4.I1.i3.p1.4.m4.1.1.2">subscript</csymbol><ci id="S4.I1.i3.p1.4.m4.1.1.2.2.cmml" xref="S4.I1.i3.p1.4.m4.1.1.2.2">𝑝</ci><cn id="S4.I1.i3.p1.4.m4.1.1.2.3.cmml" type="integer" xref="S4.I1.i3.p1.4.m4.1.1.2.3">1</cn></apply><cn id="S4.I1.i3.p1.4.m4.1.1.3.cmml" type="integer" xref="S4.I1.i3.p1.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i3.p1.4.m4.1c">p_{1}=2</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i3.p1.4.m4.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2</annotation></semantics></math>, respectively;</p> </div> </li> <li class="ltx_item" id="S4.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i4.p1"> <p class="ltx_p" id="S4.I1.i4.p1.4">the test for the regression of one set of variables on the other under the column <span class="ltx_text ltx_font_italic" id="S4.I1.i4.p1.4.1">Regression</span> and selected <math alttext="\boldsymbol{\Sigma}_{3}" class="ltx_Math" display="inline" id="S4.I1.i4.p1.1.m1.1"><semantics id="S4.I1.i4.p1.1.m1.1a"><msub id="S4.I1.i4.p1.1.m1.1.1" xref="S4.I1.i4.p1.1.m1.1.1.cmml"><mi id="S4.I1.i4.p1.1.m1.1.1.2" xref="S4.I1.i4.p1.1.m1.1.1.2.cmml">𝚺</mi><mn id="S4.I1.i4.p1.1.m1.1.1.3" xref="S4.I1.i4.p1.1.m1.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i4.p1.1.m1.1b"><apply id="S4.I1.i4.p1.1.m1.1.1.cmml" xref="S4.I1.i4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.I1.i4.p1.1.m1.1.1.1.cmml" xref="S4.I1.i4.p1.1.m1.1.1">subscript</csymbol><ci id="S4.I1.i4.p1.1.m1.1.1.2.cmml" xref="S4.I1.i4.p1.1.m1.1.1.2">𝚺</ci><cn id="S4.I1.i4.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.I1.i4.p1.1.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i4.p1.1.m1.1c">\boldsymbol{\Sigma}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i4.p1.1.m1.1d">bold_Σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.I1.i4.p1.2.m2.1"><semantics id="S4.I1.i4.p1.2.m2.1a"><msub id="S4.I1.i4.p1.2.m2.1.1" xref="S4.I1.i4.p1.2.m2.1.1.cmml"><mi id="S4.I1.i4.p1.2.m2.1.1.2" xref="S4.I1.i4.p1.2.m2.1.1.2.cmml">𝚺</mi><mn id="S4.I1.i4.p1.2.m2.1.1.3" xref="S4.I1.i4.p1.2.m2.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i4.p1.2.m2.1b"><apply id="S4.I1.i4.p1.2.m2.1.1.cmml" xref="S4.I1.i4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.I1.i4.p1.2.m2.1.1.1.cmml" xref="S4.I1.i4.p1.2.m2.1.1">subscript</csymbol><ci id="S4.I1.i4.p1.2.m2.1.1.2.cmml" xref="S4.I1.i4.p1.2.m2.1.1.2">𝚺</ci><cn id="S4.I1.i4.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.I1.i4.p1.2.m2.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i4.p1.2.m2.1c">\boldsymbol{\Sigma}_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i4.p1.2.m2.1d">bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math>, for <math alttext="p_{1}=2" class="ltx_Math" display="inline" id="S4.I1.i4.p1.3.m3.1"><semantics id="S4.I1.i4.p1.3.m3.1a"><mrow id="S4.I1.i4.p1.3.m3.1.1" xref="S4.I1.i4.p1.3.m3.1.1.cmml"><msub id="S4.I1.i4.p1.3.m3.1.1.2" xref="S4.I1.i4.p1.3.m3.1.1.2.cmml"><mi id="S4.I1.i4.p1.3.m3.1.1.2.2" xref="S4.I1.i4.p1.3.m3.1.1.2.2.cmml">p</mi><mn id="S4.I1.i4.p1.3.m3.1.1.2.3" xref="S4.I1.i4.p1.3.m3.1.1.2.3.cmml">1</mn></msub><mo id="S4.I1.i4.p1.3.m3.1.1.1" xref="S4.I1.i4.p1.3.m3.1.1.1.cmml">=</mo><mn id="S4.I1.i4.p1.3.m3.1.1.3" xref="S4.I1.i4.p1.3.m3.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i4.p1.3.m3.1b"><apply id="S4.I1.i4.p1.3.m3.1.1.cmml" xref="S4.I1.i4.p1.3.m3.1.1"><eq id="S4.I1.i4.p1.3.m3.1.1.1.cmml" xref="S4.I1.i4.p1.3.m3.1.1.1"></eq><apply id="S4.I1.i4.p1.3.m3.1.1.2.cmml" xref="S4.I1.i4.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.I1.i4.p1.3.m3.1.1.2.1.cmml" xref="S4.I1.i4.p1.3.m3.1.1.2">subscript</csymbol><ci id="S4.I1.i4.p1.3.m3.1.1.2.2.cmml" xref="S4.I1.i4.p1.3.m3.1.1.2.2">𝑝</ci><cn id="S4.I1.i4.p1.3.m3.1.1.2.3.cmml" type="integer" xref="S4.I1.i4.p1.3.m3.1.1.2.3">1</cn></apply><cn id="S4.I1.i4.p1.3.m3.1.1.3.cmml" type="integer" xref="S4.I1.i4.p1.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i4.p1.3.m3.1c">p_{1}=2</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i4.p1.3.m3.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2</annotation></semantics></math> and <math alttext="p_{1}=1" class="ltx_Math" display="inline" id="S4.I1.i4.p1.4.m4.1"><semantics id="S4.I1.i4.p1.4.m4.1a"><mrow id="S4.I1.i4.p1.4.m4.1.1" xref="S4.I1.i4.p1.4.m4.1.1.cmml"><msub id="S4.I1.i4.p1.4.m4.1.1.2" xref="S4.I1.i4.p1.4.m4.1.1.2.cmml"><mi id="S4.I1.i4.p1.4.m4.1.1.2.2" xref="S4.I1.i4.p1.4.m4.1.1.2.2.cmml">p</mi><mn id="S4.I1.i4.p1.4.m4.1.1.2.3" xref="S4.I1.i4.p1.4.m4.1.1.2.3.cmml">1</mn></msub><mo id="S4.I1.i4.p1.4.m4.1.1.1" xref="S4.I1.i4.p1.4.m4.1.1.1.cmml">=</mo><mn id="S4.I1.i4.p1.4.m4.1.1.3" xref="S4.I1.i4.p1.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i4.p1.4.m4.1b"><apply id="S4.I1.i4.p1.4.m4.1.1.cmml" xref="S4.I1.i4.p1.4.m4.1.1"><eq id="S4.I1.i4.p1.4.m4.1.1.1.cmml" xref="S4.I1.i4.p1.4.m4.1.1.1"></eq><apply id="S4.I1.i4.p1.4.m4.1.1.2.cmml" xref="S4.I1.i4.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.I1.i4.p1.4.m4.1.1.2.1.cmml" xref="S4.I1.i4.p1.4.m4.1.1.2">subscript</csymbol><ci id="S4.I1.i4.p1.4.m4.1.1.2.2.cmml" xref="S4.I1.i4.p1.4.m4.1.1.2.2">𝑝</ci><cn id="S4.I1.i4.p1.4.m4.1.1.2.3.cmml" type="integer" xref="S4.I1.i4.p1.4.m4.1.1.2.3">1</cn></apply><cn id="S4.I1.i4.p1.4.m4.1.1.3.cmml" type="integer" xref="S4.I1.i4.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i4.p1.4.m4.1c">p_{1}=1</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i4.p1.4.m4.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math>, respectively.</p> </div> </li> </ul> </div> <figure class="ltx_table" id="S4.T2"> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S4.T2.12"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T2.12.13.1"> <th class="ltx_td ltx_th ltx_th_row" id="S4.T2.12.13.1.1"></th> <td class="ltx_td ltx_align_center" colspan="2" id="S4.T2.12.13.1.2"><span class="ltx_text ltx_font_italic" id="S4.T2.12.13.1.2.1">Gener. Variance</span></td> <td class="ltx_td ltx_align_center" colspan="2" id="S4.T2.12.13.1.3"><span class="ltx_text ltx_font_italic" id="S4.T2.12.13.1.3.1">Sphericity</span></td> <td class="ltx_td ltx_align_center" colspan="2" id="S4.T2.12.13.1.4"><span class="ltx_text ltx_font_italic" id="S4.T2.12.13.1.4.1">Independence</span></td> <td class="ltx_td ltx_align_center" colspan="2" id="S4.T2.12.13.1.5"><span class="ltx_text ltx_font_italic" id="S4.T2.12.13.1.5.1">Regression</span></td> </tr> <tr class="ltx_tr" id="S4.T2.8.8"> <th class="ltx_td ltx_th ltx_th_row ltx_border_r" id="S4.T2.8.8.9"></th> <td class="ltx_td ltx_align_center" id="S4.T2.1.1.1"><math alttext="\boldsymbol{\Sigma}_{3}" class="ltx_Math" display="inline" id="S4.T2.1.1.1.m1.1"><semantics id="S4.T2.1.1.1.m1.1a"><msub id="S4.T2.1.1.1.m1.1.1" xref="S4.T2.1.1.1.m1.1.1.cmml"><mi id="S4.T2.1.1.1.m1.1.1.2" xref="S4.T2.1.1.1.m1.1.1.2.cmml">𝚺</mi><mn id="S4.T2.1.1.1.m1.1.1.3" xref="S4.T2.1.1.1.m1.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S4.T2.1.1.1.m1.1b"><apply id="S4.T2.1.1.1.m1.1.1.cmml" xref="S4.T2.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.T2.1.1.1.m1.1.1.1.cmml" xref="S4.T2.1.1.1.m1.1.1">subscript</csymbol><ci id="S4.T2.1.1.1.m1.1.1.2.cmml" xref="S4.T2.1.1.1.m1.1.1.2">𝚺</ci><cn id="S4.T2.1.1.1.m1.1.1.3.cmml" type="integer" xref="S4.T2.1.1.1.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.1.1.1.m1.1c">\boldsymbol{\Sigma}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.1.1.1.m1.1d">bold_Σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.2.2.2"><math alttext="\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.T2.2.2.2.m1.1"><semantics id="S4.T2.2.2.2.m1.1a"><msub id="S4.T2.2.2.2.m1.1.1" xref="S4.T2.2.2.2.m1.1.1.cmml"><mi id="S4.T2.2.2.2.m1.1.1.2" 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id="S4.T2.4.4.4.m1.1"><semantics id="S4.T2.4.4.4.m1.1a"><msub id="S4.T2.4.4.4.m1.1.1" xref="S4.T2.4.4.4.m1.1.1.cmml"><mi id="S4.T2.4.4.4.m1.1.1.2" xref="S4.T2.4.4.4.m1.1.1.2.cmml">𝚺</mi><mn id="S4.T2.4.4.4.m1.1.1.3" xref="S4.T2.4.4.4.m1.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.T2.4.4.4.m1.1b"><apply id="S4.T2.4.4.4.m1.1.1.cmml" xref="S4.T2.4.4.4.m1.1.1"><csymbol cd="ambiguous" id="S4.T2.4.4.4.m1.1.1.1.cmml" xref="S4.T2.4.4.4.m1.1.1">subscript</csymbol><ci id="S4.T2.4.4.4.m1.1.1.2.cmml" xref="S4.T2.4.4.4.m1.1.1.2">𝚺</ci><cn id="S4.T2.4.4.4.m1.1.1.3.cmml" type="integer" xref="S4.T2.4.4.4.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.4.4.4.m1.1c">\boldsymbol{\Sigma}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.4.4.4.m1.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T2.5.5.5"><math 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end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T2.7.7.7"><math alttext="\boldsymbol{\Sigma}_{3}" class="ltx_Math" display="inline" id="S4.T2.7.7.7.m1.1"><semantics id="S4.T2.7.7.7.m1.1a"><msub id="S4.T2.7.7.7.m1.1.1" xref="S4.T2.7.7.7.m1.1.1.cmml"><mi id="S4.T2.7.7.7.m1.1.1.2" xref="S4.T2.7.7.7.m1.1.1.2.cmml">𝚺</mi><mn id="S4.T2.7.7.7.m1.1.1.3" xref="S4.T2.7.7.7.m1.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S4.T2.7.7.7.m1.1b"><apply id="S4.T2.7.7.7.m1.1.1.cmml" xref="S4.T2.7.7.7.m1.1.1"><csymbol cd="ambiguous" id="S4.T2.7.7.7.m1.1.1.1.cmml" xref="S4.T2.7.7.7.m1.1.1">subscript</csymbol><ci id="S4.T2.7.7.7.m1.1.1.2.cmml" xref="S4.T2.7.7.7.m1.1.1.2">𝚺</ci><cn id="S4.T2.7.7.7.m1.1.1.3.cmml" type="integer" xref="S4.T2.7.7.7.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.7.7.7.m1.1c">\boldsymbol{\Sigma}_{3}</annotation><annotation encoding="application/x-llamapun" 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id="S4.T2.8.8.8.m1.1c">\boldsymbol{\Sigma}_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.8.8.8.m1.1d">bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T2.12.12"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S4.T2.12.12.5">n</th> <td class="ltx_td" id="S4.T2.12.12.6"></td> <td class="ltx_td ltx_border_r" id="S4.T2.12.12.7"></td> <td class="ltx_td" id="S4.T2.12.12.8"></td> <td class="ltx_td ltx_border_r" id="S4.T2.12.12.9"></td> <td class="ltx_td ltx_align_center" id="S4.T2.9.9.1"><math alttext="p_{1}=1" class="ltx_Math" display="inline" id="S4.T2.9.9.1.m1.1"><semantics id="S4.T2.9.9.1.m1.1a"><mrow id="S4.T2.9.9.1.m1.1.1" xref="S4.T2.9.9.1.m1.1.1.cmml"><msub id="S4.T2.9.9.1.m1.1.1.2" xref="S4.T2.9.9.1.m1.1.1.2.cmml"><mi id="S4.T2.9.9.1.m1.1.1.2.2" xref="S4.T2.9.9.1.m1.1.1.2.2.cmml">p</mi><mn id="S4.T2.9.9.1.m1.1.1.2.3" xref="S4.T2.9.9.1.m1.1.1.2.3.cmml">1</mn></msub><mo id="S4.T2.9.9.1.m1.1.1.1" xref="S4.T2.9.9.1.m1.1.1.1.cmml">=</mo><mn id="S4.T2.9.9.1.m1.1.1.3" xref="S4.T2.9.9.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.9.9.1.m1.1b"><apply id="S4.T2.9.9.1.m1.1.1.cmml" xref="S4.T2.9.9.1.m1.1.1"><eq id="S4.T2.9.9.1.m1.1.1.1.cmml" xref="S4.T2.9.9.1.m1.1.1.1"></eq><apply id="S4.T2.9.9.1.m1.1.1.2.cmml" xref="S4.T2.9.9.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T2.9.9.1.m1.1.1.2.1.cmml" xref="S4.T2.9.9.1.m1.1.1.2">subscript</csymbol><ci id="S4.T2.9.9.1.m1.1.1.2.2.cmml" xref="S4.T2.9.9.1.m1.1.1.2.2">𝑝</ci><cn id="S4.T2.9.9.1.m1.1.1.2.3.cmml" type="integer" xref="S4.T2.9.9.1.m1.1.1.2.3">1</cn></apply><cn id="S4.T2.9.9.1.m1.1.1.3.cmml" type="integer" xref="S4.T2.9.9.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.9.9.1.m1.1c">p_{1}=1</annotation><annotation encoding="application/x-llamapun" id="S4.T2.9.9.1.m1.