Stein's method for comparison of univariate distributions

By Christophe Ley, Gesine Reinert, Yvik Swan. 2014

161 citations recorded by the source corpus.

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, Frechet and Gumbel.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c698e7043a46e0d702a961e5f76803ab02c0f652

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In Artificial Intelligence

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

Measuring Sample Quality with Kernels (2017, methodology): “…widely applicable univariate Stein operator is the density method operator (see Stein et al., 2004; Chat-terjee & Shao, 2011; Chen et al., 2011; Ley et al., 2017), Inspired by the generator method of Barbour (1988; 1990) and G ¨ otze (1991), Gorham & Mackey (2015) generalized this operator to…”
Measuring Sample Quality with Diffusions (2016, background): “, [12, 11] for explicit bounds or [53] for a recent review).”
Convergence rates for a class of estimators based on Stein’s method (2016, methodology): “rator method of Barbour (1988) applied to an overdamped Langevin diusion (Gorham and Mackey, 2015). Our denition generalises the original construction of Stein (1972) and also generalises Def. 1 in Ley et al. (2016) to consider multi-variate random variables. For functional approximation we follow A”