Stein's method of exchangeable pairs for the Beta distribution and generalizations

By C. Dobler. 2014

68 citations recorded by the source corpus.

Abstract. We propose a version of Stein’s method of exchangeable pairs, which, given a suitable exchangeable pair (W,W ) of real-valued random variables, suggests the approximation of the law of W by a certain absolutely continuous distribution, which is characterized by a first order differential Stein operator, whose coefficients γ and η are motivated by two regression properties satisfied by the pair (W,W ). Furthermore, the general theory of Stein’s method for such an absolutely continuous distribution is developed and a general characterization result as well as general bounds on the solution to the Stein equation are given. This abstract approach is a certain extension of the theory developed in the papers [CS11] and [EL10], which only consider the framework of the density approach, i.e. η ≡ 1. As an illustration of our technique we prove a general plug-in result, which bounds a certain distance of the distribution of a given random variable W to a Beta distribution in terms of a given exchangeable pair (W,W ) and provide new bounds on the solution to the Stein equation for the Beta distribution, which in a way complement the existing bounds from [GR13]. The abstract plug-in result is then applied to derive bounds of order n for the distance between the distribution of the relative number of drawn red balls after n drawings in a Pólya urn model and the limiting Beta distribution.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9235492bf2c6f282cb65035e004ba1e9cabeb75f

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Citations in the corpus

Chi-square approximation by Stein's method with application to Pearson's statistic (2015, methodology): “2 of [10] (the technique has since been further developed in [11]).”
Approximation of stable law in Wasserstein-1 distance by Stein’s method (2017, methodology): “Nowadays, Stein’s method has been extended and refined by many authors and become a very important tool for getting bounds of measure approximations, see [6, 10, 14, 22, 24, 29, 39, 46, 26, 27, 28, 33, 42, 43, 44, 47].”
On Stein's method for products of normal random variables and zero bias couplings (2013, methodology): “Since then, Stein’s method has been extended to many other distributions, such as the Poisson [5], beta [8], [25], gamma [21], [29], [31], exponential [4], [34], Laplace [9], [36] and variance-gamma [13], [19]; for an overview see Reinert [37].”
Measuring Sample Quality with Stein's Method (2015, unknown): “5, 1) for P = Unif(0, 1) and c1:3 = (1, 4, 2) for P = N (0, 1) [10, 24].”
Stein's method for comparison of univariate distributions (2014, methodology): “See also [30, 58, 28, 65] for computations under quite general assumptions on the density of X1.”