Barbour introduced a probabilistic view of Stein's method for estimating the error in probability approximations. However, in the case of approximations by general distributions on the integers, there have been no purely probabilistic proofs of Stein's bounds till this paper. Furthermore, the methods introduced here apply to a very large class of approximating distributions on the non-negative integers, among which there is a natural class for higher-order approximations by probability distributions rather than signed measures (as previously). The methods also produce Stein magic factors for process approximations which do not increase with the window of observation and which are simpler to apply than those in Brown, Weinberg and Xia.

/pdf/stein-s-method-and-birth-death-processes-51crrfmsvy.pdf

Stein's Method and Birth-Death Processes

Using Stein's method techniques, we develop a framework which allows one to bound the error terms arising from approximation by the Laplace distribution and apply it to the study of random sums of mean zero random variables. As a corollary, we deduce a Berry-Esseen type theorem for the convergence of certain geometric sums. Our results make use of a second order characterizing equation and a distributional transformation which is related to zero-biasing. 1. Background and Introduction Beginning with the publication of Charles Stein's seminal paper Stein (1972), probabilists and statisticians have developed a wide range of techniques based on characterizing equations for bounding the distance between the distribution of a random variable X and that of a random variable Z having some specied

Stein's Method and the Laplace Distribution

Using a characterizing equation for the beta distribution, Stein's method is applied to obtain bounds of the optimal order for the Wasserstein distance between the distribution of the scaled number of white balls drawn from a Polya-Eggenberger urn and its limiting beta distribution. The bound is computed by making a direct comparison between characterizing operators of the target and the beta distribution, the former derived by extending Stein's density approach to discrete distributions. In addition, refinements are given to Dobler's (2012) result for the arcsine approximation for the fraction of time a simple random walk of even length spends positive, and so also to the distributions of its last return time to 0 and its first visit to its terminal point, by supplying explicit constants to the present Wasserstein bound and also demonstrating that its rate is of the optimal order.

/pdf/stein-s-method-for-the-beta-distribution-and-the-polya-4z0jeh49t0.pdf

Stein's method for the Beta distribution and the Pólya-Eggenberger urn

Several characterizations of the Generalized Inverse Gaussian (GIG) distribution on the positive real line have been proposed in the literature, especially over the past two decades. These characterization theorems are surveyed, and two new characterizations are established, one based on maximum likelihood estimation and the other is a Stein characterization.

/pdf/characterizations-of-gig-laws-a-survey-10kul90a00.pdf

Characterizations of GIG laws: A survey

A stochastic ordering approach is applied with Stein's method for approximation by the equilibrium distribution of a birth-death process. The usual stochastic order and the more general s-convex orders are discussed. Attention is focused on Poisson and translated Poisson approximations of a sum of dependent Bernoulli random variables, for example, k-runs in independent and identically distributed Bernoulli trials. Other applications include approximation by polynomial birth-death distributions.

Stein's method and stochastic orderings

Bounds on the rate of convergence to the negative binomial distribution are found, where this rate is measured by the total variation distance between probability laws. For an arbitrary discrete random variable written as a sum of indicators, an upper bound of coupling form is expressed as an average of terms each of which measures the difference between the effect of particular indicator being one and the value of a geometrically distributed random variable. When a monotone coupling exists a lower bound can also be shown. Application of these results is illustrated with the example of the Po´lya distribution for which the rate of approach to the negative binomial limit is found.

M. J. Phillips

Papers

Negative Binomial Approximation with Stein's Method