A unified approach to Stein characterizations

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.

References in corpus (4)

• Exponential Approximation by Stein's Method and Spectral Graph Theory
• Compound Poisson Approximation via Information Functionals
• Stein's method for discrete Gibbs measures
• On a connection between Stein characterizations and Fisher information

Cited by in corpus (4)

• Fixed point characterizations of continuous univariate probability distributions and their applications
• Characterizations of non-normalized discrete probability distributions and their application in statistics
• Discrete Stein characterizations and discrete information distances
• Variance-Gamma approximation via Stein's method