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arXiv · 2607.12208

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

Edgar Dobriban
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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 $\alpha=0.01$, 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.

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Edgar Dobriban. 2026-07-13. The Benjamini--Hochberg Procedure Can Fail to Control the FDR for Correlated Two-Sided Gaussian Tests. https://arxiv.org/abs/2607.12208

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