{
  "id": "2306.17014",
  "version": "v1",
  "published": "2023-06-29T15:07:57.000Z",
  "updated": "2023-06-29T15:07:57.000Z",
  "title": "Poisson and Gaussian approximations of the power divergence family of statistics",
  "authors": [
    "Fraser Daly"
  ],
  "comment": "14 pages",
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Consider the family of power divergence statistics based on $n$ trials, each leading to one of $r$ possible outcomes. This includes the log-likelihood ratio and Pearson's statistic as important special cases. It is known that in certain regimes (e.g., when $r$ is of order $n^2$ and the allocation is asymptotically uniform as $n\\to\\infty$) the power divergence statistic converges in distribution to a linear transformation of a Poisson random variable. We establish explicit error bounds in the Kolmogorov (or uniform) metric to complement this convergence result, which may be applied for any values of $n$, $r$ and the index parameter $\\lambda$ for which such a finite-sample bound is meaningful. We further use this Poisson approximation result to derive error bounds in Gaussian approximation of the power divergence statistics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-29T15:07:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62E17",
      "60F05"
    ],
    "keywords": [
      "gaussian approximation",
      "power divergence family",
      "power divergence statistic converges",
      "establish explicit error bounds",
      "important special cases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}