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The Bayes oracle and asymptotic optimality of multiple testing procedures under sparsity | | Malgorzata Bogdan
; Arijit Chakrabarti
; Florian Frommlet
; Jayanta K. Ghosh
; | | Date: |
18 Feb 2010 | | Abstract: | We investigate the asymptotic optimality of a large class of multiple testing
rules using the framework of Bayesian Decision Theory. We consider a parametric
setup, in which observations come from a normal scale mixture model and assume
that the total loss is the sum of losses for individual tests. Our model can be
used for testing point null hypotheses of no signals (zero effects), as well as
to distinguish large signals from a multitude of very small effects. The
optimality of a rule is proved by showing that, within our chosen asymptotic
framework, the ratio of its Bayes risk and that of the Bayes oracle (a rule
which minimizes the Bayes risk) converges to one. Our main interest is in the
asymptotic scheme under which the proportion p of "true" alternatives converges
to zero. We fully characterize the class of fixed threshold multiple testing
rules which are asymptotically optimal and hence derive conditions for the
asymptotic optimality of rules controlling the Bayesian False Discovery Rate
(BFDR). We also provide conditions under which the popular Benjamini-Hochberg
and Bonferroni procedures are asymptotically optimal and show that for a wide
class of sparsity levels, the threshold of the former can be approximated very
well by a non-random threshold. As far as we know, this is the first proof of
the decision theoretic asymptotic optimality of the Benjamini-Hochberg rule in
the context of hypothesis testing. | | Source: | arXiv, 1002.3501 | | Services: | Forum | Review | PDF | Favorites | |
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