Strong consistency and asymptotic normality for quantities related to the Benjamini–Hochberg false discovery rate procedure

Abstract

The Benjamini–Hochberg (BH) false discovery rate (FDR) procedure enjoys widespread use in multiple testing scenarios when the cost of false positives is low and the number of simultaneous tests prohibits use of the Bonferroni procedure. The FDR and the average power are each expected values, of the false discovery proportion (FDP) and true positive proportion (TPP), respectively. Chi (2007), proved a law of the iterated logarithm (LIL) for the positive proportion (PP), FDP and TPP, and discusses a criticality phenomenon whereby a minimal FDR corresponding to the effect size is required for a nonzero rate of positive calls. While almost sure convergence follows as a corollary, no direct simple proof exists in the literature. We provide this result under the same set of conditions. Of greater consequence, we prove central limit results (CLT) for the PP, FPF and TPP under these weak conditions, providing full characterization of the limits.