{
  "id": "2206.10236",
  "version": "v1",
  "published": "2022-06-21T10:21:40.000Z",
  "updated": "2022-06-21T10:21:40.000Z",
  "title": "Mixture representations of noncentral distributions",
  "authors": [
    "Ludwig Baringhaus",
    "Rudolf Grübel"
  ],
  "journal": "Commun. Stat., Theory Methods 50, No. 24, 5997-6013 (2021)",
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "With any symmetric distribution $\\mu$ on the real line we may associate a parametric family of noncentral distributions as the distributions of $(X+\\delta)^2$, $\\delta\\not=0$, where $X$ is a random variable with distribution $\\mu$. The classical case arises if $\\mu$ is the standard normal distribution, leading to the noncentral chi-squared distributions. It is well-known that these may be written as Poisson mixtures of the central chi-squared distributions with odd degrees of freedom. We obtain such mixture representations for the logistic distribution and for the hyperbolic secant distribution. We also derive alternative representations for chi-squared distributions and relate these to representations of the Poisson family. While such questions originated in parametric statistics they also appear in the context of the generalized second Ray-Knight theorem, which connects Gaussian processes and local times of Markov processes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-21T10:21:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62E10",
      "60E05"
    ],
    "keywords": [
      "mixture representations",
      "noncentral distributions",
      "connects gaussian processes",
      "generalized second ray-knight theorem",
      "standard normal distribution"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}