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 Probability Surveys > Vol. 14 (2017) open journal systems 

Stein's method for comparison of univariate distributions

Christophe Ley, Ghent University
Gesine Reinert, University of Oxford
Yvik Swan, Université de Liège

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.

AMS 2000 subject classifications: Primary 60B10; secondary 60E15, 60E05, 60F05.

Keywords: Density approach, Stein’s method, comparison of distributions.

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Ley, Christophe, Reinert, Gesine, Swan, Yvik, Stein's method for comparison of univariate distributions, Probability Surveys, 14, (2017), 1-52 (electronic). DOI: 10.1214/16-PS278.


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