{
  "id": "math/0410170",
  "version": "v1",
  "published": "2004-10-06T15:52:22.000Z",
  "updated": "2004-10-06T15:52:22.000Z",
  "title": "Weighted uniform consistency of kernel density estimators",
  "authors": [
    "Evarist Gine",
    "Vladimir Koltchinskii",
    "Joel Zinn"
  ],
  "comment": "Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000063",
  "journal": "Annals of Probability 2004, Vol. 32, No. 3B, 2570-2605",
  "doi": "10.1214/009117904000000063",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let f_n denote a kernel density estimator of a continuous density f in d dimensions, bounded and positive. Let \\Psi(t) be a positive continuous function such that \\|\\Psi f^{\\beta}\\|_{\\infty}<\\infty for some 0<\\beta<1/2. Under natural smoothness conditions, necessary and sufficient conditions for the sequence \\sqrt\\frac{nh_n^d}{2|\\log h_n^d|}\\|\\Psi(t)(f_n(t)-Ef_n(t))\\|_{\\infty} to be stochastically bounded and to converge a.s. to a constant are obtained. Also, the case of larger values of \\beta is studied where a similar sequence with a different norming converges a.s. either to 0 or to +\\infty, depending on convergence or divergence of a certain integral involving the tail probabilities of \\Psi(X). The results apply as well to some discontinuous not strictly positive densities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-10-06T15:52:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62G07",
      "60F15",
      "62G20"
    ],
    "keywords": [
      "kernel density estimator",
      "weighted uniform consistency",
      "natural smoothness conditions",
      "similar sequence",
      "larger values"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....10170G"
    }
  }
}