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

Abstract

We propose a new version of Stein’s method of exchangeable pairs, which, given a suitable exchangeable pair (W,Wʹ) 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 γ and η are motivated by two regression properties satisfied by the pair (W,Wʹ). 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 [5] and [13], 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,Wʹ) and provide new bounds on the solution to the Stein equation for the Beta distribution, which complement the existing bounds from [18]. 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.