{
  "abstract": "Natural exponential families (NEFs) are characterized by their variance functions (VFs), namely by the pair \\((V,M)\\), where \\(V\\) expresses the variance as a function of the mean and \\(M\\) is the mean domain. This characterization naturally raises the inverse question: when is a given function \\(V\\) the VF of an NEF, and what structural properties of the family can be read directly from \\(V\\)? Bar-Lev (1987), as part of a more general result for absolutely monotone functions, showed that every nonzero polynomial \\[ V(m)=\\sum_{j=1}^{r} a_j m^j,\\qquad a_j\\ge 0, \\] is the VF of an infinitely divisible NEF on a positive mean domain. This polynomial class splits exhaustively into the cases \\(a_1>0\\) and \\(a_1=0\\). Bar-Lev, Letac and Ridder (2024) proved that, after scaling so that \\(a_1=1\\), the first case yields a counting NEF supported on \\(\\mathbb N_0\\). We establish the complementary result: when \\(a_1=0\\), the corresponding NEF is absolutely continuous with respect to Lebesgue measure on the positive half-line. More generally, we prove absolute continuity whenever \\[ V(m)=m^2G(m), \\] where \\(G\\) is a nonzero power series with nonnegative coefficients. When \\(G\\) is a polynomial, the mean domain is the entire positive half-line. We also record several consequences for the cumulant structure and for statistical inference.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05478",
  "author_total": 1,
  "authors": [
    {
      "name": "Shaul K. Bar-Lev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/enwr8ywaypf60.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05478",
  "fulltext_url": null,
  "id": "enwr8ywaypf60",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05478",
  "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",
      "stat.AP"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7220745724",
  "pdf_url": "https://arxiv.org/pdf/2610.05478",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/enwr8ywaypf60.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:16Z",
      "native_id": "2610.05478",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05478"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.AP",
      "subfield": "statistics"
    }
  ],
  "title": "From Counting to Continuous Natural Exponential Families",
  "title_text": "From Counting to Continuous Natural Exponential Families",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:16Z",
  "url": "https://research.draconitic.com/paper/enwr8ywaypf60",
  "venue": "arXiv",
  "versions": []
}