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.
Funding Statement
DMK was supported by a Stanford Graduate Fellowship and a Stanford Graduate Interdisciplinary Fellowship. ABO was supported by the National Science Foundation under grants IIS-1837931 and DMS-2152780.
Acknowledgements
The authors wish to thank Will Fithian, Kevin Guo, Grant Izmirlian, Lihua Lei, Kenneth Tay, Marius Tirlea, Jingshu Wang, and anonymous reviewers for helpful comments and discussions. The authors also thank Kevin Guo for a proof sketch on how to remove a condition from an earlier version of this paper and Jingshu Wang for sharing the DMD data. Finally, the authors would like to thank two anonymous referees, an Associate Editor and the Editor for their constructive comments that helped improve the quality of this paper.
Citation
Dan M. Kluger. Art B. Owen. "A central limit theorem for the Benjamini-Hochberg false discovery proportion under a factor model." Bernoulli 30 (1) 743 - 769, February 2024. https://doi.org/10.3150/23-BEJ1615
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