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

TL;DR

This paper establishes a central limit theorem for the false discovery proportion in the BH procedure under various correlation structures, revealing conditions that lead to bursty false discoveries.

Contribution

It provides the first CLT for FDP under a factor model, clarifying when BH is prone to bursty false discoveries due to correlation patterns.

Findings

BH exhibits burstiness with strong, long-range correlations

CLT for FDP under factor models is established

Simulations confirm theoretical predictions

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.