{
  "id": "1907.02006",
  "version": "v1",
  "published": "2019-07-03T16:01:16.000Z",
  "updated": "2019-07-03T16:01:16.000Z",
  "title": "Bounding quantiles of Wasserstein distance between true and empirical measure",
  "authors": [
    "Samuel N. Cohen",
    "Martin N. A. Tegnér",
    "Johannes Wiesel"
  ],
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Consider the empirical measure, $\\hat{\\mathbb{P}}_N$, associated to $N$ i.i.d. samples of a given probability distribution $\\mathbb{P}$ on the unit interval. For fixed $\\mathbb{P}$ the Wasserstein distance between $\\hat{\\mathbb{P}}_N$ and $\\mathbb{P}$ is a random variable on the sample space $[0,1]^N$. Our main result is that its normalised quantiles are asymptotically maximised when $\\mathbb{P}$ is a convex combination between the uniform distribution supported on the two points $\\{0,1\\}$ and the uniform distribution on the unit interval $[0,1]$. This allows us to obtain explicit asymptotic confidence regions for the underlying measure $\\mathbb{P}$. We also suggest extensions to higher dimensions with numerical evidence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-03T16:01:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "62G15",
      "60E15"
    ],
    "keywords": [
      "wasserstein distance",
      "empirical measure",
      "bounding quantiles",
      "unit interval",
      "explicit asymptotic confidence regions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}