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.10.10.2"><math alttext="p_{1}=2" class="ltx_Math" display="inline" id="S4.T2.10.10.2.m1.1"><semantics id="S4.T2.10.10.2.m1.1a"><mrow id="S4.T2.10.10.2.m1.1.1" xref="S4.T2.10.10.2.m1.1.1.cmml"><msub id="S4.T2.10.10.2.m1.1.1.2" xref="S4.T2.10.10.2.m1.1.1.2.cmml"><mi id="S4.T2.10.10.2.m1.1.1.2.2" xref="S4.T2.10.10.2.m1.1.1.2.2.cmml">p</mi><mn id="S4.T2.10.10.2.m1.1.1.2.3" xref="S4.T2.10.10.2.m1.1.1.2.3.cmml">1</mn></msub><mo id="S4.T2.10.10.2.m1.1.1.1" xref="S4.T2.10.10.2.m1.1.1.1.cmml">=</mo><mn id="S4.T2.10.10.2.m1.1.1.3" xref="S4.T2.10.10.2.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.10.10.2.m1.1b"><apply id="S4.T2.10.10.2.m1.1.1.cmml" xref="S4.T2.10.10.2.m1.1.1"><eq id="S4.T2.10.10.2.m1.1.1.1.cmml" xref="S4.T2.10.10.2.m1.1.1.1"></eq><apply id="S4.T2.10.10.2.m1.1.1.2.cmml" xref="S4.T2.10.10.2.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T2.10.10.2.m1.1.1.2.1.cmml" xref="S4.T2.10.10.2.m1.1.1.2">subscript</csymbol><ci id="S4.T2.10.10.2.m1.1.1.2.2.cmml" xref="S4.T2.10.10.2.m1.1.1.2.2">𝑝</ci><cn id="S4.T2.10.10.2.m1.1.1.2.3.cmml" type="integer" xref="S4.T2.10.10.2.m1.1.1.2.3">1</cn></apply><cn id="S4.T2.10.10.2.m1.1.1.3.cmml" type="integer" xref="S4.T2.10.10.2.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.10.10.2.m1.1c">p_{1}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T2.10.10.2.m1.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S4.T2.11.11.3"><math alttext="p_{1}=2" class="ltx_Math" display="inline" id="S4.T2.11.11.3.m1.1"><semantics id="S4.T2.11.11.3.m1.1a"><mrow id="S4.T2.11.11.3.m1.1.1" xref="S4.T2.11.11.3.m1.1.1.cmml"><msub id="S4.T2.11.11.3.m1.1.1.2" xref="S4.T2.11.11.3.m1.1.1.2.cmml"><mi id="S4.T2.11.11.3.m1.1.1.2.2" xref="S4.T2.11.11.3.m1.1.1.2.2.cmml">p</mi><mn id="S4.T2.11.11.3.m1.1.1.2.3" xref="S4.T2.11.11.3.m1.1.1.2.3.cmml">1</mn></msub><mo id="S4.T2.11.11.3.m1.1.1.1" xref="S4.T2.11.11.3.m1.1.1.1.cmml">=</mo><mn id="S4.T2.11.11.3.m1.1.1.3" xref="S4.T2.11.11.3.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.11.11.3.m1.1b"><apply id="S4.T2.11.11.3.m1.1.1.cmml" xref="S4.T2.11.11.3.m1.1.1"><eq id="S4.T2.11.11.3.m1.1.1.1.cmml" xref="S4.T2.11.11.3.m1.1.1.1"></eq><apply id="S4.T2.11.11.3.m1.1.1.2.cmml" xref="S4.T2.11.11.3.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T2.11.11.3.m1.1.1.2.1.cmml" xref="S4.T2.11.11.3.m1.1.1.2">subscript</csymbol><ci id="S4.T2.11.11.3.m1.1.1.2.2.cmml" xref="S4.T2.11.11.3.m1.1.1.2.2">𝑝</ci><cn id="S4.T2.11.11.3.m1.1.1.2.3.cmml" type="integer" xref="S4.T2.11.11.3.m1.1.1.2.3">1</cn></apply><cn id="S4.T2.11.11.3.m1.1.1.3.cmml" type="integer" xref="S4.T2.11.11.3.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.11.11.3.m1.1c">p_{1}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T2.11.11.3.m1.1d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.12.12.4"><math alttext="p_{1}=1" class="ltx_Math" display="inline" id="S4.T2.12.12.4.m1.1"><semantics id="S4.T2.12.12.4.m1.1a"><mrow id="S4.T2.12.12.4.m1.1.1" xref="S4.T2.12.12.4.m1.1.1.cmml"><msub id="S4.T2.12.12.4.m1.1.1.2" xref="S4.T2.12.12.4.m1.1.1.2.cmml"><mi id="S4.T2.12.12.4.m1.1.1.2.2" xref="S4.T2.12.12.4.m1.1.1.2.2.cmml">p</mi><mn id="S4.T2.12.12.4.m1.1.1.2.3" xref="S4.T2.12.12.4.m1.1.1.2.3.cmml">1</mn></msub><mo id="S4.T2.12.12.4.m1.1.1.1" xref="S4.T2.12.12.4.m1.1.1.1.cmml">=</mo><mn id="S4.T2.12.12.4.m1.1.1.3" xref="S4.T2.12.12.4.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.12.12.4.m1.1b"><apply id="S4.T2.12.12.4.m1.1.1.cmml" xref="S4.T2.12.12.4.m1.1.1"><eq 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id="S4.T2.12.14.2.2">0.948</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.12.14.2.3">0.950</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T2.12.14.2.4">0.951</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.12.14.2.5">0.952</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T2.12.14.2.6">0.951</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.12.14.2.7">0.948</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T2.12.14.2.8">0.949</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.12.14.2.9">0.950</td> </tr> <tr class="ltx_tr" id="S4.T2.12.15.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S4.T2.12.15.3.1">20</th> <td class="ltx_td ltx_align_center" id="S4.T2.12.15.3.2">0.949</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.12.15.3.3">0.949</td> <td class="ltx_td ltx_align_center" id="S4.T2.12.15.3.4">0.949</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.12.15.3.5">0.949</td> <td class="ltx_td ltx_align_center" id="S4.T2.12.15.3.6">0.949</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.12.15.3.7">0.950</td> <td class="ltx_td ltx_align_center" id="S4.T2.12.15.3.8">0.949</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.12.15.3.9">0.950</td> </tr> <tr class="ltx_tr" id="S4.T2.12.16.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S4.T2.12.16.4.1">100</th> <td class="ltx_td ltx_align_center" id="S4.T2.12.16.4.2">0.951</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.12.16.4.3">0.948</td> <td class="ltx_td ltx_align_center" id="S4.T2.12.16.4.4">0.951</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.12.16.4.5">0.950</td> <td class="ltx_td ltx_align_center" id="S4.T2.12.16.4.6">0.949</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.12.16.4.7">0.948</td> <td class="ltx_td ltx_align_center" id="S4.T2.12.16.4.8">0.950</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.12.16.4.9">0.952</td> </tr> <tr class="ltx_tr" id="S4.T2.12.17.5"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S4.T2.12.17.5.1">500</th> <td class="ltx_td ltx_align_center" id="S4.T2.12.17.5.2">0.951</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.12.17.5.3">0.948</td> <td class="ltx_td ltx_align_center" id="S4.T2.12.17.5.4">0.950</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.12.17.5.5">0.951</td> <td class="ltx_td ltx_align_center" id="S4.T2.12.17.5.6">0.949</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.12.17.5.7">0.948</td> <td class="ltx_td ltx_align_center" id="S4.T2.12.17.5.8">0.951</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T2.12.17.5.9">0.950</td> </tr> </tbody> </table> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S4.T2.24.6.1" style="font-size:90%;">Table 2</span>: </span><span class="ltx_text" id="S4.T2.22.5" style="font-size:90%;">Estimated coverage probability for the tests of section <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2" title="2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">2</span></a> for <math alttext="n=10,20,100,500,\ p_{1}=1,2,\ \boldsymbol{\mu}=(1,2,3,4)^{\prime}" class="ltx_Math" display="inline" id="S4.T2.18.1.m1.12"><semantics id="S4.T2.18.1.m1.12b"><mrow id="S4.T2.18.1.m1.12.12.2" xref="S4.T2.18.1.m1.12.12.3.cmml"><mrow id="S4.T2.18.1.m1.11.11.1.1" xref="S4.T2.18.1.m1.11.11.1.1.cmml"><mi id="S4.T2.18.1.m1.11.11.1.1.2" xref="S4.T2.18.1.m1.11.11.1.1.2.cmml">n</mi><mo id="S4.T2.18.1.m1.11.11.1.1.1" xref="S4.T2.18.1.m1.11.11.1.1.1.cmml">=</mo><mrow id="S4.T2.18.1.m1.11.11.1.1.3.2" xref="S4.T2.18.1.m1.11.11.1.1.3.1.cmml"><mn 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id="S4.T2.20.3.m3.1d">\boldsymbol{\Sigma}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.20.3.m3.1e">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\boldsymbol{\Sigma}_{3}" class="ltx_Math" display="inline" id="S4.T2.21.4.m4.1"><semantics id="S4.T2.21.4.m4.1b"><msub id="S4.T2.21.4.m4.1.1" xref="S4.T2.21.4.m4.1.1.cmml"><mi id="S4.T2.21.4.m4.1.1.2" xref="S4.T2.21.4.m4.1.1.2.cmml">𝚺</mi><mn id="S4.T2.21.4.m4.1.1.3" xref="S4.T2.21.4.m4.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S4.T2.21.4.m4.1c"><apply id="S4.T2.21.4.m4.1.1.cmml" xref="S4.T2.21.4.m4.1.1"><csymbol cd="ambiguous" id="S4.T2.21.4.m4.1.1.1.cmml" xref="S4.T2.21.4.m4.1.1">subscript</csymbol><ci id="S4.T2.21.4.m4.1.1.2.cmml" xref="S4.T2.21.4.m4.1.1.2">𝚺</ci><cn id="S4.T2.21.4.m4.1.1.3.cmml" type="integer" xref="S4.T2.21.4.m4.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.21.4.m4.1d">\boldsymbol{\Sigma}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.21.4.m4.1e">bold_Σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.T2.22.5.m5.1"><semantics id="S4.T2.22.5.m5.1b"><msub id="S4.T2.22.5.m5.1.1" xref="S4.T2.22.5.m5.1.1.cmml"><mi id="S4.T2.22.5.m5.1.1.2" xref="S4.T2.22.5.m5.1.1.2.cmml">𝚺</mi><mn id="S4.T2.22.5.m5.1.1.3" xref="S4.T2.22.5.m5.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S4.T2.22.5.m5.1c"><apply id="S4.T2.22.5.m5.1.1.cmml" xref="S4.T2.22.5.m5.1.1"><csymbol cd="ambiguous" id="S4.T2.22.5.m5.1.1.1.cmml" xref="S4.T2.22.5.m5.1.1">subscript</csymbol><ci id="S4.T2.22.5.m5.1.1.2.cmml" xref="S4.T2.22.5.m5.1.1.2">𝚺</ci><cn id="S4.T2.22.5.m5.1.1.3.cmml" type="integer" xref="S4.T2.22.5.m5.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.22.5.m5.1d">\boldsymbol{\Sigma}_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.22.5.m5.1e">bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math> defined in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4.E16" title="In 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">16</span></a>).</span></figcaption> </figure> <div class="ltx_para" id="S4.SS4.p4"> <p class="ltx_p" id="S4.SS4.p4.3">From Table <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4.T2" title="Table 2 ‣ 4.4 Average coverage probability for the Regression Test ‣ 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">2</span></a>, we observe that all values of <math alttext="cov" class="ltx_Math" display="inline" id="S4.SS4.p4.1.m1.1"><semantics id="S4.SS4.p4.1.m1.1a"><mrow id="S4.SS4.p4.1.m1.1.1" xref="S4.SS4.p4.1.m1.1.1.cmml"><mi id="S4.SS4.p4.1.m1.1.1.2" xref="S4.SS4.p4.1.m1.1.1.2.cmml">c</mi><mo id="S4.SS4.p4.1.m1.1.1.1" xref="S4.SS4.p4.1.m1.1.1.1.cmml"></mo><mi id="S4.SS4.p4.1.m1.1.1.3" xref="S4.SS4.p4.1.m1.1.1.3.cmml">o</mi><mo id="S4.SS4.p4.1.m1.1.1.1a" xref="S4.SS4.p4.1.m1.1.1.1.cmml"></mo><mi id="S4.SS4.p4.1.m1.1.1.4" xref="S4.SS4.p4.1.m1.1.1.4.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p4.1.m1.1b"><apply id="S4.SS4.p4.1.m1.1.1.cmml" xref="S4.SS4.p4.1.m1.1.1"><times id="S4.SS4.p4.1.m1.1.1.1.cmml" xref="S4.SS4.p4.1.m1.1.1.1"></times><ci id="S4.SS4.p4.1.m1.1.1.2.cmml" xref="S4.SS4.p4.1.m1.1.1.2">𝑐</ci><ci id="S4.SS4.p4.1.m1.1.1.3.cmml" xref="S4.SS4.p4.1.m1.1.1.3">𝑜</ci><ci id="S4.SS4.p4.1.m1.1.1.4.cmml" xref="S4.SS4.p4.1.m1.1.1.4">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p4.1.m1.1c">cov</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p4.1.m1.1d">italic_c italic_o italic_v</annotation></semantics></math> are approximately equal to the nominal value of 0.95, as expected, due to the exact nature of the inference procedures across all tests and under different conditions. Furthermore, to better illustrate the behavior of the random variables discussed in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S2" title="2 Notation and Models ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">2</span></a>, Figures <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4.F1" title="Figure 1 ‣ 4.4 Average coverage probability for the Regression Test ‣ 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">1</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4.F2" title="Figure 2 ‣ 4.4 Average coverage probability for the Regression Test ‣ 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">2</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4.F3" title="Figure 3 ‣ 4.4 Average coverage probability for the Regression Test ‣ 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">3</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4.F4" title="Figure 4 ‣ 4.4 Average coverage probability for the Regression Test ‣ 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">4</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4.F5" title="Figure 5 ‣ 4.4 Average coverage probability for the Regression Test ‣ 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">5</span></a>, and <a class="ltx_ref" href="https://arxiv.org/html/2503.14711v1#S4.F6" title="Figure 6 ‣ 4.4 Average coverage probability for the Regression Test ‣ 4 Numerical Studies and Demonstration of Programming Code ‣ PSInference: A Package to Draw Inference for Released Plug-in Sampling Single Synthetic Dataset"><span class="ltx_text ltx_ref_tag">6</span></a> display the empirical distributions of the observed values from the synthetic datasets, generated under varying simulation conditions. To simplify we only include the distributions for <math alttext="n=10" class="ltx_Math" display="inline" id="S4.SS4.p4.2.m2.1"><semantics id="S4.SS4.p4.2.m2.1a"><mrow id="S4.SS4.p4.2.m2.1.1" xref="S4.SS4.p4.2.m2.1.1.cmml"><mi id="S4.SS4.p4.2.m2.1.1.2" xref="S4.SS4.p4.2.m2.1.1.2.cmml">n</mi><mo id="S4.SS4.p4.2.m2.1.1.1" xref="S4.SS4.p4.2.m2.1.1.1.cmml">=</mo><mn id="S4.SS4.p4.2.m2.1.1.3" xref="S4.SS4.p4.2.m2.