Stein's method and stochastic orderings

TL;DR

This paper combines Stein's method with stochastic orderings to improve approximation techniques for distributions like Poisson and polynomial birth-death, especially for dependent Bernoulli sums.

Contribution

It introduces a stochastic ordering approach within Stein's method framework, extending approximation techniques to dependent variables and more general distributions.

Findings

Enhanced Poisson approximation for dependent Bernoulli sums

Application to k-runs in i.i.d. Bernoulli trials

Extension to polynomial birth-death distributions

Abstract

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 approximation of a sum of dependent Bernoulli random variables, for example k-runs in i.i.d. Bernoulli trials. Other applications include approximation by polynomial birth--death distributions.