Some optimality properties of FDR controlling rules under sparsity

Abstract

False Discovery Rate (FDR) and the Bayes risk are two different statistical measures, which can be used to evaluate and compare multiple testing procedures. Recent results show that under sparsity FDR controlling procedures, like the popular Benjamini-Hochberg (BH) procedure, perform also very well in terms of the Bayes risk. In particular asymptotic Bayes optimality under sparsity (ABOS) of BH was shown previously for location and scale models based on log-concave densities. This article extends previous work to a substantially larger set of distributions of effect sizes under the alternative, where the alternative distribution of true signals does not change with the number of tests $m$, while the sample size $n$ slowly increases. ABOS of BH and the corresponding step-down procedure based on FDR levels proportional to $n^{-1/2}$ are proved. A simulation study shows that these asymptotic results are relevant already for relatively small values of $m$ and $n$. Apart from showing asymptotic optimality of BH, our results on the optimal FDR level provide a natural extension of the well known results on the significance levels of Bayesian tests.