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p4.2.m2.1b"><apply id="S4.SS4.p4.2.m2.1.1.cmml" xref="S4.SS4.p4.2.m2.1.1"><eq id="S4.SS4.p4.2.m2.1.1.1.cmml" xref="S4.SS4.p4.2.m2.1.1.1"></eq><ci id="S4.SS4.p4.2.m2.1.1.2.cmml" xref="S4.SS4.p4.2.m2.1.1.2">𝑛</ci><cn id="S4.SS4.p4.2.m2.1.1.3.cmml" type="integer" xref="S4.SS4.p4.2.m2.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p4.2.m2.1c">n=10</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p4.2.m2.1d">italic_n = 10</annotation></semantics></math> and <math alttext="n=500" class="ltx_Math" display="inline" id="S4.SS4.p4.3.m3.1"><semantics id="S4.SS4.p4.3.m3.1a"><mrow id="S4.SS4.p4.3.m3.1.1" xref="S4.SS4.p4.3.m3.1.1.cmml"><mi id="S4.SS4.p4.3.m3.1.1.2" xref="S4.SS4.p4.3.m3.1.1.2.cmml">n</mi><mo id="S4.SS4.p4.3.m3.1.1.1" xref="S4.SS4.p4.3.m3.1.1.1.cmml">=</mo><mn id="S4.SS4.p4.3.m3.1.1.3" xref="S4.SS4.p4.3.m3.1.1.3.cmml">500</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p4.3.m3.1b"><apply id="S4.SS4.p4.3.m3.1.1.cmml" xref="S4.SS4.p4.3.m3.1.1"><eq id="S4.SS4.p4.3.m3.1.1.1.cmml" xref="S4.SS4.p4.3.m3.1.1.1"></eq><ci id="S4.SS4.p4.3.m3.1.1.2.cmml" xref="S4.SS4.p4.3.m3.1.1.2">𝑛</ci><cn id="S4.SS4.p4.3.m3.1.1.3.cmml" type="integer" xref="S4.SS4.p4.3.m3.1.1.3">500</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p4.3.m3.1c">n=500</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p4.3.m3.1d">italic_n = 500</annotation></semantics></math>. These empirical distributions are presented alongside the corresponding theoretical distributions created using the functions <span class="ltx_text ltx_font_typewriter" id="S4.SS4.p4.3.1">GVdist</span>, <span class="ltx_text ltx_font_typewriter" id="S4.SS4.p4.3.2">Sphdist</span>, <span class="ltx_text ltx_font_typewriter" id="S4.SS4.p4.3.3">Inddist</span>, and <span class="ltx_text ltx_font_typewriter" id="S4.SS4.p4.3.4">Canodist</span>, allowing for the comparison and the validation of the theoretical results.</p> </div> <figure class="ltx_figure" id="S4.F1"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F1.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="421" id="S4.F1.sf1.g1" src="extracted/6290084/GV_10.png" width="698"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F1.sf1.4.1.1" style="font-size:90%;">(a)</span> </span><math alttext="n=10" class="ltx_Math" display="inline" id="S4.F1.sf1.2.m1.1"><semantics id="S4.F1.sf1.2.m1.1b"><mrow id="S4.F1.sf1.2.m1.1.1" xref="S4.F1.sf1.2.m1.1.1.cmml"><mi id="S4.F1.sf1.2.m1.1.1.2" mathsize="90%" xref="S4.F1.sf1.2.m1.1.1.2.cmml">n</mi><mo id="S4.F1.sf1.2.m1.1.1.1" mathsize="90%" xref="S4.F1.sf1.2.m1.1.1.1.cmml">=</mo><mn id="S4.F1.sf1.2.m1.1.1.3" mathsize="90%" xref="S4.F1.sf1.2.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F1.sf1.2.m1.1c"><apply id="S4.F1.sf1.2.m1.1.1.cmml" xref="S4.F1.sf1.2.m1.1.1"><eq id="S4.F1.sf1.2.m1.1.1.1.cmml" xref="S4.F1.sf1.2.m1.1.1.1"></eq><ci id="S4.F1.sf1.2.m1.1.1.2.cmml" xref="S4.F1.sf1.2.m1.1.1.2">𝑛</ci><cn id="S4.F1.sf1.2.m1.1.1.3.cmml" type="integer" xref="S4.F1.sf1.2.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F1.sf1.2.m1.1d">n=10</annotation><annotation encoding="application/x-llamapun" id="S4.F1.sf1.2.m1.1e">italic_n = 10</annotation></semantics></math></figcaption> </figure> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F1.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="469" id="S4.F1.sf2.g1" src="extracted/6290084/GV_500.png" width="698"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F1.sf2.4.1.1" style="font-size:90%;">(b)</span> </span><math alttext="n=500" class="ltx_Math" display="inline" id="S4.F1.sf2.2.m1.1"><semantics id="S4.F1.sf2.2.m1.1b"><mrow id="S4.F1.sf2.2.m1.1.1" xref="S4.F1.sf2.2.m1.1.1.cmml"><mi id="S4.F1.sf2.2.m1.1.1.2" mathsize="90%" xref="S4.F1.sf2.2.m1.1.1.2.cmml">n</mi><mo id="S4.F1.sf2.2.m1.1.1.1" mathsize="90%" xref="S4.F1.sf2.2.m1.1.1.1.cmml">=</mo><mn id="S4.F1.sf2.2.m1.1.1.3" mathsize="90%" xref="S4.F1.sf2.2.m1.1.1.3.cmml">500</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F1.sf2.2.m1.1c"><apply id="S4.F1.sf2.2.m1.1.1.cmml" xref="S4.F1.sf2.2.m1.1.1"><eq id="S4.F1.sf2.2.m1.1.1.1.cmml" xref="S4.F1.sf2.2.m1.1.1.1"></eq><ci id="S4.F1.sf2.2.m1.1.1.2.cmml" xref="S4.F1.sf2.2.m1.1.1.2">𝑛</ci><cn id="S4.F1.sf2.2.m1.1.1.3.cmml" type="integer" xref="S4.F1.sf2.2.m1.1.1.3">500</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F1.sf2.2.m1.1d">n=500</annotation><annotation encoding="application/x-llamapun" id="S4.F1.sf2.2.m1.1e">italic_n = 500</annotation></semantics></math></figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F1.14.7.1" style="font-size:90%;">Figure 1</span>: </span><span class="ltx_text" id="S4.F1.12.6" style="font-size:90%;">Graph for the Generalized Variance distribution along with the empirical distribution of the observed values of <math alttext="T_{1}^{\star}" class="ltx_Math" display="inline" id="S4.F1.7.1.m1.1"><semantics id="S4.F1.7.1.m1.1b"><msubsup id="S4.F1.7.1.m1.1.1" xref="S4.F1.7.1.m1.1.1.cmml"><mi id="S4.F1.7.1.m1.1.1.2.2" xref="S4.F1.7.1.m1.1.1.2.2.cmml">T</mi><mn id="S4.F1.7.1.m1.1.1.2.3" xref="S4.F1.7.1.m1.1.1.2.3.cmml">1</mn><mo id="S4.F1.7.1.m1.1.1.3" xref="S4.F1.7.1.m1.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.F1.7.1.m1.1c"><apply id="S4.F1.7.1.m1.1.1.cmml" xref="S4.F1.7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.F1.7.1.m1.1.1.1.cmml" xref="S4.F1.7.1.m1.1.1">superscript</csymbol><apply id="S4.F1.7.1.m1.1.1.2.cmml" xref="S4.F1.7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.F1.7.1.m1.1.1.2.1.cmml" xref="S4.F1.7.1.m1.1.1">subscript</csymbol><ci id="S4.F1.7.1.m1.1.1.2.2.cmml" xref="S4.F1.7.1.m1.1.1.2.2">𝑇</ci><cn id="S4.F1.7.1.m1.1.1.2.3.cmml" type="integer" xref="S4.F1.7.1.m1.1.1.2.3">1</cn></apply><ci id="S4.F1.7.1.m1.1.1.3.cmml" xref="S4.F1.7.1.m1.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F1.7.1.m1.1d">T_{1}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S4.F1.7.1.m1.1e">italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> for given <math alttext="n=10,500" class="ltx_Math" display="inline" id="S4.F1.8.2.m2.2"><semantics id="S4.F1.8.2.m2.2b"><mrow id="S4.F1.8.2.m2.2.3" xref="S4.F1.8.2.m2.2.3.cmml"><mi id="S4.F1.8.2.m2.2.3.2" xref="S4.F1.8.2.m2.2.3.2.cmml">n</mi><mo id="S4.F1.8.2.m2.2.3.1" xref="S4.F1.8.2.m2.2.3.1.cmml">=</mo><mrow id="S4.F1.8.2.m2.2.3.3.2" xref="S4.F1.8.2.m2.2.3.3.1.cmml"><mn id="S4.F1.8.2.m2.1.1" xref="S4.F1.8.2.m2.1.1.cmml">10</mn><mo id="S4.F1.8.2.m2.2.3.3.2.1" xref="S4.F1.8.2.m2.2.3.3.1.cmml">,</mo><mn id="S4.F1.8.2.m2.2.2" xref="S4.F1.8.2.m2.2.2.cmml">500</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F1.8.2.m2.2c"><apply id="S4.F1.8.2.m2.2.3.cmml" xref="S4.F1.8.2.m2.2.3"><eq id="S4.F1.8.2.m2.2.3.1.cmml" xref="S4.F1.8.2.m2.2.3.1"></eq><ci id="S4.F1.8.2.m2.2.3.2.cmml" xref="S4.F1.8.2.m2.2.3.2">𝑛</ci><list id="S4.F1.8.2.m2.2.3.3.1.cmml" xref="S4.F1.8.2.m2.2.3.3.2"><cn id="S4.F1.8.2.m2.1.1.cmml" type="integer" xref="S4.F1.8.2.m2.1.1">10</cn><cn id="S4.F1.8.2.m2.2.2.cmml" type="integer" xref="S4.F1.8.2.m2.2.2">500</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F1.8.2.m2.2d">n=10,500</annotation><annotation encoding="application/x-llamapun" id="S4.F1.8.2.m2.2e">italic_n = 10 , 500</annotation></semantics></math>, <math alttext="p=4" class="ltx_Math" display="inline" id="S4.F1.9.3.m3.1"><semantics id="S4.F1.9.3.m3.1b"><mrow id="S4.F1.9.3.m3.1.1" xref="S4.F1.9.3.m3.1.1.cmml"><mi id="S4.F1.9.3.m3.1.1.2" xref="S4.F1.9.3.m3.1.1.2.cmml">p</mi><mo id="S4.F1.9.3.m3.1.1.1" xref="S4.F1.9.3.m3.1.1.1.cmml">=</mo><mn id="S4.F1.9.3.m3.1.1.3" xref="S4.F1.9.3.m3.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F1.9.3.m3.1c"><apply id="S4.F1.9.3.m3.1.1.cmml" xref="S4.F1.9.3.m3.1.1"><eq id="S4.F1.9.3.m3.1.1.1.cmml" xref="S4.F1.9.3.m3.1.1.1"></eq><ci id="S4.F1.9.3.m3.1.1.2.cmml" xref="S4.F1.9.3.m3.1.1.2">𝑝</ci><cn id="S4.F1.9.3.m3.1.1.3.cmml" type="integer" xref="S4.F1.9.3.m3.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F1.9.3.m3.1d">p=4</annotation><annotation encoding="application/x-llamapun" id="S4.F1.9.3.m3.1e">italic_p = 4</annotation></semantics></math>, <math alttext="\boldsymbol{\mu}" class="ltx_Math" display="inline" id="S4.F1.10.4.m4.1"><semantics id="S4.F1.10.4.m4.1b"><mi id="S4.F1.10.4.m4.1.1" xref="S4.F1.10.4.m4.1.1.cmml">𝝁</mi><annotation-xml encoding="MathML-Content" id="S4.F1.10.4.m4.1c"><ci id="S4.F1.10.4.m4.1.1.cmml" xref="S4.F1.10.4.m4.1.1">𝝁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F1.10.4.m4.1d">\boldsymbol{\mu}</annotation><annotation encoding="application/x-llamapun" id="S4.F1.10.4.m4.1e">bold_italic_μ</annotation></semantics></math>, <math alttext="\boldsymbol{\Sigma}_{3}" class="ltx_Math" display="inline" id="S4.F1.11.5.m5.1"><semantics id="S4.F1.11.5.m5.1b"><msub id="S4.F1.11.5.m5.1.1" xref="S4.F1.11.5.m5.1.1.cmml"><mi id="S4.F1.11.5.m5.1.1.2" xref="S4.F1.11.5.m5.1.1.2.cmml">𝚺</mi><mn id="S4.F1.11.5.m5.1.1.3" xref="S4.F1.11.5.m5.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F1.11.5.m5.1c"><apply id="S4.F1.11.5.m5.1.1.cmml" xref="S4.F1.11.5.m5.1.1"><csymbol cd="ambiguous" id="S4.F1.11.5.m5.1.1.1.cmml" xref="S4.F1.11.5.m5.1.1">subscript</csymbol><ci id="S4.F1.11.5.m5.1.1.2.cmml" xref="S4.F1.11.5.m5.1.1.2">𝚺</ci><cn id="S4.F1.11.5.m5.1.1.3.cmml" type="integer" xref="S4.F1.11.5.m5.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F1.11.5.m5.1d">\boldsymbol{\Sigma}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.F1.11.5.m5.1e">bold_Σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> (T1star_obs1) and <math alttext="\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.F1.12.6.m6.1"><semantics id="S4.F1.12.6.m6.1b"><msub id="S4.F1.12.6.m6.1.1" xref="S4.F1.12.6.m6.1.1.cmml"><mi id="S4.F1.12.6.m6.1.1.2" xref="S4.F1.12.6.m6.1.1.2.cmml">𝚺</mi><mn id="S4.F1.12.6.m6.1.1.3" xref="S4.F1.12.6.m6.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F1.12.6.m6.1c"><apply id="S4.F1.12.6.m6.1.1.cmml" xref="S4.F1.12.6.m6.1.1"><csymbol cd="ambiguous" id="S4.F1.12.6.m6.1.1.1.cmml" xref="S4.F1.12.6.m6.1.1">subscript</csymbol><ci id="S4.F1.12.6.m6.1.1.2.cmml" xref="S4.F1.12.6.m6.1.1.2">𝚺</ci><cn id="S4.F1.12.6.m6.1.1.3.cmml" type="integer" xref="S4.F1.12.6.m6.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F1.12.6.m6.1d">\boldsymbol{\Sigma}_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.F1.12.6.m6.1e">bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math> (T1star_obs2).</span></figcaption> </figure> <figure class="ltx_figure" id="S4.F2"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_figure_panel" id="S4.F2.