{
  "id": "1201.5507",
  "version": "v1",
  "published": "2012-01-26T13:20:26.000Z",
  "updated": "2012-01-26T13:20:26.000Z",
  "title": "Uniform in bandwidth exact rates for a class of kernel estimators",
  "authors": [
    "Davit Varron",
    "Ingrid Van Keilegom"
  ],
  "comment": "Published in the Annals of the Institute of Statistical Mathematics Volume 63, p. 1077-1102 (2011)",
  "categories": [
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Given an i.i.d sample $(Y_i,Z_i)$, taking values in $\\RRR^{d'}\\times \\RRR^d$, we consider a collection Nadarya-Watson kernel estimators of the conditional expectations $\\EEE(<c_g(z),g(Y)>+d_g(z)\\mid Z=z)$, where $z$ belongs to a compact set $H\\subset \\RRR^d$, $g$ a Borel function on $\\RRR^{d'}$ and $c_g(\\cdot),d_g(\\cdot)$ are continuous functions on $\\RRR^d$. Given two bandwidth sequences $h_n<\\wth_n$ fulfilling mild conditions, we obtain an exact and explicit almost sure limit bounds for the deviations of these estimators around their expectations, uniformly in $g\\in\\GG,\\;z\\in H$ and $h_n\\le h\\le \\wth_n$ under mild conditions on the density $f_Z$, the class $\\GG$, the kernel $K$ and the functions $c_g(\\cdot),d_g(\\cdot)$. We apply this result to prove that smoothed empirical likelihood can be used to build confidence intervals for conditional probabilities $\\PPP(Y\\in C\\mid Z=z)$, that hold uniformly in $z\\in H,\\; C\\in \\CC,\\; h\\in [h_n,\\wth_n]$. Here $\\CC$ is a Vapnik-Chervonenkis class of sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-01-26T13:20:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bandwidth exact rates",
      "collection nadarya-watson kernel estimators",
      "mild conditions",
      "sure limit bounds",
      "build confidence intervals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.5507V"
    }
  }
}