A Central Limit Theorem for the Benjamini-Hochberg False Discovery Proportion Under a Factor Model

• Activity Number: 444 - Recent Advances in Statistical Methodology for Big Data
• Type: Contributed
• Date/Time: Thursday, August 12, 2021 : 4:00 PM to 5:50 PM
• Sponsor: IMS
• Author(s): Dan M. Kluger* and Art Owen
• Companies: Stanford University and Stanford University
• Keywords: multiple hypothesis testing; functional central limit theorem; functional delta method; empirical cumulative distribution function; Simes line

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 exhibits bursty behavior; that is, it typically yields no false discoveries but occasionally yields 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, using a functional central limit theorem for dependent empirical processes coupled with the functional delta method, 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 bursts when the test statistics have many strong, long-range correlations, but not otherwise.