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="402" id="S4.F2.sf1.g1" src="extracted/6290084/Sph_10.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F2.sf1.4.1.1" style="font-size:90%;">(a)</span> </span><math alttext="n=10" class="ltx_Math" display="inline" id="S4.F2.sf1.2.m1.1"><semantics id="S4.F2.sf1.2.m1.1b"><mrow id="S4.F2.sf1.2.m1.1.1" xref="S4.F2.sf1.2.m1.1.1.cmml"><mi id="S4.F2.sf1.2.m1.1.1.2" mathsize="90%" xref="S4.F2.sf1.2.m1.1.1.2.cmml">n</mi><mo id="S4.F2.sf1.2.m1.1.1.1" mathsize="90%" xref="S4.F2.sf1.2.m1.1.1.1.cmml">=</mo><mn id="S4.F2.sf1.2.m1.1.1.3" mathsize="90%" xref="S4.F2.sf1.2.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F2.sf1.2.m1.1c"><apply id="S4.F2.sf1.2.m1.1.1.cmml" xref="S4.F2.sf1.2.m1.1.1"><eq id="S4.F2.sf1.2.m1.1.1.1.cmml" xref="S4.F2.sf1.2.m1.1.1.1"></eq><ci id="S4.F2.sf1.2.m1.1.1.2.cmml" xref="S4.F2.sf1.2.m1.1.1.2">𝑛</ci><cn id="S4.F2.sf1.2.m1.1.1.3.cmml" type="integer" xref="S4.F2.sf1.2.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F2.sf1.2.m1.1d">n=10</annotation><annotation encoding="application/x-llamapun" id="S4.F2.sf1.2.m1.1e">italic_n = 10</annotation></semantics></math></figcaption> </figure> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_figure_panel" id="S4.F2.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="402" id="S4.F2.sf2.g1" src="extracted/6290084/Sph_500.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F2.sf2.4.1.1" style="font-size:90%;">(b)</span> </span><math alttext="n=500" class="ltx_Math" display="inline" id="S4.F2.sf2.2.m1.1"><semantics id="S4.F2.sf2.2.m1.1b"><mrow id="S4.F2.sf2.2.m1.1.1" xref="S4.F2.sf2.2.m1.1.1.cmml"><mi id="S4.F2.sf2.2.m1.1.1.2" mathsize="90%" xref="S4.F2.sf2.2.m1.1.1.2.cmml">n</mi><mo id="S4.F2.sf2.2.m1.1.1.1" mathsize="90%" xref="S4.F2.sf2.2.m1.1.1.1.cmml">=</mo><mn id="S4.F2.sf2.2.m1.1.1.3" mathsize="90%" xref="S4.F2.sf2.2.m1.1.1.3.cmml">500</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F2.sf2.2.m1.1c"><apply id="S4.F2.sf2.2.m1.1.1.cmml" xref="S4.F2.sf2.2.m1.1.1"><eq id="S4.F2.sf2.2.m1.1.1.1.cmml" xref="S4.F2.sf2.2.m1.1.1.1"></eq><ci id="S4.F2.sf2.2.m1.1.1.2.cmml" xref="S4.F2.sf2.2.m1.1.1.2">𝑛</ci><cn id="S4.F2.sf2.2.m1.1.1.3.cmml" type="integer" xref="S4.F2.sf2.2.m1.1.1.3">500</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F2.sf2.2.m1.1d">n=500</annotation><annotation encoding="application/x-llamapun" id="S4.F2.sf2.2.m1.1e">italic_n = 500</annotation></semantics></math></figcaption> </figure> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F2.14.7.1" style="font-size:90%;">Figure 2</span>: </span><span class="ltx_text" id="S4.F2.12.6" style="font-size:90%;">Graph for the Sphericity Test statistic’s distribution along with the empirical distribution of the observed values of <math alttext="T_{2}^{\star}" class="ltx_Math" display="inline" id="S4.F2.7.1.m1.1"><semantics id="S4.F2.7.1.m1.1b"><msubsup id="S4.F2.7.1.m1.1.1" xref="S4.F2.7.1.m1.1.1.cmml"><mi id="S4.F2.7.1.m1.1.1.2.2" xref="S4.F2.7.1.m1.1.1.2.2.cmml">T</mi><mn id="S4.F2.7.1.m1.1.1.2.3" xref="S4.F2.7.1.m1.1.1.2.3.cmml">2</mn><mo id="S4.F2.7.1.m1.1.1.3" xref="S4.F2.7.1.m1.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.F2.7.1.m1.1c"><apply id="S4.F2.7.1.m1.1.1.cmml" xref="S4.F2.7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.F2.7.1.m1.1.1.1.cmml" xref="S4.F2.7.1.m1.1.1">superscript</csymbol><apply id="S4.F2.7.1.m1.1.1.2.cmml" xref="S4.F2.7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.F2.7.1.m1.1.1.2.1.cmml" xref="S4.F2.7.1.m1.1.1">subscript</csymbol><ci id="S4.F2.7.1.m1.1.1.2.2.cmml" xref="S4.F2.7.1.m1.1.1.2.2">𝑇</ci><cn id="S4.F2.7.1.m1.1.1.2.3.cmml" type="integer" xref="S4.F2.7.1.m1.1.1.2.3">2</cn></apply><ci id="S4.F2.7.1.m1.1.1.3.cmml" xref="S4.F2.7.1.m1.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F2.7.1.m1.1d">T_{2}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S4.F2.7.1.m1.1e">italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> for given <math alttext="n=10,500" class="ltx_Math" display="inline" id="S4.F2.8.2.m2.2"><semantics id="S4.F2.8.2.m2.2b"><mrow id="S4.F2.8.2.m2.2.3" xref="S4.F2.8.2.m2.2.3.cmml"><mi id="S4.F2.8.2.m2.2.3.2" xref="S4.F2.8.2.m2.2.3.2.cmml">n</mi><mo id="S4.F2.8.2.m2.2.3.1" xref="S4.F2.8.2.m2.2.3.1.cmml">=</mo><mrow id="S4.F2.8.2.m2.2.3.3.2" xref="S4.F2.8.2.m2.2.3.3.1.cmml"><mn id="S4.F2.8.2.m2.1.1" xref="S4.F2.8.2.m2.1.1.cmml">10</mn><mo id="S4.F2.8.2.m2.2.3.3.2.1" xref="S4.F2.8.2.m2.2.3.3.1.cmml">,</mo><mn id="S4.F2.8.2.m2.2.2" xref="S4.F2.8.2.m2.2.2.cmml">500</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F2.8.2.m2.2c"><apply id="S4.F2.8.2.m2.2.3.cmml" xref="S4.F2.8.2.m2.2.3"><eq id="S4.F2.8.2.m2.2.3.1.cmml" xref="S4.F2.8.2.m2.2.3.1"></eq><ci id="S4.F2.8.2.m2.2.3.2.cmml" xref="S4.F2.8.2.m2.2.3.2">𝑛</ci><list id="S4.F2.8.2.m2.2.3.3.1.cmml" xref="S4.F2.8.2.m2.2.3.3.2"><cn id="S4.F2.8.2.m2.1.1.cmml" type="integer" xref="S4.F2.8.2.m2.1.1">10</cn><cn id="S4.F2.8.2.m2.2.2.cmml" type="integer" xref="S4.F2.8.2.m2.2.2">500</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F2.8.2.m2.2d">n=10,500</annotation><annotation encoding="application/x-llamapun" id="S4.F2.8.2.m2.2e">italic_n = 10 , 500</annotation></semantics></math>, <math alttext="p=4" class="ltx_Math" display="inline" id="S4.F2.9.3.m3.1"><semantics id="S4.F2.9.3.m3.1b"><mrow id="S4.F2.9.3.m3.1.1" xref="S4.F2.9.3.m3.1.1.cmml"><mi id="S4.F2.9.3.m3.1.1.2" xref="S4.F2.9.3.m3.1.1.2.cmml">p</mi><mo id="S4.F2.9.3.m3.1.1.1" xref="S4.F2.9.3.m3.1.1.1.cmml">=</mo><mn id="S4.F2.9.3.m3.1.1.3" xref="S4.F2.9.3.m3.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F2.9.3.m3.1c"><apply id="S4.F2.9.3.m3.1.1.cmml" xref="S4.F2.9.3.m3.1.1"><eq id="S4.F2.9.3.m3.1.1.1.cmml" xref="S4.F2.9.3.m3.1.1.1"></eq><ci id="S4.F2.9.3.m3.1.1.2.cmml" xref="S4.F2.9.3.m3.1.1.2">𝑝</ci><cn id="S4.F2.9.3.m3.1.1.3.cmml" type="integer" xref="S4.F2.9.3.m3.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F2.9.3.m3.1d">p=4</annotation><annotation encoding="application/x-llamapun" id="S4.F2.9.3.m3.1e">italic_p = 4</annotation></semantics></math>, <math alttext="\boldsymbol{\mu}" class="ltx_Math" display="inline" id="S4.F2.10.4.m4.1"><semantics id="S4.F2.10.4.m4.1b"><mi id="S4.F2.10.4.m4.1.1" xref="S4.F2.10.4.m4.1.1.cmml">𝝁</mi><annotation-xml encoding="MathML-Content" id="S4.F2.10.4.m4.1c"><ci id="S4.F2.10.4.m4.1.1.cmml" xref="S4.F2.10.4.m4.1.1">𝝁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F2.10.4.m4.1d">\boldsymbol{\mu}</annotation><annotation encoding="application/x-llamapun" id="S4.F2.10.4.m4.1e">bold_italic_μ</annotation></semantics></math>, <math alttext="\boldsymbol{\Sigma}_{1}" class="ltx_Math" display="inline" id="S4.F2.11.5.m5.1"><semantics id="S4.F2.11.5.m5.1b"><msub id="S4.F2.11.5.m5.1.1" xref="S4.F2.11.5.m5.1.1.cmml"><mi id="S4.F2.11.5.m5.1.1.2" xref="S4.F2.11.5.m5.1.1.2.cmml">𝚺</mi><mn id="S4.F2.11.5.m5.1.1.3" xref="S4.F2.11.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F2.11.5.m5.1c"><apply id="S4.F2.11.5.m5.1.1.cmml" xref="S4.F2.11.5.m5.1.1"><csymbol cd="ambiguous" id="S4.F2.11.5.m5.1.1.1.cmml" xref="S4.F2.11.5.m5.1.1">subscript</csymbol><ci id="S4.F2.11.5.m5.1.1.2.cmml" xref="S4.F2.11.5.m5.1.1.2">𝚺</ci><cn id="S4.F2.11.5.m5.1.1.3.cmml" type="integer" xref="S4.F2.11.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F2.11.5.m5.1d">\boldsymbol{\Sigma}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.F2.11.5.m5.1e">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> (T2star_obs1) and <math alttext="\boldsymbol{\Sigma}_{2}" class="ltx_Math" display="inline" id="S4.F2.12.6.m6.1"><semantics id="S4.F2.12.6.m6.1b"><msub id="S4.F2.12.6.m6.1.1" xref="S4.F2.12.6.m6.1.1.cmml"><mi id="S4.F2.12.6.m6.1.1.2" xref="S4.F2.12.6.m6.1.1.2.cmml">𝚺</mi><mn id="S4.F2.12.6.m6.1.1.3" xref="S4.F2.12.6.m6.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F2.12.6.m6.1c"><apply id="S4.F2.12.6.m6.1.1.cmml" xref="S4.F2.12.6.m6.1.1"><csymbol cd="ambiguous" id="S4.F2.12.6.m6.1.1.1.cmml" xref="S4.F2.12.6.m6.1.1">subscript</csymbol><ci id="S4.F2.12.6.m6.1.1.2.cmml" xref="S4.F2.12.6.m6.1.1.2">𝚺</ci><cn id="S4.F2.12.6.m6.1.1.3.cmml" type="integer" xref="S4.F2.12.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F2.12.6.m6.1d">\boldsymbol{\Sigma}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.F2.12.6.m6.1e">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> (T2star_obs2).</span></figcaption> </figure> <figure class="ltx_figure" id="S4.F3"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_figure_panel" id="S4.F3.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="402" id="S4.F3.sf1.g1" src="extracted/6290084/Ind_T1_10.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F3.sf1.6.2.1" style="font-size:90%;">(a)</span> </span><math alttext="p_{1}=1" class="ltx_Math" display="inline" id="S4.F3.sf1.3.m1.1"><semantics id="S4.F3.sf1.3.m1.1b"><mrow id="S4.F3.sf1.3.m1.1.1" xref="S4.F3.sf1.3.m1.1.1.cmml"><msub id="S4.F3.sf1.3.m1.1.1.2" xref="S4.F3.sf1.3.m1.1.1.2.cmml"><mi id="S4.F3.sf1.3.m1.1.1.2.2" mathsize="90%" xref="S4.F3.sf1.3.m1.1.1.2.2.cmml">p</mi><mn id="S4.F3.sf1.3.m1.1.1.2.3" mathsize="90%" xref="S4.F3.sf1.3.m1.1.1.2.3.cmml">1</mn></msub><mo id="S4.F3.sf1.3.m1.1.1.1" mathsize="90%" xref="S4.F3.sf1.3.m1.1.1.1.cmml">=</mo><mn id="S4.F3.sf1.3.m1.1.1.3" mathsize="90%" xref="S4.F3.sf1.3.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.sf1.3.m1.1c"><apply id="S4.F3.sf1.3.m1.1.1.cmml" xref="S4.F3.sf1.3.m1.1.1"><eq id="S4.F3.sf1.3.m1.1.1.1.cmml" xref="S4.F3.sf1.3.m1.1.1.1"></eq><apply id="S4.F3.sf1.3.m1.1.1.2.cmml" xref="S4.F3.sf1.3.m1.1.1.2"><csymbol cd="ambiguous" id="S4.F3.sf1.3.m1.1.1.2.1.cmml" xref="S4.F3.sf1.3.m1.1.1.2">subscript</csymbol><ci id="S4.F3.sf1.3.m1.1.1.2.2.cmml" xref="S4.F3.sf1.3.m1.1.1.2.2">𝑝</ci><cn id="S4.F3.sf1.3.m1.1.1.2.3.cmml" type="integer" xref="S4.F3.sf1.3.m1.1.1.2.3">1</cn></apply><cn id="S4.F3.sf1.3.m1.1.1.3.cmml" type="integer" xref="S4.F3.sf1.3.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.sf1.3.m1.1d">p_{1}=1</annotation><annotation encoding="application/x-llamapun" id="S4.F3.sf1.3.m1.1e">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math><span class="ltx_text" id="S4.F3.sf1.4.1" style="font-size:90%;"> and <math alttext="\boldsymbol{\Sigma}=\boldsymbol{\Sigma}_{1}" class="ltx_Math" display="inline" id="S4.F3.sf1.4.1.m1.1"><semantics id="S4.F3.sf1.4.1.m1.1b"><mrow id="S4.F3.sf1.4.1.m1.1.1" xref="S4.F3.sf1.4.1.m1.1.1.cmml"><mi id="S4.F3.sf1.4.1.m1.1.1.2" xref="S4.F3.sf1.4.1.m1.1.1.2.cmml">𝚺</mi><mo id="S4.F3.sf1.4.1.m1.1.1.1" xref="S4.F3.sf1.4.1.m1.1.1.1.cmml">=</mo><msub id="S4.F3.sf1.4.1.m1.1.1.3" xref="S4.F3.sf1.4.1.m1.1.1.3.cmml"><mi id="S4.F3.sf1.4.1.m1.1.1.3.2" xref="S4.F3.sf1.4.1.m1.1.1.3.2.cmml">𝚺</mi><mn id="S4.F3.sf1.4.1.m1.1.1.3.3" xref="S4.F3.sf1.4.1.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.sf1.4.1.m1.1c"><apply id="S4.F3.sf1.4.1.m1.1.1.cmml" xref="S4.F3.sf1.4.1.m1.1.1"><eq id="S4.F3.sf1.4.1.m1.1.1.1.cmml" xref="S4.F3.sf1.4.1.m1.1.1.1"></eq><ci id="S4.F3.sf1.4.1.m1.1.1.2.cmml" xref="S4.F3.sf1.4.1.m1.1.1.2">𝚺</ci><apply id="S4.F3.sf1.4.1.m1.1.1.3.cmml" xref="S4.F3.sf1.4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.F3.sf1.4.1.m1.1.1.3.1.cmml" xref="S4.F3.sf1.4.1.m1.1.1.3">subscript</csymbol><ci id="S4.F3.sf1.4.1.m1.1.1.3.2.cmml" xref="S4.F3.sf1.4.1.m1.1.1.3.2">𝚺</ci><cn id="S4.F3.sf1.4.1.m1.1.1.3.3.cmml" type="integer" xref="S4.F3.sf1.4.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.sf1.4.1.m1.1d">\boldsymbol{\Sigma}=\boldsymbol{\Sigma}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.sf1.4.1.m1.1e">bold_Σ = bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math></span></figcaption> </figure> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_figure_panel" id="S4.F3.