{
  "abstract": "The Baringhaus--Henze--Epps--Pulley (BHEP) tests form a widely applicable class of affine invariant and consistent tests for multivariate normality. Under the null hypothesis, the test statistic converges to a weighted sum of independent chi-squared random variables. Although closed-form expressions for the first three cumulants of this limiting distribution are known for arbitrary dimension $d$ and smoothing parameter $β$, an explicit expression for the fourth cumulant has so far been available only in a special univariate case. We derive the fourth cumulant for arbitrary $d\\geq1$ and $β>0$ and use the resulting first four moments to construct Johnson and Pearson approximations to the limiting null distribution. These yield essentially instantaneous analytic approximations to BHEP $p$-values, without numerical eigenvalue calculations. Using the recently obtained complete spectrum as a benchmark, we show that both four-moment approximations are highly accurate over a broad range of dimensions, smoothing parameters and upper-tail probabilities, and substantially improve on two- and three-parameter lognormal approximations. Monte Carlo results indicate good finite-sample calibration for many parameter combinations, although convergence to the limiting distribution may be slow in higher dimensions for extreme values of the smoothing parameter.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35037",
  "author_total": 2,
  "authors": [
    {
      "name": "Bruno Ebner"
    },
    {
      "name": "Norbert Henze"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3ybazh7jq7nk7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35037",
  "fulltext_url": null,
  "id": "3ybazh7jq7nk7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35037",
  "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 Distribution Estimation and Applications"
    ]
  ],
  "openalex": "W7214779726",
  "pdf_url": "https://arxiv.org/pdf/2609.35037",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3ybazh7jq7nk7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.35037",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35037"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Four-Moment Approximations and Fast $p$-Values for BHEP Tests of Multivariate Normality",
  "title_text": "Four-Moment Approximations and Fast p-Values for BHEP Tests of Multivariate Normality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:26Z",
  "url": "https://research.draconitic.com/paper/3ybazh7jq7nk7",
  "venue": "arXiv",
  "versions": []
}