The non-central negative binomial distribution: Further properties and applications

Abstract

The non-central negative binomial distribution is both a mixed and compound Poisson distribution with applications in photon and neural counting, statistical optics, astronomy and a stochastic reversible counter system. In this paper various important probabilistic properties of the non-central negative binomial distribution in practical applications like log-concavity, discrete self-decomposability, unimodality, asymptotic behavior and tail length of the probability distribution have been derived. The construction as a mixed Poisson process by specifying a joint distribution for the inter-arrival times and its application is illustrated by a fit to real life data set.

Keywords:

• Mixed and compound Poisson processes
• photon and neural counting
• log-concavity
• discrete self-decomposability
• unimodality
• goodness-of-fit

AMS subject classification:

• 60E05
• 62-07
• 62F03
• 62F10
• 62H12
• 62H15
• 62H20

Acknowledgments

The authors wish to thank the referees for their insightful comments and thorough reading of the manuscript which have greatly improved the paper.