{
  "abstract": "Under Gaussian sampling, removing location and scale turns the sample into a direction that is uniformly distributed on a sphere of dimension $n-2$, and sample skewness and kurtosis become a cubic-quartic polynomial image of spherical measure. We use this geometry to obtain exact finite-sample results for the Jarque-Bera statistic $\\mathrm{JB}_n$. First, for every $n\\ge 3$ the largest attainable value of $\\mathrm{JB}_n$ is $n\\{(n-2)^4+4(n-1)^2\\}/\\{24(n-1)^2\\}$, attained by a single-outlier sample, so the asymptotic $10\\%$ and $5\\%$ tests have size zero for $n\\le 6$ and the $1\\%$ test for $n\\le 7$; near this maximum the density of $\\mathrm{JB}_n$ behaves like an explicit multiple of $(J_n^+-x)^{(n-4)/2}$. Second, passing from residual coordinates to power sums writes the joint density of skewness and kurtosis as an integral of the reciprocal square root of a polynomial discriminant; for every $n\\ge 5$ the innermost integral runs over a single interval and is a Lauricella $F_D$ period. Third, this yields explicit laws: an arcsine law for $n=3$, an algebraic joint density for $n=4$, and a single complete elliptic integral, equivalently ${}_2F_1(\\tfrac12,\\tfrac12;1;\\cdot)$, for $n=5$. Residual-coordinate collisions generate the discriminant singularities of the joint law, while stationary points of $\\mathrm{JB}_n$ on the residual sphere govern the singularities of its one-dimensional density.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29976",
  "author_total": 1,
  "authors": [
    {
      "name": "Steve Lawford"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cpxy7dj48hgma.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29976",
  "fulltext_url": null,
  "id": "cpxy7dj48hgma",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29976",
  "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.ME"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Statistical Distribution Estimation and Applications"
    ]
  ],
  "openalex": "W7214301500",
  "pdf_url": "https://arxiv.org/pdf/2609.29976",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cpxy7dj48hgma.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29976",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29976"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Exact finite-sample geometry of the Jarque-Bera statistic: support, discriminants and elliptic integrals",
  "title_text": "Exact finite-sample geometry of the Jarque-Bera statistic: support, discriminants and elliptic integrals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:21Z",
  "url": "https://research.draconitic.com/paper/cpxy7dj48hgma",
  "venue": "arXiv",
  "versions": []
}