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.

翻译：本文证明了负二项分布的局部极限定理和一种精炼的连续性校正方法。我们展示了该结果的两个应用：首先，我们找到了经$\mathrm{Uniform}\hspace{0.5mm}(0,1)$扰动后的$\mathrm{Negative\hspace{0.5mm}Binomial}\hspace{0.2mm}(r,p)$随机变量中位数的渐近性质，这解答了Coeurjolly与Trépanier（2020）中遗留的未解决问题。该成果用于构建参数$p$（当$r>0$已知时）的一种简洁、稳健且相合的估计量，并简要讨论了$r$未知的情形。其次，我们给出了负二项分布实验与正态分布实验之间Le Cam距离的上界。