Electronic Journal of Probability

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

Christian Döbler

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We propose a new 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 suitable absolutely continuous distribution. This distribution is characterized by a first order linear differential Stein operator, whose coefficients $\gamma$ and $\eta$ 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 previous works, which only consider the framework of the density approach, i.e. $\eta\equiv1$. 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 complement the existing bounds from previous works. The abstract plug-in result is then applied to derive bounds of order $n^{-1}$ 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 measured by a certain class of smooth test functions.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 109, 34 pp.

Accepted: 19 October 2015
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60E99: None of the above, but in this section

Stein's method exchangeable pairs Beta distribution Pólya urn model

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Döbler, Christian. Stein's method of exchangeable pairs for the Beta distribution and generalizations. Electron. J. Probab. 20 (2015), paper no. 109, 34 pp. doi:10.1214/EJP.v20-3933. https://projecteuclid.org/euclid.ejp/1465067215

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