Polynomial Birth–Death Distribution Approximation in the Wasserstein Distance

Abstract

The polynomial birth-death distribution (abbr. as PBD) on I = {0, 1, 2, ...} or I = {0, 1, 2, ..., m} for some finite m introduced in Brown & Xia (2001) is the equilibrium distribution of the birth-death process with birth rates {α i } and death rates {β i }, where α i ≥ 0 and β i ≥ 0 are polynomial functions of i ∈ I. The family includes Poisson, negative binomial, binomial and hypergeometric distributions. In this paper, we give probabilistic proofs of various Stein's factors for the PBD approximation with α i = a and β i = i + bi(i − 1) in terms of the Wasserstein distance. The paper complements the work of Brown & Xia (2001) and generalizes the work of Barbour & Xia (2006) where Poisson approximation (b = 0) in the Wasserstein distance is investigated. As an application, we establish an upper bound for the Wasserstein distance between the PBD and Poisson binomial distribution and show that the PBD approximation to the Poisson binomial distribution is much more precise than the approximation by the Poisson or shifted Poisson distributions.

A Pólya Approximation to the Poisson-Binomial Law

“…There are several directions in which hopefully our result can be extended. In particular, previous total variation approximations to the sum of independent Bernoulli variables have been extended to the Wasserstein metric [4], [36], generalised Poisson-binomial distributions [10], [11], [12], sums of independent Bernoulli vectors [1], Bernoulli processes [35], and sums of dependent Bernoulli variables [6], [27], [30]. In addition, the Pólya distribution is a threeparameter, quadratic polynomial birth-death distribution as defined in [8] and also a generalized hypergeometric factorial moment distribution as defined in [20]; thus, in combining what is known about the apparently similar families of distributions, there is the potential to increase the accuracy of approximations through deducing similar results for higher-order polynomials (see [29] for a more detailed sketch).…”

A Pólya Approximation to the Poisson-Binomial Law

“…There are several directions in which hopefully our result can be extended. In particular, previous total variation approximations to the sum of independent Bernoulli variables have been extended to the Wasserstein metric [4], [36], generalised Poisson-binomial distributions [10], [11], [12], sums of independent Bernoulli vectors [1], Bernoulli processes [35], and sums of dependent Bernoulli variables [6], [27], [30]. In addition, the Pólya distribution is a threeparameter, quadratic polynomial birth-death distribution as defined in [8] and also a generalized hypergeometric factorial moment distribution as defined in [20]; thus, in combining what is known about the apparently similar families of distributions, there is the potential to increase the accuracy of approximations through deducing similar results for higher-order polynomials (see [29] for a more detailed sketch).…”