{
  "id": "0810.4821",
  "version": "v1",
  "published": "2008-10-27T14:27:52.000Z",
  "updated": "2008-10-27T14:27:52.000Z",
  "title": "Estimation of distributions, moments and quantiles in deconvolution problems",
  "authors": [
    "Peter Hall",
    "Soumendra N. Lahiri"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/07-AOS534 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Statistics 2008, Vol. 36, No. 5, 2110-2134",
  "doi": "10.1214/07-AOS534",
  "categories": [
    "math.ST",
    "stat.TH"
  ],
  "abstract": "When using the bootstrap in the presence of measurement error, we must first estimate the target distribution function; we cannot directly resample, since we do not have a sample from the target. These and other considerations motivate the development of estimators of distributions, and of related quantities such as moments and quantiles, in errors-in-variables settings. We show that such estimators have curious and unexpected properties. For example, if the distributions of the variable of interest, $W$, say, and of the observation error are both centered at zero, then the rate of convergence of an estimator of the distribution function of $W$ can be slower at the origin than away from the origin. This is an intrinsic characteristic of the problem, not a quirk of particular estimators; the property holds true for optimal estimators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-10-27T14:27:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62G20",
      "62C20"
    ],
    "keywords": [
      "deconvolution problems",
      "estimation",
      "target distribution function",
      "property holds true",
      "first estimate"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Institute of Mathematical Statistics",
      "journal": "Ann. Stat."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0810.4821H"
    }
  }
}