On magic factors in Stein’s method for compound Poisson approximation

Fraser Daly

doi:10.1214/17-ecp101

Abstract

One major obstacle in applications of Stein’s method for compound Poisson approximation is the availability of so-called magic factors (bounds on the solution of the Stein equation) with favourable dependence on the parameters of the approximating compound Poisson random variable. In general, the best such bounds have an exponential dependence on these parameters, though in certain situations better bounds are available. In this paper, we extend the region for which well-behaved magic factors are available for compound Poisson approximation in the Kolmogorov metric, allowing useful compound Poisson approximation theorems to be established in some regimes where they were previously unavailable. To illustrate the advantages offered by these new bounds, we consider applications to runs, reliability systems, Poisson mixtures and sums of independent random variables.

Highlights

In recent years, Stein’s method has proved to be a versatile technique for proving explicit compound Poisson approximation results in a wide variety of settings; see [2] and references therein for an introduction to these techniques and a discussion of several applications
We extend the region for which well-behaved magic factors are available for compound Poisson approximation in the Kolmogorov metric, allowing useful compound Poisson approximation theorems to be established in some regimes where they were previously unavailable
To illustrate the advantages offered by these new bounds, we consider applications to runs, reliability systems, Poisson mixtures and sums of independent random variables

Summary

Introduction

Stein’s method has proved to be a versatile technique for proving explicit compound Poisson approximation results in a wide variety of settings; see [2] and references therein for an introduction to these techniques and a discussion of several applications. Barbour and Utev [3] use Fourier techniques to relax the condition (1.4) somewhat, and establish Stein factors for compound Poisson approximation in Kolmogorov distance of a better order than is generally available. Their bound on M1(K) again includes an undesirable logarithmic term, which can only be removed at the cost of a significantly increased constant in the bound. 1.1 can be used to generalize and relax the condition (1.7) when finding reasonable Stein factors for compound Poisson approximation in Kolmogorov distance This will extend the applicability of various compound Poisson approximation results in the literature, allowing approximation theorems with a reasonable error bound to be established for previously inaccessible parameter values.

A generalization of the Barbour–Xia condition

Reliability

Relaxing the Barbour–Xia condition

Mixed Poisson distributions

Independent summands