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

2026-07-13

AI Summary

This study demonstrates that the Benjamini–Hochberg (BH) procedure may fail to control the nominal false discovery rate (FDR) in correlated two-sample Gaussian testing, thereby refuting the long-standing conjecture that BH maintains FDR control under dependence among Gaussian p-values. By constructing a factor model and combining interval arithmetic certification with rigorous mathematical analysis—supplemented by verification using GPT-5.6 Pro—the authors prove that, for sufficiently large numbers of hypotheses and a significance level α = 0.01, the actual FDR strictly exceeds 0.0104. These theoretical findings are corroborated by Monte Carlo simulations, providing the first conclusive evidence that the BH procedure can indeed lose FDR control under specific correlation structures.

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.

Problem

• Benjamini–Hochberg procedure
• false discovery rate
• correlated tests
• Gaussian p-values
• FDR control

Innovation

• Benjamini–Hochberg procedure
• false discovery rate (FDR)
• correlated Gaussian tests
• interval arithmetic
• AI-assisted proof