A Non-uniform Bound on Negative Binomial Approximation via Stein’s Method and z-functions

In this article, Stein’s method and z-functions are used to determine a non-uniform bound for approximating the cumulative distribution function of a nonnegative integer-valued random variable X by the negative binomial cumulative distribution function with parameters r∈R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\in {\mathbb {R}}^+$$\end{document} and p=1-q∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=1-q\in (0,1)$$\end{document}. This bound is an appropriate criterion for evaluating the accuracy of this approximation. The result obtained in this study is used to approximate cumulative distribution functions including the negative hypergeometric cumulative distribution function, the Pólya cumulative distribution function, and the beta negative binomial cumulative distribution function.