  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.
