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

2026/07/13 by Edgar Dobriban · 2 voices
Mathematics · Computer Science · #math.ST #cs.AI #stat.ME

paper · pdf

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.

Citations

• Dependence-Aware False Discovery Rate Control in Two-Sided Gaussian Mean Testing
• Solving a Research Problem in Mathematical Statistics with AI Assistance
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• Accelerating mathematical research with language models: A case study of an interaction with GPT-5-Pro on a convex analysis problem
• Point Convergence of Nesterov's Accelerated Gradient Method: An AI-Assisted Proof
• Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof
• Gödel Test: Can Large Language Models Solve Easy Conjectures?
• On Controlling the False Discovery Rate in Multiple Testing of the Means of Correlated Normals Against Two-Sided Alternatives
• Conditional calibration for false discovery rate control under dependence
• Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic
• A proof of the Cycle Double Cover Conjecture
• Two simple sufficient conditions for FDR control
• Dependency and false discovery rate: Asymptotics
• A stochastic process approach to false discovery control
• Strong Control, Conservative Point Estimation and Simultaneous Conservative Consistency of False Discovery Rates: A Unified Approach
• The control of the false discovery rate in multiple testing under dependency

Discussions

• arxiv.org/abs/2607.12208 [bsky, 1 points, 1 comments]
• This is about one of the major unproven conjectures in stat theory (generally presumed true.) It took GPT-5.6 90 minutes to find a counterexample showing* it's (very slightly) wrong. Not sure what to [bsky, 0 points, 0 comments]