A refined continuity correction for the negative binomial distribution and asymptotics of the median

TL;DR

This paper develops a refined continuity correction and local limit theorem for the negative binomial distribution, providing new asymptotics for the median and bounds on experiment distances, with applications in parameter estimation.

Contribution

It introduces a refined continuity correction and local limit theorem for the negative binomial distribution, solving an open problem on median asymptotics and establishing bounds on experiment distances.

Findings

Asymptotics of the median for jittered negative binomial variables derived.

A simple estimator for the parameter p is constructed and shown to be consistent.

An upper bound on the Le Cam distance between negative binomial and normal experiments established.

Abstract

In this paper, we prove a local limit theorem and a refined continuity correction for the negative binomial distribution. We present two applications of the results. First, we find the asymptotics of the median for a $\mathrm{Negative\hspace{0.5mm}Binomial}\hspace{0.2mm}(r,p)$ random variable jittered by a $\mathrm{Uniform}\hspace{0.2mm}(0,1)$, which answers a problem left open in Coeurjolly & Tr\'epanier (2020). This is used to construct a simple, robust and consistent estimator of the parameter $p$, when $r > 0$ is known. The case where $r$ is unknown is also briefly covered. Second, we find an upper bound on the Le Cam distance between negative binomial and normal experiments.