The Benjamini--Hochberg Procedure Can Fail to Control the FDR for Correlated Two-Sided Gaussian Tests

Edgar Dobriban

arXiv:2607.12208v1 Primary: math.ST Secondary: cs.AI, stat.ME

Abstract

We show that the Benjamini--Hochberg procedure can fail to control the false discovery rate (FDR) at its nominal level for correlated two-sided Gaussian $p$-values. We construct a factor model for which, at level $α=0.01$, a rigorous interval-arithmetic certificate proves $FDR>0.0104$ for all sufficiently large numbers of hypotheses. This disproves a conjecture widely believed to be true for twenty years. Monte Carlo experiments are consistent with the theoretical result. The proof was obtained by GPT-5.6 Pro and carefully checked by the author.

Disclosure

“te proves $FDR>0.0104$ for all sufficiently large numbers of hypotheses. This disproves a conjecture widely believed to be true for twenty years. Monte Carlo experiments are consistent with the theoretical result. The proof was obtained by GPT-5.6 Pro and carefully checked by the author.”
arXiv metadata: abstract

Classification: Substantial proof generation
Multiplier: 10
Verified: 2026-08-23T18:52:10.926003Z

Structural counts

Pages 16 pdf

Theorems 1 source

Lemmas 2 source

Propositions 0 source

Corollaries 0 source

Definitions 0 source

Displayed equations 37 source

Bibliography entries 645 source

Appendix pages 14 estimated

Count notes

Source counts use the expanded primary TeX file main.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.