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

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arxiv 2103.08846 v2 pith:KMXLMIZP submitted 2021-03-16 math.ST math.PRstat.TH

classification math.STmath.PRstat.TH
keywords binomialnegativehspaceasymptoticscontinuitycorrectiondistributionfind
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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.

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  1. The mean absolute deviation of the classical discrete distributions: collapse identities, complete asymptotic expansions, and enveloping series

    math.PR 2026-08 conditional novelty 6.0 of 10

    A common telescoping identity yields closed forms, complete asymptotics, and enveloping series for the mean absolute deviation of the four classical discrete laws.

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