{
  "id": "math/0609020",
  "version": "v2",
  "published": "2006-09-01T08:41:37.000Z",
  "updated": "2008-06-17T14:12:12.000Z",
  "title": "Current status data with competing risks: Consistency and rates of Convergence of the MLE",
  "authors": [
    "Piet Groeneboom",
    "Marloes H. Maathuis",
    "Jon A. Wellner"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/009053607000000974 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. 3, 1031-1063",
  "doi": "10.1214/009053607000000974",
  "categories": [
    "math.ST",
    "stat.TH"
  ],
  "abstract": "We study nonparametric estimation of the sub-distribution functions for current status data with competing risks. Our main interest is in the nonparametric maximum likelihood estimator (MLE), and for comparison we also consider a simpler ``naive estimator.'' Both types of estimators were studied by Jewell, van der Laan and Henneman [Biometrika (2003) 90 183--197], but little was known about their large sample properties. We have started to fill this gap, by proving that the estimators are consistent and converge globally and locally at rate $n^{1/3}$. We also show that this local rate of convergence is optimal in a minimax sense. The proof of the local rate of convergence of the MLE uses new methods, and relies on a rate result for the sum of the MLEs of the sub-distribution functions which holds uniformly on a fixed neighborhood of a point. Our results are used in Groeneboom, Maathuis and Wellner [Ann. Statist. (2008) 36 1064--1089] to obtain the local limiting distributions of the estimators.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-06-17T14:12:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62N01",
      "62G20",
      "62G05"
    ],
    "keywords": [
      "current status data",
      "competing risks",
      "convergence",
      "local rate",
      "consistency"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Institute of Mathematical Statistics",
      "journal": "Ann. Stat."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......9020G"
    }
  }
}