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Characterizations of distributions by variance bounds

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  • Cacoullos, T.
  • Papathanasiou, V.

Abstract

The distribution of a continuous r.v. X is characterized by the function w appearing in the lower bound [sigma]2E2[w(X)g'(X)] for the variance of a function g(X); for a discrete X, g'(x) is replaced by [Delta]g(x) = g(x + 1) - g(x). The same characterizations are obtained by considering the upper bound [sigma]2E{w(X)[g'(X)]2} [greater-or-equal, slanted] Var[g(X)]. The special case w(x) = 1 gives the normal, Borovkov and Utev (1983), and the Poisson, Prakasa Rao and Sreehari (1987). The results extend to independent random variables.

Suggested Citation

  • Cacoullos, T. & Papathanasiou, V., 1989. "Characterizations of distributions by variance bounds," Statistics & Probability Letters, Elsevier, vol. 7(5), pages 351-356, April.
    • Handle: RePEc:eee:stapro:v:7:y:1989:i:5:p:351-356
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    Cited by:

    1. Mikami, Toshio, 2004. "Covariance kernel and the central limit theorem in the total variation distance," Journal of Multivariate Analysis, Elsevier, vol. 90(2), pages 257-268, August.
    2. Mikami, Toshio, 1998. "Equivalent conditions on the central limit theorem for a sequence of probability measures on R," Statistics & Probability Letters, Elsevier, vol. 37(3), pages 237-242, March.
    3. Salinelli, Ernesto, 2009. "Nonlinear principal components, II: Characterization of normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 652-660, April.
    4. Arras, Benjamin & Azmoodeh, Ehsan & Poly, Guillaume & Swan, Yvik, 2019. "A bound on the Wasserstein-2 distance between linear combinations of independent random variables," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2341-2375.
    5. Christophe Ley & Gesine Reinert & Yvik Swan, 2014. "Approximate Computation of Expectations: the Canonical Stein Operator," Working Papers ECARES ECARES 2014-36, ULB -- Universite Libre de Bruxelles.
    6. Giorgos Afendras & Vassilis Papathanasiou, 2014. "A note on a variance bound for the multinomial and the negative multinomial distribution," Naval Research Logistics (NRL), John Wiley & Sons, vol. 61(3), pages 179-183, April.
    7. Papadatos, N. & Papathanasiou, V., 1996. "A generalization of variance bounds," Statistics & Probability Letters, Elsevier, vol. 28(2), pages 191-194, June.
    8. Balakrishnan, N. & Cramer, E. & Kamps, U., 2001. "Bounds for means and variances of progressive type II censored order statistics," Statistics & Probability Letters, Elsevier, vol. 54(3), pages 301-315, October.
    9. Landsman, Zinoviy & Vanduffel, Steven & Yao, Jing, 2015. "Some Stein-type inequalities for multivariate elliptical distributions and applications," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 54-62.
    10. Arakelian, V. & Papathanasiou, V., 2004. "On bounding the absolute mean value," Statistics & Probability Letters, Elsevier, vol. 69(4), pages 447-450, October.

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    Keywords

    characterizations variance bounds;

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