Distributional Transformations, Orthogonal Polynomials, and Stein Characterizations

Larry Goldstein and Gesine Reinert ()
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Larry Goldstein: University of Southern California
Gesine Reinert: University of Oxford

Journal of Theoretical Probability, 2005, vol. 18, issue 1, 237-260

Abstract: A new class of distributional transformations is introduced, characterized by equations relating function weighted expectations of test functions on a given distribution to expectations of the transformed distribution on the test function’s higher order derivatives. The class includes the size and zero bias transformations, and when specializing to weighting by polynomial functions, relates distributional families closed under independent addition, and in particular the infinitely divisible distributions, to the family of transformations induced by their associated orthogonal polynomial systems. For these families, generalizing a well known property of size biasing, sums of independent variables are transformed by replacing summands chosen according to a multivariate distribution on its index set by independent variables whose distributions are transformed by members of that same family. A variety of the transformations associated with the classical orthogonal polynomial systems have as fixed points the original distribution, or a member of the same family with different parameter.

Keywords: Orthogonal polynomials; size bias; zero bias; Stein’s method; distributional fixed points. (search for similar items in EconPapers)
Date: 2005
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10959-004-2602-6

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