{
  "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 $q\\downarrow0$. 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 ${\\ell_{=}}(q)>q$: \\[ \\ell_{=}(q)=\\frac{q\\sqrt{\\log(1/q)}}{2\\sqrtπ}+c_\\ell q+o(q), \\qquad c_\\ell=0.6492828\\ldots. \\] For one-sided tests, we show a lower bound $\\ell_{\\le}(q)>q$: \\[ \\ell_{\\le}(q)=\\frac{q\\sqrt{\\log(1/q)}}{\\sqrtπ} +\\frac q2+o(q). \\] Finally, we prove the leading constants are sharp for a class of correlation matrices for both one- and two-sided tests.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.14812",
  "author_total": 1,
  "authors": [
    {
      "name": "Lihua Lei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f40ms12636gzn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.14812",
  "fulltext_url": null,
  "id": "f40ms12636gzn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.14812",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Statistical Methods in Clinical Trials"
    ]
  ],
  "openalex": "W7169510948",
  "pdf_url": "https://arxiv.org/pdf/2607.14812",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f40ms12636gzn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2607.14812",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.14812"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "How Much Can Gaussian Dependence Inflate the Benjamini-Hochberg Procedure's FDR?",
  "title_text": "How Much Can Gaussian Dependence Inflate the Benjamini-Hochberg Procedure's FDR?",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:46Z",
  "url": "https://research.draconitic.com/paper/f40ms12636gzn",
  "venue": "arXiv",
  "versions": []
}