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="402" id="S4.F3.sf2.g1" src="extracted/6290084/Ind_T2_10.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F3.sf2.6.2.1" style="font-size:90%;">(b)</span> </span><math alttext="p_{1}=2" class="ltx_Math" display="inline" id="S4.F3.sf2.3.m1.1"><semantics id="S4.F3.sf2.3.m1.1b"><mrow id="S4.F3.sf2.3.m1.1.1" xref="S4.F3.sf2.3.m1.1.1.cmml"><msub id="S4.F3.sf2.3.m1.1.1.2" xref="S4.F3.sf2.3.m1.1.1.2.cmml"><mi id="S4.F3.sf2.3.m1.1.1.2.2" mathsize="90%" xref="S4.F3.sf2.3.m1.1.1.2.2.cmml">p</mi><mn id="S4.F3.sf2.3.m1.1.1.2.3" mathsize="90%" xref="S4.F3.sf2.3.m1.1.1.2.3.cmml">1</mn></msub><mo id="S4.F3.sf2.3.m1.1.1.1" mathsize="90%" xref="S4.F3.sf2.3.m1.1.1.1.cmml">=</mo><mn id="S4.F3.sf2.3.m1.1.1.3" mathsize="90%" xref="S4.F3.sf2.3.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.sf2.3.m1.1c"><apply id="S4.F3.sf2.3.m1.1.1.cmml" xref="S4.F3.sf2.3.m1.1.1"><eq id="S4.F3.sf2.3.m1.1.1.1.cmml" xref="S4.F3.sf2.3.m1.1.1.1"></eq><apply id="S4.F3.sf2.3.m1.1.1.2.cmml" xref="S4.F3.sf2.3.m1.1.1.2"><csymbol cd="ambiguous" id="S4.F3.sf2.3.m1.1.1.2.1.cmml" xref="S4.F3.sf2.3.m1.1.1.2">subscript</csymbol><ci id="S4.F3.sf2.3.m1.1.1.2.2.cmml" xref="S4.F3.sf2.3.m1.1.1.2.2">𝑝</ci><cn id="S4.F3.sf2.3.m1.1.1.2.3.cmml" type="integer" xref="S4.F3.sf2.3.m1.1.1.2.3">1</cn></apply><cn id="S4.F3.sf2.3.m1.1.1.3.cmml" type="integer" xref="S4.F3.sf2.3.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.sf2.3.m1.1d">p_{1}=2</annotation><annotation encoding="application/x-llamapun" id="S4.F3.sf2.3.m1.1e">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2</annotation></semantics></math><span class="ltx_text" id="S4.F3.sf2.4.1" style="font-size:90%;"> and <math alttext="\boldsymbol{\Sigma}=\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.F3.sf2.4.1.m1.1"><semantics id="S4.F3.sf2.4.1.m1.1b"><mrow id="S4.F3.sf2.4.1.m1.1.1" xref="S4.F3.sf2.4.1.m1.1.1.cmml"><mi id="S4.F3.sf2.4.1.m1.1.1.2" xref="S4.F3.sf2.4.1.m1.1.1.2.cmml">𝚺</mi><mo id="S4.F3.sf2.4.1.m1.1.1.1" xref="S4.F3.sf2.4.1.m1.1.1.1.cmml">=</mo><msub id="S4.F3.sf2.4.1.m1.1.1.3" xref="S4.F3.sf2.4.1.m1.1.1.3.cmml"><mi id="S4.F3.sf2.4.1.m1.1.1.3.2" xref="S4.F3.sf2.4.1.m1.1.1.3.2.cmml">𝚺</mi><mn id="S4.F3.sf2.4.1.m1.1.1.3.3" xref="S4.F3.sf2.4.1.m1.1.1.3.3.cmml">4</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.sf2.4.1.m1.1c"><apply id="S4.F3.sf2.4.1.m1.1.1.cmml" xref="S4.F3.sf2.4.1.m1.1.1"><eq id="S4.F3.sf2.4.1.m1.1.1.1.cmml" xref="S4.F3.sf2.4.1.m1.1.1.1"></eq><ci id="S4.F3.sf2.4.1.m1.1.1.2.cmml" xref="S4.F3.sf2.4.1.m1.1.1.2">𝚺</ci><apply id="S4.F3.sf2.4.1.m1.1.1.3.cmml" xref="S4.F3.sf2.4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.F3.sf2.4.1.m1.1.1.3.1.cmml" xref="S4.F3.sf2.4.1.m1.1.1.3">subscript</csymbol><ci id="S4.F3.sf2.4.1.m1.1.1.3.2.cmml" xref="S4.F3.sf2.4.1.m1.1.1.3.2">𝚺</ci><cn id="S4.F3.sf2.4.1.m1.1.1.3.3.cmml" type="integer" xref="S4.F3.sf2.4.1.m1.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.sf2.4.1.m1.1d">\boldsymbol{\Sigma}=\boldsymbol{\Sigma}_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.sf2.4.1.m1.1e">bold_Σ = bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math></span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F3.14.7.1" style="font-size:90%;">Figure 3</span>: </span><span class="ltx_text" id="S4.F3.12.6" style="font-size:90%;">Graph for the Independence Test statistic’s distribution along with the empirical distribution of the observed values of <math alttext="T_{3}^{\star}" class="ltx_Math" display="inline" id="S4.F3.7.1.m1.1"><semantics id="S4.F3.7.1.m1.1b"><msubsup id="S4.F3.7.1.m1.1.1" xref="S4.F3.7.1.m1.1.1.cmml"><mi id="S4.F3.7.1.m1.1.1.2.2" xref="S4.F3.7.1.m1.1.1.2.2.cmml">T</mi><mn id="S4.F3.7.1.m1.1.1.2.3" xref="S4.F3.7.1.m1.1.1.2.3.cmml">3</mn><mo id="S4.F3.7.1.m1.1.1.3" xref="S4.F3.7.1.m1.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.F3.7.1.m1.1c"><apply id="S4.F3.7.1.m1.1.1.cmml" xref="S4.F3.7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.F3.7.1.m1.1.1.1.cmml" xref="S4.F3.7.1.m1.1.1">superscript</csymbol><apply id="S4.F3.7.1.m1.1.1.2.cmml" xref="S4.F3.7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.F3.7.1.m1.1.1.2.1.cmml" xref="S4.F3.7.1.m1.1.1">subscript</csymbol><ci id="S4.F3.7.1.m1.1.1.2.2.cmml" xref="S4.F3.7.1.m1.1.1.2.2">𝑇</ci><cn id="S4.F3.7.1.m1.1.1.2.3.cmml" type="integer" xref="S4.F3.7.1.m1.1.1.2.3">3</cn></apply><ci id="S4.F3.7.1.m1.1.1.3.cmml" xref="S4.F3.7.1.m1.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.7.1.m1.1d">T_{3}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.7.1.m1.1e">italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> for given <math alttext="n=10" class="ltx_Math" display="inline" id="S4.F3.8.2.m2.1"><semantics id="S4.F3.8.2.m2.1b"><mrow id="S4.F3.8.2.m2.1.1" xref="S4.F3.8.2.m2.1.1.cmml"><mi id="S4.F3.8.2.m2.1.1.2" xref="S4.F3.8.2.m2.1.1.2.cmml">n</mi><mo id="S4.F3.8.2.m2.1.1.1" xref="S4.F3.8.2.m2.1.1.1.cmml">=</mo><mn id="S4.F3.8.2.m2.1.1.3" xref="S4.F3.8.2.m2.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.8.2.m2.1c"><apply id="S4.F3.8.2.m2.1.1.cmml" xref="S4.F3.8.2.m2.1.1"><eq id="S4.F3.8.2.m2.1.1.1.cmml" xref="S4.F3.8.2.m2.1.1.1"></eq><ci id="S4.F3.8.2.m2.1.1.2.cmml" xref="S4.F3.8.2.m2.1.1.2">𝑛</ci><cn id="S4.F3.8.2.m2.1.1.3.cmml" type="integer" xref="S4.F3.8.2.m2.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.8.2.m2.1d">n=10</annotation><annotation encoding="application/x-llamapun" id="S4.F3.8.2.m2.1e">italic_n = 10</annotation></semantics></math>, <math alttext="p=4" class="ltx_Math" display="inline" id="S4.F3.9.3.m3.1"><semantics id="S4.F3.9.3.m3.1b"><mrow id="S4.F3.9.3.m3.1.1" xref="S4.F3.9.3.m3.1.1.cmml"><mi id="S4.F3.9.3.m3.1.1.2" xref="S4.F3.9.3.m3.1.1.2.cmml">p</mi><mo id="S4.F3.9.3.m3.1.1.1" xref="S4.F3.9.3.m3.1.1.1.cmml">=</mo><mn id="S4.F3.9.3.m3.1.1.3" xref="S4.F3.9.3.m3.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.9.3.m3.1c"><apply id="S4.F3.9.3.m3.1.1.cmml" xref="S4.F3.9.3.m3.1.1"><eq id="S4.F3.9.3.m3.1.1.1.cmml" xref="S4.F3.9.3.m3.1.1.1"></eq><ci id="S4.F3.9.3.m3.1.1.2.cmml" xref="S4.F3.9.3.m3.1.1.2">𝑝</ci><cn id="S4.F3.9.3.m3.1.1.3.cmml" type="integer" xref="S4.F3.9.3.m3.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.9.3.m3.1d">p=4</annotation><annotation encoding="application/x-llamapun" id="S4.F3.9.3.m3.1e">italic_p = 4</annotation></semantics></math>, <math alttext="\boldsymbol{\mu}" class="ltx_Math" display="inline" id="S4.F3.10.4.m4.1"><semantics id="S4.F3.10.4.m4.1b"><mi id="S4.F3.10.4.m4.1.1" xref="S4.F3.10.4.m4.1.1.cmml">𝝁</mi><annotation-xml encoding="MathML-Content" id="S4.F3.10.4.m4.1c"><ci id="S4.F3.10.4.m4.1.1.cmml" xref="S4.F3.10.4.m4.1.1">𝝁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.10.4.m4.1d">\boldsymbol{\mu}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.10.4.m4.1e">bold_italic_μ</annotation></semantics></math>, <math alttext="\boldsymbol{\Sigma}_{1}" class="ltx_Math" display="inline" id="S4.F3.11.5.m5.1"><semantics id="S4.F3.11.5.m5.1b"><msub id="S4.F3.11.5.m5.1.1" xref="S4.F3.11.5.m5.1.1.cmml"><mi id="S4.F3.11.5.m5.1.1.2" xref="S4.F3.11.5.m5.1.1.2.cmml">𝚺</mi><mn id="S4.F3.11.5.m5.1.1.3" xref="S4.F3.11.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F3.11.5.m5.1c"><apply id="S4.F3.11.5.m5.1.1.cmml" xref="S4.F3.11.5.m5.1.1"><csymbol cd="ambiguous" id="S4.F3.11.5.m5.1.1.1.cmml" xref="S4.F3.11.5.m5.1.1">subscript</csymbol><ci id="S4.F3.11.5.m5.1.1.2.cmml" xref="S4.F3.11.5.m5.1.1.2">𝚺</ci><cn id="S4.F3.11.5.m5.1.1.3.cmml" type="integer" xref="S4.F3.11.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.11.5.m5.1d">\boldsymbol{\Sigma}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.11.5.m5.1e">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> (T3star_obs1) and <math alttext="\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.F3.12.6.m6.1"><semantics id="S4.F3.12.6.m6.1b"><msub id="S4.F3.12.6.m6.1.1" xref="S4.F3.12.6.m6.1.1.cmml"><mi id="S4.F3.12.6.m6.1.1.2" xref="S4.F3.12.6.m6.1.1.2.cmml">𝚺</mi><mn id="S4.F3.12.6.m6.1.1.3" xref="S4.F3.12.6.m6.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F3.12.6.m6.1c"><apply id="S4.F3.12.6.m6.1.1.cmml" xref="S4.F3.12.6.m6.1.1"><csymbol cd="ambiguous" id="S4.F3.12.6.m6.1.1.1.cmml" xref="S4.F3.12.6.m6.1.1">subscript</csymbol><ci id="S4.F3.12.6.m6.1.1.2.cmml" xref="S4.F3.12.6.m6.1.1.2">𝚺</ci><cn id="S4.F3.12.6.m6.1.1.3.cmml" type="integer" xref="S4.F3.12.6.m6.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.12.6.m6.1d">\boldsymbol{\Sigma}_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.12.6.m6.1e">bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math> (T3star_obs2).</span></figcaption> </figure> <figure class="ltx_figure" id="S4.F4"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_figure_panel" id="S4.F4.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="402" id="S4.F4.sf1.g1" src="extracted/6290084/Ind_T1_500.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F4.sf1.6.2.1" style="font-size:90%;">(a)</span> </span><math alttext="p_{1}=1" class="ltx_Math" display="inline" id="S4.F4.sf1.3.m1.1"><semantics id="S4.F4.sf1.3.m1.1b"><mrow id="S4.F4.sf1.3.m1.1.1" xref="S4.F4.sf1.3.m1.1.1.cmml"><msub id="S4.F4.sf1.3.m1.1.1.2" xref="S4.F4.sf1.3.m1.1.1.2.cmml"><mi id="S4.F4.sf1.3.m1.1.1.2.2" mathsize="90%" xref="S4.F4.sf1.3.m1.1.1.2.2.cmml">p</mi><mn id="S4.F4.sf1.3.m1.1.1.2.3" mathsize="90%" xref="S4.F4.sf1.3.m1.1.1.2.3.cmml">1</mn></msub><mo id="S4.F4.sf1.3.m1.1.1.1" mathsize="90%" xref="S4.F4.sf1.3.m1.1.1.1.cmml">=</mo><mn id="S4.F4.sf1.3.m1.1.1.3" mathsize="90%" xref="S4.F4.sf1.3.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.sf1.3.m1.1c"><apply id="S4.F4.sf1.3.m1.1.1.cmml" xref="S4.F4.sf1.3.m1.1.1"><eq id="S4.F4.sf1.3.m1.1.1.1.cmml" xref="S4.F4.sf1.3.m1.1.1.1"></eq><apply id="S4.F4.sf1.3.m1.1.1.2.cmml" xref="S4.F4.sf1.3.m1.1.1.2"><csymbol cd="ambiguous" id="S4.F4.sf1.3.m1.1.1.2.1.cmml" xref="S4.F4.sf1.3.m1.1.1.2">subscript</csymbol><ci id="S4.F4.sf1.3.m1.1.1.2.2.cmml" xref="S4.F4.sf1.3.m1.1.1.2.2">𝑝</ci><cn id="S4.F4.sf1.3.m1.1.1.2.3.cmml" type="integer" xref="S4.F4.sf1.3.m1.1.1.2.3">1</cn></apply><cn id="S4.F4.sf1.3.m1.1.1.3.cmml" type="integer" xref="S4.F4.sf1.3.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf1.3.m1.1d">p_{1}=1</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf1.3.m1.1e">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math><span class="ltx_text" id="S4.F4.sf1.4.1" style="font-size:90%;"> and <math alttext="\boldsymbol{\Sigma}=\boldsymbol{\Sigma}_{1}" class="ltx_Math" display="inline" id="S4.F4.sf1.4.1.m1.1"><semantics id="S4.F4.sf1.4.1.m1.1b"><mrow id="S4.F4.sf1.4.1.m1.1.1" xref="S4.F4.sf1.4.1.m1.1.1.cmml"><mi id="S4.F4.sf1.4.1.m1.1.1.2" xref="S4.F4.sf1.4.1.m1.1.1.2.cmml">𝚺</mi><mo id="S4.F4.sf1.4.1.m1.1.1.1" xref="S4.F4.sf1.4.1.m1.1.1.1.cmml">=</mo><msub id="S4.F4.sf1.4.1.m1.1.1.3" xref="S4.F4.sf1.4.1.m1.1.1.3.cmml"><mi id="S4.F4.sf1.4.1.m1.1.1.3.2" xref="S4.F4.sf1.4.1.m1.1.1.3.2.cmml">𝚺</mi><mn id="S4.F4.sf1.4.1.m1.1.1.3.3" xref="S4.F4.sf1.4.1.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.sf1.4.1.m1.1c"><apply id="S4.F4.sf1.4.1.m1.1.1.cmml" 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id="S4.F4.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="402" id="S4.F4.sf2.g1" src="extracted/6290084/Ind_T2_500.