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

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 $\mathrm{Negative}\mathrm{Binomial}(r,p)$ random variable jittered by a $\mathrm{Uniform}(0,1)$, which answers a problem left open in Coeurjolly & Trépanier (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.

21 pages, 3 figures

References in corpus (2)

• On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators
• Closed-form Tight Bounds and Approximations for the Median of a Gamma Distribution

Cited by in corpus (4)

• Refined normal approximations for the central and noncentral chi-square distributions and some applications
• Asymptotic comparison of negative multinomial and multivariate normal experiments
• Moments of the negative multinomial distribution
• Normal approximations for the multivariate inverse Gaussian distribution and asymmetric kernel smoothing on $d$-dimensional half-spaces