{
  "id": "2205.09266",
  "version": "v1",
  "published": "2022-05-19T01:02:27.000Z",
  "updated": "2022-05-19T01:02:27.000Z",
  "title": "Exact lower and upper bounds for shifts of Gaussian measures",
  "authors": [
    "Iosif Pinelis"
  ],
  "comment": "10 pages; to appear in Theory of Probability and Its Applications",
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Exact upper and lower bounds on the ratio $\\mathsf{E}w(\\mathbf{X}-\\mathbf{v})/\\mathsf{E}w(\\mathbf{X})$ for a centered Gaussian random vector $\\mathbf{X}$ in $\\mathbb{R}^n$, as well as bounds on the rate of change of $\\mathsf{E}w(\\mathbf{X}-t\\mathbf{v})$ in $t$, where $w\\colon\\mathbb{R}^n\\to[0,\\infty)$ is any even unimodal function and $\\mathbf{v}$ is any vector in $\\mathbb{R}^n$. As a corollary of such results, exact upper and lower bounds on the power function of statistical tests for the mean of a multivariate normal distribution are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-19T01:02:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E15",
      "62E17",
      "62H10",
      "62H15",
      "26B25",
      "26D10",
      "26D15",
      "28C20"
    ],
    "keywords": [
      "gaussian measures",
      "exact lower",
      "upper bounds",
      "lower bounds",
      "exact upper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}