A central limit theorem for the Benjamini-Hochberg false discovery proportion under a factor model

Abstract

The Benjamini-Hochberg (BH) procedure remains widely popular despite having limited theoretical guarantees in the commonly encountered scenario of correlated test statistics. Of particular concern is the possibility that the method could exhibit bursty behavior, meaning that it might typically yield no false discoveries while occasionally yielding both a large number of false discoveries and a false discovery proportion (FDP) that far exceeds its own well controlled mean. In this paper, we investigate which test statistic correlation structures lead to bursty behavior and which ones lead to well controlled FDPs. To this end, we develop a central limit theorem for the FDP in a multiple testing setup where the test statistic correlations can be either short-range or long-range as well as either weak or strong. The theorem and our simulations from a data-driven factor model suggest that the BH procedure exhibits severe burstiness when the test statistics have many strong, long-range correlations, but does not otherwise.Comment: Main changes in version 2: i) restated Corollary 1 in a way that is clearer and easier to use, ii) removed a regularity condition for our theorems (in particular we removed Condition 2 from version 1), and iii) we added a couple of remarks (namely, Remark 1 and 6 in version 2). Throughout the text we also fixed typos, improved clarity, and added a some additional commentary and reference