Distributional Transformations, Orthogonal Polynomials, and Stein Characterizations

By Larry Goldstein, Gesine Reinert. 2005

67 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f781c445d15cc60de97d6b9a12257ae7956aa33a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Discrete Mathematics and Combinatorics

In Statistics and Probability

Citations in the corpus

Stein's Method and the Laplace Distribution (2012, methodology): “Specifically, using the notation of Theorem 3, one can show that X ) is of the form U1Y where U1 and Y are independent with U1 ∼ Uniform(0, 1) and Y having density fY (y) = |y| E|X|fX(y), and XL is of the form U2Z where U2 and Z are independent with U2 ∼ Uniform(0, 1) and Z having density fZ(z) = |z”
Stein couplings for normal approximation (2010, methodology): “Based on an exchangeable pair, Goldstein and Reinert (2005b) propose a way to construct the zero bias W z.”
Berry Esseen Bounds for Combinatorial Central Limit Theorems and Pattern Occurrences, using Zero and Size Biasing (2005, background): “In general, such a Y (m) exists when H and Y satisfy certain sign change and orthogonality properties, as discussed in [16].”
Stein’s density approach and information inequalities (2012, background): “8) belongs to the family of so-called “Stein-type identities” discussed for instance in [17, 8, 1].”