On Perturbations of Stein Operator

TL;DR

This paper develops a Stein operator for sums of independent variables as a perturbation of the negative binomial operator, providing improved error bounds and applications to waiting time distributions.

Contribution

It introduces a novel Stein operator for sums of independent variables as a perturbation of the NB operator and improves existing approximation bounds.

Findings

Derived error bounds for total variation distance

Proposed a three-parameter approximation method

Applied results to waiting time functions for (k1,k2)-events

Abstract

In this paper, we obtain Stein operator for sum of $n$ independent random variables (rvs) which is shown as perturbation of negative binomial (NB) operator. Comparing the operator with NB operator, we derive the error bounds for total variation distance by matching parameters. Also, three parameters approximation for such a sum is considered and is shown to improve the existing bounds in the literature. Finally, an application of our results to a function of waiting time for $(k_1,k_2)$-events is given.