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


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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Christophe Ley. Gesine Reinert. Yvik Swan. "Stein’s method for comparison of univariate distributions." Probab. Surveys 14 1 - 52, 2017.


Received: 1 April 2016; Published: 2017
First available in Project Euclid: 9 January 2017

zbMATH: 06673431
MathSciNet: MR3595350
Digital Object Identifier: 10.1214/16-PS278

Primary: 60B10
Secondary: 60E05 , 60E15 , 60F05

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

Rights: Copyright © 2017 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.14 • 2017
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