We propose a new general version of Stein's method for univariatedistributions. In particular we propose a canonical definition of the Steinoperator of a probability distribution {which is based on a linear differenceor differential-type operator}. The resulting Stein identity highlights theunifying theme behind the literature on Stein's method (both for continuous anddiscrete distributions). Viewing the Stein operator as an operator acting onpairs of functions, we provide an extensive toolkit for distributionalcomparisons. Several abstract approximation theorems are provided. Our approachis illustrated for comparison of several pairs of distributions : normal vsnormal, sums of independent Rademacher vs normal, normal vs Student, andmaximum of random variables vs exponential, Frechet and Gumbel.
展开▼
