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</div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.13621">arXiv:2405.13621</a> <span> [<a href="https://arxiv.org/pdf/2405.13621">pdf</a>, <a href="https://arxiv.org/format/2405.13621">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> Interval identification of natural effects in the presence of outcome-related unmeasured confounding </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Doretti%2C+M">Marco Doretti</a>, <a href="/search/stat?searchtype=author&query=Stanghellini%2C+E">Elena Stanghellini</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.13621v1-abstract-short" style="display: inline;"> With reference to a binary outcome and a binary mediator, we derive identification bounds for natural effects under a reduced set of assumptions. Specifically, no assumptions about confounding are made that involve the outcome; we only assume no unobserved exposure-mediator confounding as well as a condition termed partially constant cross-world dependence (PC-CWD), which poses fewer constraints o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.13621v1-abstract-full').style.display = 'inline'; document.getElementById('2405.13621v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.13621v1-abstract-full" style="display: none;"> With reference to a binary outcome and a binary mediator, we derive identification bounds for natural effects under a reduced set of assumptions. Specifically, no assumptions about confounding are made that involve the outcome; we only assume no unobserved exposure-mediator confounding as well as a condition termed partially constant cross-world dependence (PC-CWD), which poses fewer constraints on the counterfactual probabilities than the usual cross-world independence assumption. The proposed strategy can be used also to achieve interval identification of the total effect, which is no longer point identified under the considered set of assumptions. Our derivations are based on postulating a logistic regression model for the mediator as well as for the outcome. However, in both cases the functional form governing the dependence on the explanatory variables is allowed to be arbitrary, thereby resulting in a semi-parametric approach. To account for sampling variability, we provide delta-method approximations of standard errors in order to build uncertainty intervals from identification bounds. The proposed method is applied to a dataset gathered from a Spanish prospective cohort study. The aim is to evaluate whether the effect of smoking on lung cancer risk is mediated by the onset of pulmonary emphysema. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.13621v1-abstract-full').style.display = 'none'; document.getElementById('2405.13621v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">14 pages, 2 figures, 2 tables</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2306.09969">arXiv:2306.09969</a> <span> [<a href="https://arxiv.org/pdf/2306.09969">pdf</a>, <a href="https://arxiv.org/format/2306.09969">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> Omitting continuous covariates in binary regression models: implications for sensitivity and mediation analysis </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Gasparin%2C+M">Matteo Gasparin</a>, <a href="/search/stat?searchtype=author&query=Scarpa%2C+B">Bruno Scarpa</a>, <a href="/search/stat?searchtype=author&query=Stanghellini%2C+E">Elena Stanghellini</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2306.09969v1-abstract-short" style="display: inline;"> By exploiting the theory of skew-symmetric distributions, we generalise existing results in sensitivity analysis by providing the analytic expression of the bias induced by marginalization over an unobserved continuous confounder in a logistic regression model. The expression is approximated and mimics Cochran's formula under some simplifying assumptions. Other link functions and error distributio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2306.09969v1-abstract-full').style.display = 'inline'; document.getElementById('2306.09969v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2306.09969v1-abstract-full" style="display: none;"> By exploiting the theory of skew-symmetric distributions, we generalise existing results in sensitivity analysis by providing the analytic expression of the bias induced by marginalization over an unobserved continuous confounder in a logistic regression model. The expression is approximated and mimics Cochran's formula under some simplifying assumptions. Other link functions and error distributions are also considered. A simulation study is performed to assess its properties. The derivations can also be applied in causal mediation analysis, thereby enlarging the number of circumstances where simple parametric formulations can be used to evaluate causal direct and indirect effects. Standard errors of the causal effect estimators are provided via the first-order Delta method. Simulations show that our proposed estimators perform equally well as others based on numerical methods and that the additional interpretability of the explicit formulas does not compromise their precision. The new estimator has been applied to measure the effect of humidity on upper airways diseases mediated by the presence of common aeroallergens in the air. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2306.09969v1-abstract-full').style.display = 'none'; document.getElementById('2306.09969v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 June, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages, 3 figures, 4 tables</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2211.09420">arXiv:2211.09420</a> <span> [<a href="https://arxiv.org/pdf/2211.09420">pdf</a>, <a href="https://arxiv.org/ps/2211.09420">ps</a>, <a href="https://arxiv.org/format/2211.09420">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1002/bimj.202300089">10.1002/bimj.202300089 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Mediation analysis with case-control sampling: Identification and estimation in the presence of a binary mediator </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Doretti%2C+M">Marco Doretti</a>, <a href="/search/stat?searchtype=author&query=Genb%C3%A4ck%2C+M">Minna Genb盲ck</a>, <a href="/search/stat?searchtype=author&query=Stanghellini%2C+E">Elena