A Discrete Distribution Arising as a Solution of a Linear Difference Equation: Extension of the Non Central Negative Binomial Distribution

Abstract

This article considers a discrete distribution that arises as the dominant solution of a linear difference equation. Basic properties and various chance mechanisms that lead to this distribution are given. In particular, its formulation as a weighted distribution and a mixed Poisson process are proposed. Parameter estimation by (a) using a combination of observed frequencies and moments and (b) maximum likelihood are examined. An example of goodness of fit is considered.

Keywords:

• Goodness of fit
• Index of dispersion
• Mixed, compound, generalized distributions
• Mixed Poisson and birth–death with immigration processes
• Parameter estimation
• Weighted distribution

Mathematics Subject Classification:

• Primary 60E05
• Secondary 62F10, 62P99

Acknowledgment

The authors wish to thank the referee and Associate Editor for their insightful comments which led to a greatly improved article.

Notes

NNBD:  = 0.2403,  = 0.0001,  = 2.3811. Extended NNBD:  = 0.1023,  = 0.0235,  = 5.1605,  = 14.7001.