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F4.sf2.6.2.1" style="font-size:90%;">(b)</span> </span><math alttext="p_{1}=2" class="ltx_Math" display="inline" id="S4.F4.sf2.3.m1.1"><semantics id="S4.F4.sf2.3.m1.1b"><mrow id="S4.F4.sf2.3.m1.1.1" xref="S4.F4.sf2.3.m1.1.1.cmml"><msub id="S4.F4.sf2.3.m1.1.1.2" xref="S4.F4.sf2.3.m1.1.1.2.cmml"><mi id="S4.F4.sf2.3.m1.1.1.2.2" mathsize="90%" xref="S4.F4.sf2.3.m1.1.1.2.2.cmml">p</mi><mn id="S4.F4.sf2.3.m1.1.1.2.3" mathsize="90%" xref="S4.F4.sf2.3.m1.1.1.2.3.cmml">1</mn></msub><mo id="S4.F4.sf2.3.m1.1.1.1" mathsize="90%" xref="S4.F4.sf2.3.m1.1.1.1.cmml">=</mo><mn id="S4.F4.sf2.3.m1.1.1.3" mathsize="90%" xref="S4.F4.sf2.3.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.sf2.3.m1.1c"><apply id="S4.F4.sf2.3.m1.1.1.cmml" xref="S4.F4.sf2.3.m1.1.1"><eq id="S4.F4.sf2.3.m1.1.1.1.cmml" xref="S4.F4.sf2.3.m1.1.1.1"></eq><apply id="S4.F4.sf2.3.m1.1.1.2.cmml" xref="S4.F4.sf2.3.m1.1.1.2"><csymbol cd="ambiguous" id="S4.F4.sf2.3.m1.1.1.2.1.cmml" xref="S4.F4.sf2.3.m1.1.1.2">subscript</csymbol><ci id="S4.F4.sf2.3.m1.1.1.2.2.cmml" xref="S4.F4.sf2.3.m1.1.1.2.2">𝑝</ci><cn id="S4.F4.sf2.3.m1.1.1.2.3.cmml" type="integer" xref="S4.F4.sf2.3.m1.1.1.2.3">1</cn></apply><cn id="S4.F4.sf2.3.m1.1.1.3.cmml" type="integer" xref="S4.F4.sf2.3.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf2.3.m1.1d">p_{1}=2</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf2.3.m1.1e">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2</annotation></semantics></math><span class="ltx_text" id="S4.F4.sf2.4.1" style="font-size:90%;"> and <math alttext="\boldsymbol{\Sigma}=\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.F4.sf2.4.1.m1.1"><semantics id="S4.F4.sf2.4.1.m1.1b"><mrow id="S4.F4.sf2.4.1.m1.1.1" xref="S4.F4.sf2.4.1.m1.1.1.cmml"><mi id="S4.F4.sf2.4.1.m1.1.1.2" xref="S4.F4.sf2.4.1.m1.1.1.2.cmml">𝚺</mi><mo id="S4.F4.sf2.4.1.m1.1.1.1" xref="S4.F4.sf2.4.1.m1.1.1.1.cmml">=</mo><msub id="S4.F4.sf2.4.1.m1.1.1.3" xref="S4.F4.sf2.4.1.m1.1.1.3.cmml"><mi id="S4.F4.sf2.4.1.m1.1.1.3.2" xref="S4.F4.sf2.4.1.m1.1.1.3.2.cmml">𝚺</mi><mn id="S4.F4.sf2.4.1.m1.1.1.3.3" xref="S4.F4.sf2.4.1.m1.1.1.3.3.cmml">4</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.sf2.4.1.m1.1c"><apply id="S4.F4.sf2.4.1.m1.1.1.cmml" xref="S4.F4.sf2.4.1.m1.1.1"><eq id="S4.F4.sf2.4.1.m1.1.1.1.cmml" xref="S4.F4.sf2.4.1.m1.1.1.1"></eq><ci id="S4.F4.sf2.4.1.m1.1.1.2.cmml" xref="S4.F4.sf2.4.1.m1.1.1.2">𝚺</ci><apply id="S4.F4.sf2.4.1.m1.1.1.3.cmml" xref="S4.F4.sf2.4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.F4.sf2.4.1.m1.1.1.3.1.cmml" xref="S4.F4.sf2.4.1.m1.1.1.3">subscript</csymbol><ci id="S4.F4.sf2.4.1.m1.1.1.3.2.cmml" xref="S4.F4.sf2.4.1.m1.1.1.3.2">𝚺</ci><cn id="S4.F4.sf2.4.1.m1.1.1.3.3.cmml" type="integer" xref="S4.F4.sf2.4.1.m1.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf2.4.1.m1.1d">\boldsymbol{\Sigma}=\boldsymbol{\Sigma}_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf2.4.1.m1.1e">bold_Σ = bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math></span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F4.14.7.1" style="font-size:90%;">Figure 4</span>: </span><span class="ltx_text" id="S4.F4.12.6" style="font-size:90%;">Graph for the Independence Test statistic’s distribution along with the empirical distribution of the observed values of <math alttext="T_{3}^{\star}" class="ltx_Math" display="inline" id="S4.F4.7.1.m1.1"><semantics id="S4.F4.7.1.m1.1b"><msubsup id="S4.F4.7.1.m1.1.1" xref="S4.F4.7.1.m1.1.1.cmml"><mi id="S4.F4.7.1.m1.1.1.2.2" xref="S4.F4.7.1.m1.1.1.2.2.cmml">T</mi><mn id="S4.F4.7.1.m1.1.1.2.3" xref="S4.F4.7.1.m1.1.1.2.3.cmml">3</mn><mo id="S4.F4.7.1.m1.1.1.3" xref="S4.F4.7.1.m1.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.F4.7.1.m1.1c"><apply id="S4.F4.7.1.m1.1.1.cmml" xref="S4.F4.7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.F4.7.1.m1.1.1.1.cmml" xref="S4.F4.7.1.m1.1.1">superscript</csymbol><apply id="S4.F4.7.1.m1.1.1.2.cmml" xref="S4.F4.7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.F4.7.1.m1.1.1.2.1.cmml" xref="S4.F4.7.1.m1.1.1">subscript</csymbol><ci id="S4.F4.7.1.m1.1.1.2.2.cmml" xref="S4.F4.7.1.m1.1.1.2.2">𝑇</ci><cn id="S4.F4.7.1.m1.1.1.2.3.cmml" type="integer" xref="S4.F4.7.1.m1.1.1.2.3">3</cn></apply><ci id="S4.F4.7.1.m1.1.1.3.cmml" xref="S4.F4.7.1.m1.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.7.1.m1.1d">T_{3}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.7.1.m1.1e">italic_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> for given <math alttext="n=500" class="ltx_Math" display="inline" id="S4.F4.8.2.m2.1"><semantics id="S4.F4.8.2.m2.1b"><mrow id="S4.F4.8.2.m2.1.1" xref="S4.F4.8.2.m2.1.1.cmml"><mi id="S4.F4.8.2.m2.1.1.2" xref="S4.F4.8.2.m2.1.1.2.cmml">n</mi><mo id="S4.F4.8.2.m2.1.1.1" xref="S4.F4.8.2.m2.1.1.1.cmml">=</mo><mn id="S4.F4.8.2.m2.1.1.3" xref="S4.F4.8.2.m2.1.1.3.cmml">500</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.8.2.m2.1c"><apply id="S4.F4.8.2.m2.1.1.cmml" xref="S4.F4.8.2.m2.1.1"><eq id="S4.F4.8.2.m2.1.1.1.cmml" xref="S4.F4.8.2.m2.1.1.1"></eq><ci id="S4.F4.8.2.m2.1.1.2.cmml" xref="S4.F4.8.2.m2.1.1.2">𝑛</ci><cn id="S4.F4.8.2.m2.1.1.3.cmml" type="integer" xref="S4.F4.8.2.m2.1.1.3">500</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.8.2.m2.1d">n=500</annotation><annotation encoding="application/x-llamapun" id="S4.F4.8.2.m2.1e">italic_n = 500</annotation></semantics></math>, <math alttext="p=4" class="ltx_Math" display="inline" id="S4.F4.9.3.m3.1"><semantics id="S4.F4.9.3.m3.1b"><mrow id="S4.F4.9.3.m3.1.1" xref="S4.F4.9.3.m3.1.1.cmml"><mi id="S4.F4.9.3.m3.1.1.2" xref="S4.F4.9.3.m3.1.1.2.cmml">p</mi><mo id="S4.F4.9.3.m3.1.1.1" xref="S4.F4.9.3.m3.1.1.1.cmml">=</mo><mn id="S4.F4.9.3.m3.1.1.3" xref="S4.F4.9.3.m3.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.9.3.m3.1c"><apply id="S4.F4.9.3.m3.1.1.cmml" xref="S4.F4.9.3.m3.1.1"><eq id="S4.F4.9.3.m3.1.1.1.cmml" xref="S4.F4.9.3.m3.1.1.1"></eq><ci id="S4.F4.9.3.m3.1.1.2.cmml" xref="S4.F4.9.3.m3.1.1.2">𝑝</ci><cn id="S4.F4.9.3.m3.1.1.3.cmml" type="integer" xref="S4.F4.9.3.m3.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.9.3.m3.1d">p=4</annotation><annotation encoding="application/x-llamapun" id="S4.F4.9.3.m3.1e">italic_p = 4</annotation></semantics></math>, <math alttext="\boldsymbol{\mu}" class="ltx_Math" display="inline" id="S4.F4.10.4.m4.1"><semantics id="S4.F4.10.4.m4.1b"><mi id="S4.F4.10.4.m4.1.1" xref="S4.F4.10.4.m4.1.1.cmml">𝝁</mi><annotation-xml encoding="MathML-Content" id="S4.F4.10.4.m4.1c"><ci id="S4.F4.10.4.m4.1.1.cmml" xref="S4.F4.10.4.m4.1.1">𝝁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.10.4.m4.1d">\boldsymbol{\mu}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.10.4.m4.1e">bold_italic_μ</annotation></semantics></math>, <math alttext="\boldsymbol{\Sigma}_{1}" class="ltx_Math" display="inline" id="S4.F4.11.5.m5.1"><semantics id="S4.F4.11.5.m5.1b"><msub id="S4.F4.11.5.m5.1.1" xref="S4.F4.11.5.m5.1.1.cmml"><mi id="S4.F4.11.5.m5.1.1.2" xref="S4.F4.11.5.m5.1.1.2.cmml">𝚺</mi><mn id="S4.F4.11.5.m5.1.1.3" xref="S4.F4.11.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F4.11.5.m5.1c"><apply id="S4.F4.11.5.m5.1.1.cmml" xref="S4.F4.11.5.m5.1.1"><csymbol cd="ambiguous" id="S4.F4.11.5.m5.1.1.1.cmml" xref="S4.F4.11.5.m5.1.1">subscript</csymbol><ci id="S4.F4.11.5.m5.1.1.2.cmml" xref="S4.F4.11.5.m5.1.1.2">𝚺</ci><cn id="S4.F4.11.5.m5.1.1.3.cmml" type="integer" xref="S4.F4.11.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.11.5.m5.1d">\boldsymbol{\Sigma}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.11.5.m5.1e">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> (T3star_obs1) and <math alttext="\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.F4.12.6.m6.1"><semantics id="S4.F4.12.6.m6.1b"><msub id="S4.F4.12.6.m6.1.1" xref="S4.F4.12.6.m6.1.1.cmml"><mi id="S4.F4.12.6.m6.1.1.2" xref="S4.F4.12.6.m6.1.1.2.cmml">𝚺</mi><mn id="S4.F4.12.6.m6.1.1.3" xref="S4.F4.12.6.m6.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F4.12.6.m6.1c"><apply id="S4.F4.12.6.m6.1.1.cmml" xref="S4.F4.12.6.m6.1.1"><csymbol cd="ambiguous" id="S4.F4.12.6.m6.1.1.1.cmml" xref="S4.F4.12.6.m6.1.1">subscript</csymbol><ci id="S4.F4.12.6.m6.1.1.2.cmml" xref="S4.F4.12.6.m6.1.1.2">𝚺</ci><cn id="S4.F4.12.6.m6.1.1.3.cmml" type="integer" xref="S4.F4.12.6.m6.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.12.6.m6.1d">\boldsymbol{\Sigma}_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.12.6.m6.1e">bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math> (T3star_obs2).</span></figcaption> </figure> <figure class="ltx_figure" id="S4.F5"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_figure_panel" id="S4.F5.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="402" id="S4.F5.sf1.g1" src="extracted/6290084/Reg_T1_10.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf1.6.2.1" style="font-size:90%;">(a)</span> </span><math alttext="p_{1}=2" class="ltx_Math" display="inline" id="S4.F5.sf1.3.m1.1"><semantics id="S4.F5.sf1.3.m1.1b"><mrow id="S4.F5.sf1.3.m1.1.1" xref="S4.F5.sf1.3.m1.1.1.cmml"><msub id="S4.F5.sf1.3.m1.1.1.2" xref="S4.F5.sf1.3.m1.1.1.2.cmml"><mi id="S4.F5.sf1.3.m1.1.1.2.2" mathsize="90%" xref="S4.F5.sf1.3.m1.1.1.2.2.cmml">p</mi><mn id="S4.F5.sf1.3.m1.1.1.2.3" mathsize="90%" xref="S4.F5.sf1.3.m1.1.1.2.3.cmml">1</mn></msub><mo id="S4.F5.sf1.3.m1.1.1.1" mathsize="90%" xref="S4.F5.sf1.3.m1.1.1.1.cmml">=</mo><mn id="S4.F5.sf1.3.m1.1.1.3" mathsize="90%" xref="S4.F5.sf1.3.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.sf1.3.m1.1c"><apply id="S4.F5.sf1.3.m1.1.1.cmml" xref="S4.F5.sf1.3.m1.1.1"><eq id="S4.F5.sf1.3.m1.1.1.1.cmml" xref="S4.F5.sf1.3.m1.1.1.1"></eq><apply id="S4.F5.sf1.3.m1.1.1.2.cmml" xref="S4.F5.sf1.3.m1.1.1.2"><csymbol cd="ambiguous" id="S4.F5.sf1.3.m1.1.1.2.1.cmml" xref="S4.F5.sf1.3.m1.1.1.2">subscript</csymbol><ci id="S4.F5.sf1.3.m1.1.1.2.2.cmml" xref="S4.F5.sf1.3.m1.1.1.2.2">𝑝</ci><cn id="S4.F5.sf1.3.m1.1.1.2.3.cmml" type="integer" xref="S4.F5.sf1.3.m1.1.1.2.3">1</cn></apply><cn id="S4.F5.sf1.3.m1.1.1.3.cmml" type="integer" xref="S4.F5.sf1.3.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.sf1.3.m1.1d">p_{1}=2</annotation><annotation encoding="application/x-llamapun" id="S4.F5.sf1.3.m1.1e">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2</annotation></semantics></math><span class="ltx_text" id="S4.F5.sf1.4.1" style="font-size:90%;"> and <math alttext="\boldsymbol{\Sigma}=\boldsymbol{\Sigma}_{3}" class="ltx_Math" display="inline" id="S4.F5.sf1.4.1.m1.1"><semantics id="S4.F5.sf1.4.1.m1.1b"><mrow id="S4.F5.sf1.4.1.m1.1.1" xref="S4.F5.sf1.4.1.m1.1.1.cmml"><mi id="S4.F5.sf1.4.1.m1.1.1.2" xref="S4.F5.sf1.4.1.m1.1.1.2.cmml">𝚺</mi><mo id="S4.F5.sf1.4.1.m1.1.1.1" xref="S4.F5.sf1.4.1.m1.1.1.1.cmml">=</mo><msub id="S4.F5.sf1.4.1.m1.1.1.3" xref="S4.F5.sf1.4.1.m1.1.1.3.cmml"><mi id="S4.F5.sf1.4.1.m1.1.1.3.2" xref="S4.F5.sf1.4.1.m1.1.1.3.2.cmml">𝚺</mi><mn id="S4.F5.sf1.4.1.m1.1.1.3.3" xref="S4.F5.sf1.4.1.m1.1.1.3.3.cmml">3</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.sf1.4.1.m1.1c"><apply id="S4.F5.sf1.4.1.m1.1.1.cmml" xref="S4.F5.sf1.4.1.m1.1.1"><eq id="S4.F5.sf1.4.1.m1.1.1.1.cmml" xref="S4.F5.sf1.4.1.m1.1.1.1"></eq><ci id="S4.F5.sf1.4.1.m1.1.1.2.cmml" xref="S4.F5.sf1.4.1.m1.1.1.2">𝚺</ci><apply id="S4.F5.sf1.4.1.m1.1.1.3.cmml" xref="S4.F5.sf1.4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.F5.sf1.4.1.m1.1.1.3.1.cmml" xref="S4.F5.sf1.4.1.m1.1.1.3">subscript</csymbol><ci id="S4.F5.sf1.4.1.m1.1.1.3.2.cmml" xref="S4.F5.sf1.4.1.m1.1.1.3.2">𝚺</ci><cn id="S4.F5.sf1.4.1.m1.1.1.3.3.cmml" type="integer" xref="S4.F5.sf1.4.1.m1.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.sf1.4.1.m1.1d">\boldsymbol{\Sigma}=\boldsymbol{\Sigma}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.F5.sf1.4.1.m1.1e">bold_Σ = bold_Σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math></span></figcaption> </figure> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_figure_panel" id="S4.F5.