Stein’s method for comparison of univariate distributions. (English) Zbl 1406.60010

Ley, Christophe; Reinert, Gesine; Swan, Yvik

Probab. Surv. 14, 1-52 (2017).

Summary: We propose a new general version of Stein’s method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution which is based on a linear difference or differential-type operator. The resulting Stein identity highlights the unifying theme behind the literature on Stein’s method (both for continuous and discrete distributions). Viewing the Stein operator as an operator acting on pairs of functions, we provide an extensive toolkit for distributional comparisons. Several abstract approximation theorems are provided. Our approach is illustrated for comparison of several pairs of distributions: normal vs normal, sums of independent Rademacher vs normal, normal vs Student, and maximum of random variables vs exponential, Fréchet and Gumbel.

MSC:

60B10	Convergence of probability measures
60E15	Inequalities; stochastic orderings
60E05	Probability distributions: general theory
60F05	Central limit and other weak theorems

Keywords:

density approach; Stein’s method; comparison of distributions