{
  "abstract": "Kerman's approximation to the median of the Beta$(α,β)$ distribution is the ratio $g(α)/(g(α) + g(β))$ with $g(x) = x - 1/3$, using the first two terms of the asymptotic expansion of the median of the standard gamma distribution. We extend this elegant ratio form to the one-parameter family with $g(x) = x - 1/3 + c_1/x$, in which $c_1 = 0$ gives Kerman's. For large shape parameters along any ray $β= rα$ with $r > 1$, the resulting approximation is eventually a lower bound on the median if $c_1 \\le 8/405$ and an upper bound if $c_1 \\ge 1/45$. We recommend $c_1 = 1/36$ for its simplicity: it makes $g(x) = (x - 1/6)^2/x$ a perfect square over $x$, and it is within 1% of the minimax value of $c_1$ for $α, β\\ge 1$ under several error measures, while the theoretically distinguished values $8/405$ and $1/45$ are less accurate. The approximation is then $((α- 1/6)^2/α)/((α- 1/6)^2/α+ (β- 1/6)^2/β)$. Its relative error lies between -0.171% and +0.212% for all $1 \\le α\\le β$, compared with up to -3.8% for Kerman's. It is more accurate than Kerman's approximation wherever the latter is defined, with relative error never more than 13/32 of Kerman's, the ratio approached asymptotically. The same $g(α)$ is a useful approximation to the gamma distribution median, overestimating it by at most 0.212% for shape parameter $α\\ge 1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33859",
  "author_total": 1,
  "authors": [
    {
      "name": "Richard F. Lyon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/66q7gagn1d4pj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33859",
  "fulltext_url": null,
  "id": "66q7gagn1d4pj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33859",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Statistical Distribution Estimation and Applications"
    ]
  ],
  "openalex": "W7224022749",
  "pdf_url": "https://arxiv.org/pdf/2610.33859",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/66q7gagn1d4pj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33859",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33859"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "A simple improvement on Kerman's approximation to the median of the beta distribution",
  "title_text": "A simple improvement on Kerman's approximation to the median of the beta distribution",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:49Z",
  "url": "https://research.draconitic.com/paper/66q7gagn1d4pj",
  "venue": "arXiv",
  "versions": []
}