On Stein operators for discrete approximations

TL;DR

This paper introduces a novel method using probability generating functions to derive Stein operators for various discrete distributions, enhancing approximation techniques and providing bounds for sums of indicators.

Contribution

It presents new Stein operators for related distributions and compound cases, and applies a perturbation approach to improve total variation bounds.

Findings

Derived Stein operators for Poisson, binomial, and negative binomial distributions.

Established bounds for distribution approximations using Stein's method.

Illustrated the method's effectiveness with binomial-Poisson convolutions.

Abstract

In this paper, a new method based on probability generating functions is used to obtain multiple Stein operators for various random variables closely related to Poisson, binomial and negative binomial distributions. Also, Stein operators for certain compound distributions, where the random summand satisfies Panjer's recurrence relation, are derived. A well-known perturbation approach for Stein's method is used to obtain total variation bounds for the distributions mentioned above. The importance of such approximations is illustrated, for example, by the binomial convoluted with Poisson approximation to sums of independent and dependent indicator random variables.