Recently Bogdan, Chakrabarti, Frommlet and Ghosh (2010) investigated the asymptotic Bayesian optimality of a wide range of multiple testing procedures under sparsity. Analysis was based on a two-groups model, where both the error as well as the priors for null and alternative hypothesis were assumed to be normal. The conditions were provided under which the popular Bonferroni and Benjamini-Hochberg procedures are asymptotically optimal. In this article we extend these results to the two-groups models with general priors for the alternative hypothesis. In contrast to the previous article we focus on the asymptotics of large samples, where the number of replicates for each test increases as the number of tests goes to infinity. We also address the related problem of model selection in multiple regression models. We provide the conditions for the asymptotic optimality of the mBIC criterion proposed by Bogdan, Ghosh and Doerge (2004) and define a new modification of BIC, which turns out to be asymptotically optimal for a substantially wider range of sparsity parameters.
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