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How Much Can Gaussian Dependence Inflate the Benjamini-Hochberg Procedure's FDR?

  • Lihua Lei
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Abstract

We study the worst-case false discovery rate (FDR) of the Benjamini-Hochberg procedure for both one- and two-sided Gaussian tests with general correlation matrices. For each setting and given target FDR level q, we construct a family of finite Gaussian models whose FDR divided by q diverges as q↓0. This disproves any universal multiplicative FDR bound. Moreover, we establish new FDR lower bounds that are independent of the number of hypotheses. For two-sided tests, the worst-case FDR over the number of hypotheses, mean vector, and correlation matrix is at least an explicit ℓ=(q)>q: ℓ=(q)=q√(log(1/q))/2√π+cℓ q+o(q), cℓ=0.6492828…. For one-sided tests, we show a lower bound ℓ≤(q)>q: ℓ≤(q)=q√(log(1/q))/√π +q/2+o(q). Finally, we prove the leading constants are sharp for a class of correlation matrices for both one- and two-sided tests.

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Cite

BibTeXRIS
BibTeX
@misc{lei2026much,
  title = {{How Much Can Gaussian Dependence Inflate the Benjamini-Hochberg Procedure's FDR?}},
  author = {Lihua Lei},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.14812},
  eprint = {2607.14812},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.14812},
}
RIS
TY  - UNPB
TI  - How Much Can Gaussian Dependence Inflate the Benjamini-Hochberg Procedure's FDR?
AU  - Lihua Lei
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2607.14812
UR  - https://arxiv.org/abs/2607.14812
AB  - We study the worst-case false discovery rate (FDR) of the Benjamini-Hochberg procedure for both one- and two-sided Gaussian tests with general correlation matrices. For each setting and given target FDR level q, we construct a family of finite Gaussian models whose FDR divided by q diverges as q↓0. This disproves any universal multiplicative FDR bound. Moreover, we establish new FDR lower bounds that are independent of the number of hypotheses. For two-sided tests, the worst-case FDR over the number of hypotheses, mean vector, and correlation matrix is at least an explicit ℓ_=(q)>q: ℓ_=(q)=q√(log(1/q))/2√π+c_ℓ q+o(q), c_ℓ=0.6492828…. For one-sided tests, we show a lower bound ℓ_≤(q)>q: ℓ_≤(q)=q√(log(1/q))/√π +q/2+o(q). Finally, we prove the leading constants are sharp for a class of correlation matrices for both one- and two-sided tests.
ER  -