Stanghellini</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2211.09420v1-abstract-short" style="display: inline;"> With reference to a stratified case-control procedure based on a binary variable of primary interest, we derive the expression of the distortion induced by the sampling design on the parameters of the logistic model of a secondary variable. This is particularly relevant when performing mediation analysis (possibly in a causal framework) with stratified case-control data in settings where both the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2211.09420v1-abstract-full').style.display = 'inline'; document.getElementById('2211.09420v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2211.09420v1-abstract-full" style="display: none;"> With reference to a stratified case-control procedure based on a binary variable of primary interest, we derive the expression of the distortion induced by the sampling design on the parameters of the logistic model of a secondary variable. This is particularly relevant when performing mediation analysis (possibly in a causal framework) with stratified case-control data in settings where both the outcome and the mediator are binary. Our identification result opens the way to M-estimation and Maximum Likelihood estimation. We then conduct a simulation study showing the gain in efficiency of the estimators of both the outcome and mediator model parameters w.r. to existing methods, based on weighting. As an illustrative example, we reanalyze a German case-control dataset in order to investigate whether the effect of reduced immunocompetency on listeriosis onset is mediated by the intake of gastric acid suppressors. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2211.09420v1-abstract-full').style.display = 'none'; document.getElementById('2211.09420v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 November, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">33 pages, 4 tables, 3 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2202.12171">arXiv:2202.12171</a> <span> [<a href="https://arxiv.org/pdf/2202.12171">pdf</a>, <a href="https://arxiv.org/ps/2202.12171">ps</a>, <a href="https://arxiv.org/format/2202.12171">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1097/EDE.0000000000001540">10.1097/EDE.0000000000001540 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Exact mediation analysis for ordinal outcome and binary mediator </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Stanghellini%2C+E">Elena Stanghellini</a>, <a href="/search/stat?searchtype=author&query=Kateri%2C+M">Maria Kateri</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2202.12171v2-abstract-short" style="display: inline;"> With reference to a single mediator context, this brief report presents a model-based strategy to estimate counterfactual direct and indirect effects when the response variable is ordinal and the mediator is binary. Postulating a logistic regression model for the mediator and a cumulative logit model for the outcome, the exact parametric formulation of the causal effects is presented, thereby exte… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.12171v2-abstract-full').style.display = 'inline'; document.getElementById('2202.12171v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2202.12171v2-abstract-full" style="display: none;"> With reference to a single mediator context, this brief report presents a model-based strategy to estimate counterfactual direct and indirect effects when the response variable is ordinal and the mediator is binary. Postulating a logistic regression model for the mediator and a cumulative logit model for the outcome, the exact parametric formulation of the causal effects is presented, thereby extending previous work that only contained approximated results. The identification conditions are equivalent to the ones already established in the literature. The effects can be estimated by making use of standard statistical software and standard errors can be computed via a bootstrap algorithm. To make the methodology accessible, routines to implement the proposal in R are presented in the Appendix. A natural effect model coherent with the postulated data generating mechanism is also derived. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.12171v2-abstract-full').style.display = 'none'; document.getElementById('2202.12171v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 October, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages, 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 62D20 <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> G.3 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2104.13775">arXiv:2104.13775</a> <span> [<a href="https://arxiv.org/pdf/2104.13775">pdf</a>, <a href="https://arxiv.org/ps/2104.13775">ps</a>, <a href="https://arxiv.org/format/2104.13775">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> Path Analysis for Binary Random Variables </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Raggi%2C+M">Martina Raggi</a>, <a href="/search/stat?searchtype=author&query=Stanghellini%2C+E">Elena Stanghellini</a>, <a href="/search/stat?searchtype=author&query=Doretti%2C+M">Marco Doretti</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2104.13775v1-abstract-short" style="display: inline;"> The decomposition of the overall effect of a treatment into direct and indirect effects is here investigated with reference to a recursive system of binary random variables. We show how, for the single mediator context, the marginal effect measured on the log odds scale can be written as the sum of the indirect and direct effects plus a residual term that vanishes under some specific conditions. W… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.13775v1-abstract-full').style.display = 'inline'; document.getElementById('2104.13775v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2104.13775v1-abstract-full" style="display: none;"> The decomposition of the overall effect of a treatment into direct and indirect effects is here investigated with reference to a recursive system of binary random variables. We show how, for the single mediator context, the marginal effect measured on the log odds scale can be written as the sum of the indirect and direct effects plus a residual term that vanishes under some specific conditions. We then extend our definitions to situations involving multiple mediators and address research questions concerning the decomposition of the total effect when some mediators on the pathway from the treatment to the outcome are marginalized over. Connections to the counterfactual definitions of the effects are also made. Data coming from an encouragement design on students' attitude to visit museums in Florence, Italy, are reanalyzed. The estimates of the defined quantities are reported together with their standard errors to compute p-values and form confidence intervals. