{
  "abstract": "We study the non-uniqueness of exact two-sided $p$-values in continuous one-parameter natural exponential families (NEFs). For directed one-sided problems, the tail $p$-value agrees with the $p$-values using UMP, UMPU, and likelihood-ratio (LR) tests. For a two-sided simple null, we distinguish four constructions: equal-tail, density-ordered, UMPU, and LR $p$-values. At a fixed null parameter, UMPU and equal-tail $p$-values coincide if and only if the null law is symmetric about its mean; under a regular two-branch density-level condition, the same fixed-null symmetry characterization holds for UMPU versus density ordering and equal-tail versus density ordering. Requiring any of these coincidences throughout the NEF characterizes the Gaussian family. We combine these results with the theorem of Bar-Lev, Bshouty and Letac that UMPU and LR $p$-values coincide throughout a continuous NEF precisely for the normal, gamma and inverse-Gaussian families. We also investigate the two LR pairings not covered by those results. If equal-tail and LR $p$-values coincide throughout a NEF satisfying our standing regularity assumptions, then $(V^{2/3})^{\\prime \\prime \\prime }=0$ on the mean domain. The same coincidence also forces an explicit density-at-the-mean identity. For an i.i.d.\\ sample with canonical sufficient statistic $T_n=\\sum_{i=1}^nX_i$, persistence of equal-tail-LR coincidence throughout the family along an unbounded sequence of sample sizes forces Gaussianity. A corresponding density-LR statement is given conditionally on an explicitly stated differentiated local Edgeworth expansion. Finally, inverse-Gaussian and hyperbolic-secant examples quantify numerical $p$-value differences, disagreement of rejection decisions, and differences in power.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28221",
  "author_total": 2,
  "authors": [
    {
      "name": "Shaul K. Bar-Lev"
    },
    {
      "name": "Linard Hoessly"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/441nq9kk3vp1v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28221",
  "fulltext_url": null,
  "id": "441nq9kk3vp1v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28221",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7204821232",
  "pdf_url": "https://arxiv.org/pdf/2608.28221",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/441nq9kk3vp1v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.28221",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28221"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Exact two-sided p-values in natural exponential families: coincidence, non-uniqueness, and sample-size stability",
  "title_text": "Exact two-sided p-values in natural exponential families: coincidence, non-uniqueness, and sample-size stability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:34Z",
  "url": "https://research.draconitic.com/paper/441nq9kk3vp1v",
  "venue": "arXiv",
  "versions": []
}