{
  "abstract": "Let $\\mathbf X_1,\\ldots,\\mathbf X_n$ be independent random vectors in $\\mathbb R^d$ with common mean $\\boldsymbolμ_n$, where $d\\ge2$ is a fixed integer. Let $\\bar{\\mathbf X}_n=n^{-1}\\sum_{i=1}^n\\mathbf X_i$ denote the sample mean and let \\[ \\widehatΣ_n=\\frac1n\\sum_{i=1}^n (\\mathbf X_i-\\bar{\\mathbf X}_n) (\\mathbf X_i-\\bar{\\mathbf X}_n)^\\top \\] denote the sample covariance matrix. Assume that the average covariance matrix is positive definite and that $\\widehatΣ_n$ is nonsingular almost surely. For testing $H_0:\\boldsymbolμ_n=\\boldsymbolμ_0$, the Hotelling statistic is \\[ T^2_{H,n} =n(\\bar{\\mathbf X}_n-\\boldsymbolμ_0)^\\top \\widehatΣ_n^{-1} (\\bar{\\mathbf X}_n-\\boldsymbolμ_0). \\] Under $H_0$, we establish finite-sample Cramér-type moderate deviation results and uniform Berry--Esseen bounds under finite $p$th moments for $2<p\\le3$. The moderate deviation error order is optimal over independent triangular arrays, as shown by explicit constructions. For identically distributed observations with a fixed distribution, we obtain \\[ \\sup_{0\\le x\\le b_n} \\left| \\frac{\\mathbb P(T^2_{H,n}\\ge x^2)} {\\mathbb P(χ_d^2\\ge x^2)}-1 \\right| = O\\!\\left((1+b_n)^p n^{-(p-2)/2}\\right) \\] for every nonnegative sequence $b_n=o(n^{(p-2)/(2p)})$. The uniform Berry--Esseen bound is $O(n^{-(p-2)/2})$. The proof uses a radial Stein identity and a recursive error estimate. We apply the results to chi-squared calibration in simultaneous testing.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08074",
  "author_total": 4,
  "authors": [
    {
      "name": "Song-Hao Liu"
    },
    {
      "name": "Qi-Man Shao"
    },
    {
      "name": "Hao Shi"
    },
    {
      "name": "Jing-Yu Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0fy02sn3bzxxa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08074",
  "fulltext_url": null,
  "id": "0fy02sn3bzxxa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08074",
  "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.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08074",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0fy02sn3bzxxa.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.08074",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08074"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Sharp moderate deviations and uniform Berry--Esseen bounds for Hotelling's $T^2$-statistic via Stein's method",
  "title_text": "Sharp moderate deviations and uniform Berry--Esseen bounds for Hotelling's T^2-statistic via Stein's method",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:52Z",
  "url": "https://research.draconitic.com/paper/0fy02sn3bzxxa",
  "venue": "arXiv",
  "versions": []
}