Probability Surveys

Stein’s method for comparison of univariate distributions

Christophe Ley, Gesine Reinert, and Yvik Swan

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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.

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Probab. Surveys Volume 14 (2017), 1-52.

Received: April 2016
First available in Project Euclid: 9 January 2017

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

Density approach Stein’s method comparison of distributions


Ley, Christophe; Reinert, Gesine; Swan, Yvik. Stein’s method for comparison of univariate distributions. Probab. Surveys 14 (2017), 1--52. doi:10.1214/16-PS278.

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