{
  "abstract": "In this article, we investigate the asymptotic Bayes optimality under sparsity (ABOS) of the Gavrilov-Benjamini-Sarkar (GBS) step-down multiple testing procedure (Gavrilov et al., 2009) in the sparse Gaussian sequence model. While the asymptotic optimality properties of the Benjamini-Hochberg procedure have been extensively studied, corresponding results for the GBS procedure remain unavailable despite its favorable finite-sample performance in several settings. Within the spike-and-slab Bayesian formulation and the asymptotic decision-theoretic framework of Bogdan et al. (2011), we establish that the GBS procedure is ABOS over a broad class of sparse asymptotic regimes. Existing analyses of the Benjamini-Hochberg procedure rely on approximating the random rejection threshold by a suitable deterministic surrogate. In contrast, our approach proceeds through a direct analysis of the Bayes risk by separately controlling the false discovery and false nondiscovery components. The analysis combines two key ingredients: a new finite-sample inequality for shifted order statistics associated with the GBS critical constants and a signal-crossing argument for ordered alternative $p$-values that exploits the sequential step-down structure of the procedure. For diverging expected signal counts, these tools yield ABOS without threshold localization. For bounded positive expected signal counts, a direct argument based on the first GBS critical constant establishes ABOS as well. In particular, our results cover polynomial sparsity $p_m\\asymp m^{-β}$ for every fixed $0<β\\le1$. To the best of our knowledge, this is the first asymptotic decision-theoretic analysis of the GBS procedure and the first proof of its asymptotic Bayes optimality under sparsity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.03719",
  "author_total": 2,
  "authors": [
    {
      "name": "Prasenjit Ghosh"
    },
    {
      "name": "Arijit Chakrabarti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6eqkmk9darxyr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.03719",
  "fulltext_url": null,
  "id": "6eqkmk9darxyr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.03719",
  "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 and Inference"
    ]
  ],
  "openalex": "W7167595969",
  "pdf_url": "https://arxiv.org/pdf/2607.03719",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6eqkmk9darxyr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2607.03719",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.03719"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Asymptotic Bayes Optimality Under Sparsity of the Gavrilov-Benjamini-Sarkar Step-Down Testing Procedure",
  "title_text": "Asymptotic Bayes Optimality Under Sparsity of the Gavrilov-Benjamini-Sarkar Step-Down Testing Procedure",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:49Z",
  "url": "https://research.draconitic.com/paper/6eqkmk9darxyr",
  "venue": "arXiv",
  "versions": []
}