Asymptotic theory of dependent Bayesian multiple testing procedures under possible model misspecification
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Asymptotic theory of dependent Bayesian multiple testing procedures under possible model misspecification Noirrit Kiran Chandra1 · Sourabh Bhattacharya2 Received: 13 May 2020 / Revised: 4 September 2020 / Accepted: 29 September 2020 © The Institute of Statistical Mathematics, Tokyo 2020
Abstract We study asymptotic properties of Bayesian multiple testing procedures and provide sufficient conditions for strong consistency under general dependence structure. We also consider a novel Bayesian multiple testing procedure and associated error measures that coherently accounts for the dependence structure present in the model. We advocate posterior versions of FDR and FNR as appropriate error rates and show that their asymptotic convergence rates are directly associated with the Kullback– Leibler divergence from the true model. The theories hold regardless of the class of postulated models being misspecified. We illustrate our results in a variable selection problem with autoregressive response variables and compare our procedure with some existing methods through simulation studies. Superior performance of the new procedure compared to the others indicates that proper exploitation of the dependence structure by multiple testing methods is indeed important. Moreover, we obtain encouraging results in a maize dataset, where we select influential marker variables. Keywords Bayesian multiple testing · Variable selection · False discovery rate · Kullback–Leibler · Misspecified model · Posterior convergence
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s1046 3-020-00770-3) contains supplementary material, which is available to authorized users. * Noirrit Kiran Chandra [email protected] 1
Department of Statistics and Data Science, University of Texas at Austin, 2317 Speedway D9800, Austin, TX 78712‑1823, USA
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Interdisciplinary Statistical Research Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata, WB 700108, India
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N. K. Chandra, S. Bhattacharya
1 Introduction In recent times, there has been a tremendous growth in the area of multiple hypothesis testing as simultaneous inference on several parameters is often necessary. Benjamini and Hochberg (1995) introduced a powerful approach to handle this problem in their landmark paper. However, in most real-life situations the test statistics are generally dependent. Benjamini and Yekutieli (2001) showed that the Benjamini–Hochberg procedure is valid under positive dependence. Berry and Hochberg (1999) have given a Bayesian perspective on multiple testing where the tests are related through a dependent prior. Scott and Berger (2010) discussed how empirical Bayes and fully Bayes methods adjust multiplicity. There are many works in the statistical literature on optimality and asymptotic behaviour of multiple testing methods in dependent cases. Sun and Cai (2007) have proposed an optimal adaptive procedure where the data are generated from a two-component mixture model. Finner and Roter
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