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A unified approach to Stein characterizations

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  • Christophe Ley
  • Yvik Swan

Abstract

This article deals with Stein characterizations of probability distributions. We provide a general framework for interpreting these in terms of the parameters of the underlying distribution. In order to do so we introduce two concepts (a class of functions and an operator) which generalize those which were developed in the 70’s by Charles Stein and Louis Chen for characterizing the Gaussian and the Poisson distributions. Our methodology (i) allows for writing many (if not all) known univariate Stein characterizations, (ii) permits to identify clearly minimal conditions under which these results hold and (iii) provides a straightforward tool for constructing new Stein characterizations. Our parametric interpretation of Stein characterizations also raises a number of questions which we outline at the end of the paper.

Suggested Citation

  • Christophe Ley & Yvik Swan, 2011. "A unified approach to Stein characterizations," Working Papers ECARES 2013/88988, ULB -- Universite Libre de Bruxelles.
    • Handle: RePEc:eca:wpaper:2013/88988
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    File URL: https://dipot.ulb.ac.be/dspace/bitstream/2013/88988/4/Stein.pdf
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    References listed on IDEAS

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    1. Yosef Rinott & Vladimir Rotar, 2000. "Normal approximations by Stein's method," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 23(1), pages 15-29.
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    Cited by:

    1. Steffen Betsch & Bruno Ebner, 2021. "Fixed point characterizations of continuous univariate probability distributions and their applications," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(1), pages 31-59, February.

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