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

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 Negative Binomial (r,p) random variable jittered by a Uniform (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. Second, we find an upper bound on the Le Cam distance between negative binomial and normal experiments.