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.13775v1-abstract-full').style.display = 'none'; document.getElementById('2104.13775v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 April, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">43 pages, 7 figures. A version of this paper is forthcoming in Sociological Methods & Research</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1811.00439">arXiv:1811.00439</a> <span> [<a href="https://arxiv.org/pdf/1811.00439">pdf</a>, <a href="https://arxiv.org/ps/1811.00439">ps</a>, <a href="https://arxiv.org/format/1811.00439">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s10260-021-00562-w">10.1007/s10260-021-00562-w <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Exact parametric causal mediation analysis for a binary outcome with a binary mediator </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Doretti%2C+M">Marco Doretti</a>, <a href="/search/stat?searchtype=author&query=Raggi%2C+M">Martina Raggi</a>, <a href="/search/stat?searchtype=author&query=Stanghellini%2C+E">Elena Stanghellini</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1811.00439v3-abstract-short" style="display: inline;"> A parametric expression for causal natural direct and indirect effects is derived for the setting of a binary outcome with a binary mediator. The proposed effect decomposition does not require the outcome to be rare and generalizes the existing ones, allowing for interactions between both the exposure and the mediator and confounding covariates. Further, it outlines the relationship between the ca… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.00439v3-abstract-full').style.display = 'inline'; document.getElementById('1811.00439v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1811.00439v3-abstract-full" style="display: none;"> A parametric expression for causal natural direct and indirect effects is derived for the setting of a binary outcome with a binary mediator. The proposed effect decomposition does not require the outcome to be rare and generalizes the existing ones, allowing for interactions between both the exposure and the mediator and confounding covariates. Further, it outlines the relationship between the causal effects and the correspondent pathway-specific logistic regression parameters, in parallel with results derived under the rare outcome assumption. Formulae for standard errors, obtained via the delta method, are also given. A simulation study is implemented which compares these estimators to a number of competing ones. An empirical application to data coming from a microfinance experiment performed in Bosnia and Herzegovina is illustrated as an example. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.00439v3-abstract-full').style.display = 'none'; document.getElementById('1811.00439v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">24 pages, 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Statistical Methods & Applications (2021) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1009.4279">arXiv:1009.4279</a> <span> [<a href="https://arxiv.org/pdf/1009.4279">pdf</a>, <a href="https://arxiv.org/ps/1009.4279">ps</a>, <a href="https://arxiv.org/format/1009.4279">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.3150/12-BEJ435">10.3150/12-BEJ435 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Identification of discrete concentration graph models with one hidden binary variable </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/stat?searchtype=author&query=Stanghellini%2C+E">Elena Stanghellini</a>, <a href="/search/stat?searchtype=author&query=Vantaggi%2C+B">Barbara Vantaggi</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1009.4279v2-abstract-short" style="display: inline;"> Conditions are presented for different types of identifiability of discrete variable models generated over an undirected graph in which one node represents a binary hidden variable. These models can be seen as extensions of the latent class model to allow for conditional associations between the observable random variables. Since local identification corresponds to full rank of the parametrization… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1009.4279v2-abstract-full').style.display = 'inline'; document.getElementById('1009.4279v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1009.4279v2-abstract-full" style="display: none;"> Conditions are presented for different types of identifiability of discrete variable models generated over an undirected graph in which one node represents a binary hidden variable. These models can be seen as extensions of the latent class model to allow for conditional associations between the observable random variables. Since local identification corresponds to full rank of the parametrization map, we establish a necessary and sufficient condition for the rank to be full everywhere in the parameter space. The condition is based on the topology of the undirected graph associated to the model. For non-full rank models, the obtained characterization allows us to find the subset of the parameter space where the identifiability breaks down. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1009.4279v2-abstract-full').style.display = 'none'; document.getElementById('1009.4279v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 December, 2013; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 September, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2010. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Published in at http://dx.doi.org/10.3150/12-BEJ435 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> IMS-BEJ-BEJ435 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Bernoulli 2013, Vol. 19, No. 5A, 1920-1937 </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" 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