{
  "id": "1007.3662",
  "version": "v2",
  "published": "2010-07-21T13:58:46.000Z",
  "updated": "2011-05-03T12:42:33.000Z",
  "title": "An extended Stein-type covariance identity for the Pearson family with applications to lower variance bounds",
  "authors": [
    "G. Afendras",
    "N. Papadatos",
    "V. Papathanasiou"
  ],
  "comment": "Published in at http://dx.doi.org/10.3150/10-BEJ282 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)",
  "journal": "Bernoulli 2011, Vol. 17, No. 2, 507-529",
  "doi": "10.3150/10-BEJ282",
  "categories": [
    "math.ST",
    "stat.TH"
  ],
  "abstract": "For an absolutely continuous (integer-valued) r.v. $X$ of the Pearson (Ord) family, we show that, under natural moment conditions, a Stein-type covariance identity of order $k$ holds (cf. [Goldstein and Reinert, J. Theoret. Probab. 18 (2005) 237--260]). This identity is closely related to the corresponding sequence of orthogonal polynomials, obtained by a Rodrigues-type formula, and provides convenient expressions for the Fourier coefficients of an arbitrary function. Application of the covariance identity yields some novel expressions for the corresponding lower variance bounds for a function of the r.v. $X$, expressions that seem to be known only in particular cases (for the Normal, see [Houdr\\'{e} and Kagan, J. Theoret. Probab. 8 (1995) 23--30]; see also [Houdr\\'{e} and P\\'{e}rez-Abreu, Ann. Probab. 23 (1995) 400--419] for corresponding results related to the Wiener and Poisson processes). Some applications are also given.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-05-03T12:42:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extended stein-type covariance identity",
      "application",
      "pearson family",
      "expressions",
      "natural moment conditions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.3662A"
    }
  }
}