Asymptotic Properties of False Discovery Rate Controlling Procedures Under Independence

This paper investigates the performance of a family of multiple comparison procedures which have been designed to provide strong control of the False Discovery Rate ($\mathsf{FDR}$). The $\mathsf{FDR}$ is the expected False Discovery Proportion ($\mathsf{FDP}$), that is, the expected fraction of false rejections among all rejected hypotheses. Starting from the Benjamini-Hochberg procedure [1], a number of refinements have been proposed to increase power by estimating the proportion of true null hypotheses, either explicitly, leading to plug-in procedures [3,20] or implicitly, leading to adaptive procedures [5,7]. In this paper we use a variant of the stochastic process approach proposed by Genovese and Wasserman [11] to study the fluctuations of the $\mathsf{FDP}$ achieved by each of these procedures around its expectation when tested hypotheses are independent. We introduce a framework that allows us to derive generic Central Limit Theorems for the $\mathsf{FDP}$ of these procedures, and characterize the associated regularity conditions. We interpret recently proposed adaptive procedures [5,7] as fixed points of the iteration of well-known plug-in procedures [3,20].