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="402" id="S4.F5.sf2.g1" src="extracted/6290084/Reg_T2_10.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf2.6.2.1" style="font-size:90%;">(b)</span> </span><math alttext="p_{1}=1" class="ltx_Math" display="inline" id="S4.F5.sf2.3.m1.1"><semantics id="S4.F5.sf2.3.m1.1b"><mrow id="S4.F5.sf2.3.m1.1.1" xref="S4.F5.sf2.3.m1.1.1.cmml"><msub id="S4.F5.sf2.3.m1.1.1.2" xref="S4.F5.sf2.3.m1.1.1.2.cmml"><mi id="S4.F5.sf2.3.m1.1.1.2.2" mathsize="90%" xref="S4.F5.sf2.3.m1.1.1.2.2.cmml">p</mi><mn id="S4.F5.sf2.3.m1.1.1.2.3" mathsize="90%" xref="S4.F5.sf2.3.m1.1.1.2.3.cmml">1</mn></msub><mo id="S4.F5.sf2.3.m1.1.1.1" mathsize="90%" xref="S4.F5.sf2.3.m1.1.1.1.cmml">=</mo><mn id="S4.F5.sf2.3.m1.1.1.3" mathsize="90%" xref="S4.F5.sf2.3.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.sf2.3.m1.1c"><apply id="S4.F5.sf2.3.m1.1.1.cmml" xref="S4.F5.sf2.3.m1.1.1"><eq id="S4.F5.sf2.3.m1.1.1.1.cmml" xref="S4.F5.sf2.3.m1.1.1.1"></eq><apply id="S4.F5.sf2.3.m1.1.1.2.cmml" xref="S4.F5.sf2.3.m1.1.1.2"><csymbol cd="ambiguous" id="S4.F5.sf2.3.m1.1.1.2.1.cmml" xref="S4.F5.sf2.3.m1.1.1.2">subscript</csymbol><ci id="S4.F5.sf2.3.m1.1.1.2.2.cmml" xref="S4.F5.sf2.3.m1.1.1.2.2">𝑝</ci><cn id="S4.F5.sf2.3.m1.1.1.2.3.cmml" type="integer" xref="S4.F5.sf2.3.m1.1.1.2.3">1</cn></apply><cn id="S4.F5.sf2.3.m1.1.1.3.cmml" type="integer" xref="S4.F5.sf2.3.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.sf2.3.m1.1d">p_{1}=1</annotation><annotation encoding="application/x-llamapun" id="S4.F5.sf2.3.m1.1e">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math><span class="ltx_text" id="S4.F5.sf2.4.1" style="font-size:90%;"> and <math alttext="\boldsymbol{\Sigma}=\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.F5.sf2.4.1.m1.1"><semantics id="S4.F5.sf2.4.1.m1.1b"><mrow id="S4.F5.sf2.4.1.m1.1.1" xref="S4.F5.sf2.4.1.m1.1.1.cmml"><mi id="S4.F5.sf2.4.1.m1.1.1.2" xref="S4.F5.sf2.4.1.m1.1.1.2.cmml">𝚺</mi><mo id="S4.F5.sf2.4.1.m1.1.1.1" xref="S4.F5.sf2.4.1.m1.1.1.1.cmml">=</mo><msub id="S4.F5.sf2.4.1.m1.1.1.3" xref="S4.F5.sf2.4.1.m1.1.1.3.cmml"><mi id="S4.F5.sf2.4.1.m1.1.1.3.2" xref="S4.F5.sf2.4.1.m1.1.1.3.2.cmml">𝚺</mi><mn id="S4.F5.sf2.4.1.m1.1.1.3.3" xref="S4.F5.sf2.4.1.m1.1.1.3.3.cmml">4</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.sf2.4.1.m1.1c"><apply id="S4.F5.sf2.4.1.m1.1.1.cmml" xref="S4.F5.sf2.4.1.m1.1.1"><eq id="S4.F5.sf2.4.1.m1.1.1.1.cmml" xref="S4.F5.sf2.4.1.m1.1.1.1"></eq><ci id="S4.F5.sf2.4.1.m1.1.1.2.cmml" xref="S4.F5.sf2.4.1.m1.1.1.2">𝚺</ci><apply id="S4.F5.sf2.4.1.m1.1.1.3.cmml" xref="S4.F5.sf2.4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.F5.sf2.4.1.m1.1.1.3.1.cmml" xref="S4.F5.sf2.4.1.m1.1.1.3">subscript</csymbol><ci id="S4.F5.sf2.4.1.m1.1.1.3.2.cmml" xref="S4.F5.sf2.4.1.m1.1.1.3.2">𝚺</ci><cn id="S4.F5.sf2.4.1.m1.1.1.3.3.cmml" type="integer" xref="S4.F5.sf2.4.1.m1.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.sf2.4.1.m1.1d">\boldsymbol{\Sigma}=\boldsymbol{\Sigma}_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.F5.sf2.4.1.m1.1e">bold_Σ = bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math></span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.14.7.1" style="font-size:90%;">Figure 5</span>: </span><span class="ltx_text" id="S4.F5.12.6" style="font-size:90%;">Graph for the Regression Test statistic’s distribution along with the empirical distribution of the observed values of <math alttext="T_{4}^{\star}" class="ltx_Math" display="inline" id="S4.F5.7.1.m1.1"><semantics id="S4.F5.7.1.m1.1b"><msubsup id="S4.F5.7.1.m1.1.1" xref="S4.F5.7.1.m1.1.1.cmml"><mi id="S4.F5.7.1.m1.1.1.2.2" xref="S4.F5.7.1.m1.1.1.2.2.cmml">T</mi><mn id="S4.F5.7.1.m1.1.1.2.3" xref="S4.F5.7.1.m1.1.1.2.3.cmml">4</mn><mo id="S4.F5.7.1.m1.1.1.3" xref="S4.F5.7.1.m1.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.F5.7.1.m1.1c"><apply id="S4.F5.7.1.m1.1.1.cmml" xref="S4.F5.7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.F5.7.1.m1.1.1.1.cmml" xref="S4.F5.7.1.m1.1.1">superscript</csymbol><apply id="S4.F5.7.1.m1.1.1.2.cmml" xref="S4.F5.7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.F5.7.1.m1.1.1.2.1.cmml" xref="S4.F5.7.1.m1.1.1">subscript</csymbol><ci id="S4.F5.7.1.m1.1.1.2.2.cmml" xref="S4.F5.7.1.m1.1.1.2.2">𝑇</ci><cn id="S4.F5.7.1.m1.1.1.2.3.cmml" type="integer" xref="S4.F5.7.1.m1.1.1.2.3">4</cn></apply><ci id="S4.F5.7.1.m1.1.1.3.cmml" xref="S4.F5.7.1.m1.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.7.1.m1.1d">T_{4}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S4.F5.7.1.m1.1e">italic_T start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> for given <math alttext="n=10" class="ltx_Math" display="inline" id="S4.F5.8.2.m2.1"><semantics id="S4.F5.8.2.m2.1b"><mrow id="S4.F5.8.2.m2.1.1" xref="S4.F5.8.2.m2.1.1.cmml"><mi id="S4.F5.8.2.m2.1.1.2" xref="S4.F5.8.2.m2.1.1.2.cmml">n</mi><mo id="S4.F5.8.2.m2.1.1.1" xref="S4.F5.8.2.m2.1.1.1.cmml">=</mo><mn id="S4.F5.8.2.m2.1.1.3" xref="S4.F5.8.2.m2.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.8.2.m2.1c"><apply id="S4.F5.8.2.m2.1.1.cmml" xref="S4.F5.8.2.m2.1.1"><eq id="S4.F5.8.2.m2.1.1.1.cmml" xref="S4.F5.8.2.m2.1.1.1"></eq><ci id="S4.F5.8.2.m2.1.1.2.cmml" xref="S4.F5.8.2.m2.1.1.2">𝑛</ci><cn id="S4.F5.8.2.m2.1.1.3.cmml" type="integer" xref="S4.F5.8.2.m2.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.8.2.m2.1d">n=10</annotation><annotation encoding="application/x-llamapun" id="S4.F5.8.2.m2.1e">italic_n = 10</annotation></semantics></math>, <math alttext="p=4" class="ltx_Math" display="inline" id="S4.F5.9.3.m3.1"><semantics id="S4.F5.9.3.m3.1b"><mrow id="S4.F5.9.3.m3.1.1" xref="S4.F5.9.3.m3.1.1.cmml"><mi id="S4.F5.9.3.m3.1.1.2" xref="S4.F5.9.3.m3.1.1.2.cmml">p</mi><mo id="S4.F5.9.3.m3.1.1.1" xref="S4.F5.9.3.m3.1.1.1.cmml">=</mo><mn id="S4.F5.9.3.m3.1.1.3" xref="S4.F5.9.3.m3.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.9.3.m3.1c"><apply id="S4.F5.9.3.m3.1.1.cmml" xref="S4.F5.9.3.m3.1.1"><eq id="S4.F5.9.3.m3.1.1.1.cmml" xref="S4.F5.9.3.m3.1.1.1"></eq><ci id="S4.F5.9.3.m3.1.1.2.cmml" xref="S4.F5.9.3.m3.1.1.2">𝑝</ci><cn id="S4.F5.9.3.m3.1.1.3.cmml" type="integer" xref="S4.F5.9.3.m3.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.9.3.m3.1d">p=4</annotation><annotation encoding="application/x-llamapun" id="S4.F5.9.3.m3.1e">italic_p = 4</annotation></semantics></math>, <math alttext="\boldsymbol{\mu}" class="ltx_Math" display="inline" id="S4.F5.10.4.m4.1"><semantics id="S4.F5.10.4.m4.1b"><mi id="S4.F5.10.4.m4.1.1" xref="S4.F5.10.4.m4.1.1.cmml">𝝁</mi><annotation-xml encoding="MathML-Content" id="S4.F5.10.4.m4.1c"><ci id="S4.F5.10.4.m4.1.1.cmml" xref="S4.F5.10.4.m4.1.1">𝝁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.10.4.m4.1d">\boldsymbol{\mu}</annotation><annotation encoding="application/x-llamapun" id="S4.F5.10.4.m4.1e">bold_italic_μ</annotation></semantics></math>, <math alttext="\boldsymbol{\Sigma}_{3}" class="ltx_Math" display="inline" id="S4.F5.11.5.m5.1"><semantics id="S4.F5.11.5.m5.1b"><msub id="S4.F5.11.5.m5.1.1" xref="S4.F5.11.5.m5.1.1.cmml"><mi id="S4.F5.11.5.m5.1.1.2" xref="S4.F5.11.5.m5.1.1.2.cmml">𝚺</mi><mn id="S4.F5.11.5.m5.1.1.3" xref="S4.F5.11.5.m5.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F5.11.5.m5.1c"><apply id="S4.F5.11.5.m5.1.1.cmml" xref="S4.F5.11.5.m5.1.1"><csymbol cd="ambiguous" id="S4.F5.11.5.m5.1.1.1.cmml" xref="S4.F5.11.5.m5.1.1">subscript</csymbol><ci id="S4.F5.11.5.m5.1.1.2.cmml" xref="S4.F5.11.5.m5.1.1.2">𝚺</ci><cn id="S4.F5.11.5.m5.1.1.3.cmml" type="integer" xref="S4.F5.11.5.m5.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.11.5.m5.1d">\boldsymbol{\Sigma}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.F5.11.5.m5.1e">bold_Σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> (T4star_obs1) and <math alttext="\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.F5.12.6.m6.1"><semantics id="S4.F5.12.6.m6.1b"><msub id="S4.F5.12.6.m6.1.1" xref="S4.F5.12.6.m6.1.1.cmml"><mi id="S4.F5.12.6.m6.1.1.2" xref="S4.F5.12.6.m6.1.1.2.cmml">𝚺</mi><mn id="S4.F5.12.6.m6.1.1.3" xref="S4.F5.12.6.m6.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F5.12.6.m6.1c"><apply id="S4.F5.12.6.m6.1.1.cmml" xref="S4.F5.12.6.m6.1.1"><csymbol cd="ambiguous" id="S4.F5.12.6.m6.1.1.1.cmml" xref="S4.F5.12.6.m6.1.1">subscript</csymbol><ci id="S4.F5.12.6.m6.1.1.2.cmml" xref="S4.F5.12.6.m6.1.1.2">𝚺</ci><cn id="S4.F5.12.6.m6.1.1.3.cmml" type="integer" xref="S4.F5.12.6.m6.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.12.6.m6.1d">\boldsymbol{\Sigma}_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.F5.12.6.m6.1e">bold_Σ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math> (T4star_obs2).</span></figcaption> </figure> <figure class="ltx_figure" id="S4.F6"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_figure_panel" id="S4.F6.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="402" id="S4.F6.sf1.g1" src="extracted/6290084/Reg_T1_500.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F6.sf1.6.2.1" style="font-size:90%;">(a)</span> </span><math alttext="p_{1}=2" class="ltx_Math" display="inline" id="S4.F6.sf1.3.m1.1"><semantics id="S4.F6.sf1.3.m1.1b"><mrow id="S4.F6.sf1.3.m1.1.1" xref="S4.F6.sf1.3.m1.1.1.cmml"><msub id="S4.F6.sf1.3.m1.1.1.2" xref="S4.F6.sf1.3.m1.1.1.2.cmml"><mi id="S4.F6.sf1.3.m1.1.1.2.2" mathsize="90%" xref="S4.F6.sf1.3.m1.1.1.2.2.cmml">p</mi><mn id="S4.F6.sf1.3.m1.1.1.2.3" mathsize="90%" xref="S4.F6.sf1.3.m1.1.1.2.3.cmml">1</mn></msub><mo id="S4.F6.sf1.3.m1.1.1.1" mathsize="90%" xref="S4.F6.sf1.3.m1.1.1.1.cmml">=</mo><mn id="S4.F6.sf1.3.m1.1.1.3" mathsize="90%" xref="S4.F6.sf1.3.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.sf1.3.m1.1c"><apply id="S4.F6.sf1.3.m1.1.1.cmml" xref="S4.F6.sf1.3.m1.1.1"><eq id="S4.F6.sf1.3.m1.1.1.1.cmml" xref="S4.F6.sf1.3.m1.1.1.1"></eq><apply id="S4.F6.sf1.3.m1.1.1.2.cmml" xref="S4.F6.sf1.3.m1.1.1.2"><csymbol cd="ambiguous" id="S4.F6.sf1.3.m1.1.1.2.1.cmml" xref="S4.F6.sf1.3.m1.1.1.2">subscript</csymbol><ci id="S4.F6.sf1.3.m1.1.1.2.2.cmml" xref="S4.F6.sf1.3.m1.1.1.2.2">𝑝</ci><cn id="S4.F6.sf1.3.m1.1.1.2.3.cmml" type="integer" xref="S4.F6.sf1.3.m1.1.1.2.3">1</cn></apply><cn id="S4.F6.sf1.3.m1.1.1.3.cmml" type="integer" xref="S4.F6.sf1.3.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.sf1.3.m1.1d">p_{1}=2</annotation><annotation encoding="application/x-llamapun" id="S4.F6.sf1.3.m1.1e">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2</annotation></semantics></math><span class="ltx_text" id="S4.F6.sf1.4.1" style="font-size:90%;"> and <math alttext="\boldsymbol{\Sigma}=\boldsymbol{\Sigma}_{3}" class="ltx_Math" display="inline" id="S4.F6.sf1.4.1.m1.1"><semantics id="S4.F6.sf1.4.1.m1.1b"><mrow id="S4.F6.sf1.4.1.m1.1.1" xref="S4.F6.sf1.4.1.m1.1.1.cmml"><mi id="S4.F6.sf1.4.1.m1.1.1.2" xref="S4.F6.sf1.4.1.m1.1.1.2.cmml">𝚺</mi><mo id="S4.F6.sf1.4.1.m1.1.1.1" xref="S4.F6.sf1.4.1.m1.1.1.1.cmml">=</mo><msub id="S4.F6.sf1.4.1.m1.1.1.3" xref="S4.F6.sf1.4.1.m1.1.1.3.cmml"><mi id="S4.F6.sf1.4.1.m1.1.1.3.2" xref="S4.F6.sf1.4.1.m1.1.1.3.2.cmml">𝚺</mi><mn id="S4.F6.sf1.4.1.m1.1.1.3.3" xref="S4.F6.sf1.4.1.m1.1.1.3.3.cmml">3</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.sf1.4.1.m1.1c"><apply id="S4.F6.sf1.4.1.m1.1.1.cmml" xref="S4.F6.sf1.4.1.m1.1.1"><eq id="S4.F6.sf1.4.1.m1.1.1.1.cmml" xref="S4.F6.sf1.4.1.m1.1.1.1"></eq><ci id="S4.F6.sf1.4.1.m1.1.1.2.cmml" xref="S4.F6.sf1.4.1.m1.1.1.2">𝚺</ci><apply id="S4.F6.sf1.4.1.m1.1.1.3.cmml" xref="S4.F6.sf1.4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.F6.sf1.4.1.m1.1.1.3.1.cmml" xref="S4.F6.sf1.4.1.m1.1.1.3">subscript</csymbol><ci id="S4.F6.sf1.4.1.m1.1.1.3.2.cmml" xref="S4.F6.sf1.4.1.m1.1.1.3.2">𝚺</ci><cn id="S4.F6.sf1.4.1.m1.1.1.3.3.cmml" type="integer" xref="S4.F6.sf1.4.1.m1.