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Correlated two-sided Gaussian tests can defeat the Benjamini–Hochberg guarantee

An explicit correlated Gaussian factor model, backed by an interval-arithmetic certificate, has asymptotic false discovery rate above 0.0104 when Benjamini–Hochberg is run at nominal level 0.01. A follow-on paper extends the negative result.

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Stable ID
math-benjamini-hochberg-correlated-gaussian-2026
Revision
math-benjamini-hochberg-correlated-gaussian-2026.v1
Field
Mathematics · Mathematical statistics
Evidence
Tier 1 · Peer reviewed: No
Record state
Current · Expert-supported preprint with reproducible certificate
Last checked

AI role

The paper states that GPT-5.6 Pro produced the proof and that Dobriban carefully checked it.

Record details

Problem or result
False-discovery-rate control for Benjamini–Hochberg under unrestricted Gaussian correlation
Authors
Edgar Dobriban
Institutions
University of Pennsylvania
Result date
Preprint submitted July 13, 2026

Why it matters

The result closes a precise theoretical question about one of the most widely used multiple-testing procedures.

Limits

It does not invalidate Benjamini–Hochberg under independence or accepted positive-dependence conditions. The counterexample uses a special unrestricted dependence setting, and its exhibited excess is small.

Sources

  1. Primary: Edgar Dobriban, The Benjamini--Hochberg Procedure Can Fail to Control the FDR for Correlated Two-Sided Gaussian Tests
  2. Independent: Lihua Lei, No Universal Multiplicative FDR Bound for the BH Procedure
  3. Independent: Independent technical explainer with scope caveats

Correction and revision history

  1. Added on 2026-07-22 with the dependence assumptions and small numerical excess made explicit to prevent overstatement.

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