{
  "id": "1611.03573",
  "version": "v1",
  "published": "2016-11-11T03:23:25.000Z",
  "updated": "2016-11-11T03:23:25.000Z",
  "title": "Asymptotic expansions of the inverse of the Beta distribution",
  "authors": [
    "Dimitris Askitis"
  ],
  "comment": "33 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper, we shall study the asymptotic behaviour of the $p$-inverse of the Beta distribution, i.e. the quantity $q$ defined implicitly by $\\int_0^q t^{a - 1} (1 - t)^{b - 1} \\text{d} t = p B (a, b)$, as a function of the first parameter $a$. In particular, we study the monotonicity and limits of $q (a)$, as well as of $\\varphi (a) = - a \\log q (a)$ and we derive asymptotic expansions of $\\varphi$ and $q$ at $0$ and $\\infty$. Moreover, we prove some general results on $-inverses, with some interest on their own and we provide some relations between Bell and N{\\o}rlund Polynomials, a generalisation of Bernoulli numbers. Finally, we provide Maple and Sage algorithms for computing the terms of the asymptotic expansions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-11T03:23:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A60",
      "33B15",
      "60E05",
      "11B68"
    ],
    "keywords": [
      "beta distribution",
      "sage algorithms",
      "first parameter",
      "derive asymptotic expansions",
      "general results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}