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.sf1.4.1.m1.1d">\boldsymbol{\Sigma}=\boldsymbol{\Sigma}_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.F6.sf1.4.1.m1.1e">bold_Σ = bold_Σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math></span></figcaption> </figure> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_figure_panel" id="S4.F6.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="402" id="S4.F6.sf2.g1" src="extracted/6290084/Reg_T2_500.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F6.sf2.6.2.1" style="font-size:90%;">(b)</span> </span><math alttext="p_{1}=1" class="ltx_Math" display="inline" id="S4.F6.sf2.3.m1.1"><semantics id="S4.F6.sf2.3.m1.1b"><mrow id="S4.F6.sf2.3.m1.1.1" xref="S4.F6.sf2.3.m1.1.1.cmml"><msub id="S4.F6.sf2.3.m1.1.1.2" xref="S4.F6.sf2.3.m1.1.1.2.cmml"><mi id="S4.F6.sf2.3.m1.1.1.2.2" mathsize="90%" xref="S4.F6.sf2.3.m1.1.1.2.2.cmml">p</mi><mn id="S4.F6.sf2.3.m1.1.1.2.3" mathsize="90%" xref="S4.F6.sf2.3.m1.1.1.2.3.cmml">1</mn></msub><mo id="S4.F6.sf2.3.m1.1.1.1" mathsize="90%" xref="S4.F6.sf2.3.m1.1.1.1.cmml">=</mo><mn id="S4.F6.sf2.3.m1.1.1.3" mathsize="90%" xref="S4.F6.sf2.3.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.sf2.3.m1.1c"><apply id="S4.F6.sf2.3.m1.1.1.cmml" xref="S4.F6.sf2.3.m1.1.1"><eq id="S4.F6.sf2.3.m1.1.1.1.cmml" xref="S4.F6.sf2.3.m1.1.1.1"></eq><apply id="S4.F6.sf2.3.m1.1.1.2.cmml" xref="S4.F6.sf2.3.m1.1.1.2"><csymbol cd="ambiguous" id="S4.F6.sf2.3.m1.1.1.2.1.cmml" xref="S4.F6.sf2.3.m1.1.1.2">subscript</csymbol><ci id="S4.F6.sf2.3.m1.1.1.2.2.cmml" xref="S4.F6.sf2.3.m1.1.1.2.2">𝑝</ci><cn id="S4.F6.sf2.3.m1.1.1.2.3.cmml" type="integer" xref="S4.F6.sf2.3.m1.1.1.2.3">1</cn></apply><cn id="S4.F6.sf2.3.m1.1.1.3.cmml" type="integer" xref="S4.F6.sf2.3.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.sf2.3.m1.1d">p_{1}=1</annotation><annotation encoding="application/x-llamapun" id="S4.F6.sf2.3.m1.1e">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math><span class="ltx_text" id="S4.F6.sf2.4.1" style="font-size:90%;"> and <math alttext="\boldsymbol{\Sigma}=\boldsymbol{\Sigma}_{4}" class="ltx_Math" display="inline" id="S4.F6.sf2.4.1.m1.1"><semantics id="S4.F6.sf2.4.1.m1.1b"><mrow id="S4.F6.sf2.4.1.m1.1.1" xref="S4.F6.sf2.4.1.m1.1.1.cmml"><mi id="S4.F6.sf2.4.1.m1.1.1.2" xref="S4.F6.sf2.4.1.m1.1.1.2.cmml">𝚺</mi><mo id="S4.F6.sf2.4.1.m1.1.1.1" xref="S4.F6.sf2.4.1.m1.1.1.1.cmml">=</mo><msub id="S4.F6.sf2.4.1.m1.1.1.3" xref="S4.F6.sf2.4.1.m1.1.1.3.cmml"><mi id="S4.F6.sf2.4.1.m1.1.1.3.2" xref="S4.F6.sf2.4.1.m1.1.1.3.2.cmml">𝚺</mi><mn id="S4.F6.sf2.4.1.m1.1.1.3.3" xref="S4.F6.sf2.4.1.m1.1.1.3.3.cmml">4</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.sf2.4.1.m1.1c"><apply id="S4.F6.sf2.4.1.m1.1.1.cmml" xref="S4.F6.sf2.4.1.m1.1.1"><eq id="S4.F6.sf2.4.1.m1.1.1.1.cmml" xref="S4.F6.sf2.4.1.m1.1.1.1"></eq><ci id="S4.F6.sf2.4.1.m1.1.1.2.cmml" xref="S4.F6.sf2.4.1.m1.1.1.2">𝚺</ci><apply id="S4.F6.sf2.4.1.m1.1.1.3.cmml" xref="S4.F6.sf2.4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.F6.sf2.4.1.m1.1.1.3.1.cmml" xref="S4.F6.sf2.4.1.m1.1.1.3">subscript</csymbol><ci 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id="S4.F6.7.1.m1.1.1" xref="S4.F6.7.1.m1.1.1.cmml"><mi id="S4.F6.7.1.m1.1.1.2.2" xref="S4.F6.7.1.m1.1.1.2.2.cmml">T</mi><mn id="S4.F6.7.1.m1.1.1.2.3" xref="S4.F6.7.1.m1.1.1.2.3.cmml">4</mn><mo id="S4.F6.7.1.m1.1.1.3" xref="S4.F6.7.1.m1.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.F6.7.1.m1.1c"><apply id="S4.F6.7.1.m1.1.1.cmml" xref="S4.F6.7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.F6.7.1.m1.1.1.1.cmml" xref="S4.F6.7.1.m1.1.1">superscript</csymbol><apply id="S4.F6.7.1.m1.1.1.2.cmml" xref="S4.F6.7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.F6.7.1.m1.1.1.2.1.cmml" xref="S4.F6.7.1.m1.1.1">subscript</csymbol><ci id="S4.F6.7.1.m1.1.1.2.2.cmml" xref="S4.F6.7.1.m1.1.1.2.2">𝑇</ci><cn id="S4.F6.7.1.m1.1.1.2.3.cmml" type="integer" xref="S4.F6.7.1.m1.1.1.2.3">4</cn></apply><ci id="S4.F6.7.1.m1.1.1.3.cmml" xref="S4.F6.7.1.m1.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.7.1.m1.1d">T_{4}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S4.F6.7.1.m1.1e">italic_T start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> for given <math alttext="n=500" class="ltx_Math" display="inline" id="S4.F6.8.2.m2.1"><semantics id="S4.F6.8.2.m2.1b"><mrow id="S4.F6.8.2.m2.1.1" xref="S4.F6.8.2.m2.1.1.cmml"><mi id="S4.F6.8.2.m2.1.1.2" xref="S4.F6.8.2.m2.1.1.2.cmml">n</mi><mo id="S4.F6.8.2.m2.1.1.1" xref="S4.F6.8.2.m2.1.1.1.cmml">=</mo><mn id="S4.F6.8.2.m2.1.1.3" xref="S4.F6.8.2.m2.1.1.3.cmml">500</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.8.2.m2.1c"><apply id="S4.F6.8.2.m2.1.1.cmml" xref="S4.F6.8.2.m2.1.1"><eq id="S4.F6.8.2.m2.1.1.1.cmml" xref="S4.F6.8.2.m2.1.1.1"></eq><ci id="S4.F6.8.2.m2.1.1.2.cmml" xref="S4.F6.8.2.m2.1.1.2">𝑛</ci><cn id="S4.F6.8.2.m2.1.1.3.cmml" type="integer" xref="S4.F6.8.2.m2.1.1.3">500</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.8.2.m2.1d">n=500</annotation><annotation encoding="application/x-llamapun" id="S4.F6.8.2.m2.1e">italic_n = 500</annotation></semantics></math>, <math alttext="p=4" class="ltx_Math" display="inline" id="S4.F6.9.3.m3.1"><semantics id="S4.F6.9.3.m3.1b"><mrow id="S4.F6.9.3.m3.1.1" xref="S4.F6.9.3.m3.1.1.cmml"><mi id="S4.F6.9.3.m3.1.1.2" xref="S4.F6.9.3.m3.1.1.2.cmml">p</mi><mo id="S4.F6.9.3.m3.1.1.1" xref="S4.F6.9.3.m3.1.1.1.cmml">=</mo><mn id="S4.F6.9.3.m3.1.1.3" xref="S4.F6.9.3.m3.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.9.3.m3.1c"><apply id="S4.F6.9.3.m3.1.1.cmml" xref="S4.F6.9.3.m3.1.1"><eq id="S4.F6.9.3.m3.1.1.1.cmml" xref="S4.F6.9.3.m3.1.1.1"></eq><ci id="S4.F6.9.3.m3.1.1.2.cmml" xref="S4.F6.9.3.m3.1.1.2">𝑝</ci><cn id="S4.F6.9.3.m3.1.1.3.cmml" type="integer" xref="S4.F6.9.3.m3.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.9.3.m3.1d">p=4</annotation><annotation encoding="application/x-llamapun" id="S4.F6.9.3.m3.1e">italic_p = 4</annotation></semantics></math>, <math alttext="\boldsymbol{\mu}" class="ltx_Math" display="inline" id="S4.F6.10.4.m4.1"><semantics id="S4.F6.10.4.m4.1b"><mi id="S4.F6.10.4.m4.1.1" xref="S4.F6.10.4.m4.1.1.cmml">𝝁</mi><annotation-xml encoding="MathML-Content" id="S4.F6.10.4.m4.1c"><ci id="S4.F6.10.4.m4.1.1.cmml" xref="S4.F6.10.4.m4.1.1">𝝁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.10.4.m4.1d">\boldsymbol{\mu}</annotation><annotation encoding="application/x-llamapun" id="S4.F6.10.4.m4.1e">bold_italic_μ</annotation></semantics></math>, <math alttext="\boldsymbol{\Sigma}_{3}" class="ltx_Math" display="inline" id="S4.F6.11.5.m5.1"><semantics id="S4.F6.11.5.m5.1b"><msub id="S4.F6.11.5.m5.1.1" xref="S4.F6.11.5.m5.1.1.cmml"><mi id="S4.F6.11.5.m5.1.1.2" xref="S4.F6.11.5.m5.1.1.2.cmml">𝚺</mi><mn id="S4.F6.11.5.m5.1.1.3" xref="S4.F6.11.5.m5.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F6.11.5.m5.1c"><apply 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datasets via Plug-in Sampling method and perform exact inferential procedures based on multivariate normal data. The package focuses on cases where only a single synthetic dataset is available, which is common in scenarios where multiple imputations are prohibitive. Nevertheless, the authors plan to extend the inferential procedures for the multiple imputation scenario.</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1">We illustrated the core functions of <code class="ltx_verbatim ltx_font_typewriter" id="S5.p2.1.1">PSInference</code>, including the generation of synthetic datasets using <span class="ltx_text ltx_font_typewriter" id="S5.p2.1.2">simSynthData</span> function, and the functions for inferential procedures such as computing confidence intervals for the generalized variance (<span class="ltx_text ltx_font_typewriter" id="S5.p2.1.3">GVdist</span>), tests for the sphericity (<span class="ltx_text ltx_font_typewriter" id="S5.p2.1.4">Sphdist</span>), the independence between two subsets of variables (<span class="ltx_text ltx_font_typewriter" id="S5.p2.1.5">Inddist</span>), and the regression of one set of variables on another (<span class="ltx_text ltx_font_typewriter" id="S5.p2.1.6">Canodist</span>).</p> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.1">Our numerical studies, conducted through Monte Carlo simulations, verified the accuracy of the implemented procedures by estimating the coverage probabilities (<math alttext="cov" class="ltx_Math" display="inline" id="S5.p3.1.m1.1"><semantics id="S5.p3.1.m1.1a"><mrow id="S5.p3.1.m1.1.1" xref="S5.p3.1.m1.1.1.cmml"><mi id="S5.p3.1.m1.1.1.2" xref="S5.p3.1.m1.1.1.2.cmml">c</mi><mo id="S5.p3.1.m1.1.1.1" xref="S5.p3.1.m1.1.1.1.cmml"></mo><mi id="S5.p3.1.m1.1.1.3" xref="S5.p3.1.m1.1.1.3.cmml">o</mi><mo id="S5.p3.1.m1.1.1.1a" xref="S5.p3.1.m1.1.1.1.cmml"></mo><mi id="S5.p3.1.m1.1.1.4" xref="S5.p3.1.m1.1.1.4.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.1.m1.1b"><apply id="S5.p3.1.m1.1.1.cmml" xref="S5.p3.1.m1.1.1"><times id="S5.p3.1.m1.1.1.1.cmml" xref="S5.p3.1.m1.1.1.1"></times><ci id="S5.p3.1.m1.1.1.2.cmml" xref="S5.p3.1.m1.1.1.2">𝑐</ci><ci id="S5.p3.1.m1.1.1.3.cmml" xref="S5.p3.1.m1.1.1.3">𝑜</ci><ci id="S5.p3.1.m1.1.1.4.cmml" xref="S5.p3.1.m1.1.1.4">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.1.m1.1c">cov</annotation><annotation encoding="application/x-llamapun" id="S5.p3.1.m1.1d">italic_c italic_o italic_v</annotation></semantics></math>) for different sample sizes and different covariance matrix structures. These results confirmed that the coverage probabilities closely match the nominal value of 0.95 across all tests, validating the robustness of the exact inferential procedures provided by the package.</p> </div> <div class="ltx_para" id="S5.p4"> <p class="ltx_p" id="S5.p4.1">Furthermore, we presented empirical distributions of the test statistics for each inferential procedure, computed from the synthetic datasets generated during the simulations. These empirical distributions were compared to their theoretical counterparts.</p> </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">R Software</h2> <div class="ltx_para" id="Sx1.p1"> <p class="ltx_p" id="Sx1.p1.1">The R package <a class="ltx_ref ltx_href" href="https://cran.r-project.org/web/packages/PSinference/index.html" title="">PSInference</a> is now available on the CRAN website (<a class="ltx_ref ltx_url ltx_font_typewriter" href="https://cran.r-project.org/web/packages/PSinference/index.html" title="">https://cran.r-project.org/web/packages/PSinference/index.html</a>)</p> </div> </section> <section class="ltx_section" id="Sx2"> <h2 class="ltx_title ltx_title_section">Acknowledgments</h2> <div class="ltx_para" id="Sx2.p1"> <p class="ltx_p" id="Sx2.p1.1">The work of authors is funded by national funds through the FCT – Fundaça̋o para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 (<a class="ltx_ref ltx_url ltx_font_typewriter" href="https://doi.org/10.54499/UIDB/00297/2020" title="">https://doi.org/10.54499/UIDB/00297/2020</a>) and UIDP/00297/2020 (<a class="ltx_ref ltx_url ltx_font_typewriter" href="https://doi.org/10.54499/UIDP/00297/2020" title="">https://doi.org/10.54499/UIDP/00297/2020</a>) (Center for Mathematics and Applications)”.</p> </div> </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">Abowd et al. 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