Maximum Likelihood Estimation for Generalized Inflated Power Series Distributions

In this paper we first define the class of Generalized Inflated Power Series Distributions (GIPSDs) which contain the inflated discrete distributions most often seen in practice as special cases. We describe the hitherto unkown exponential family structure of GIPSDs and use this to derive closed-form, easy to program, conditional and unconditional maximum likelihood estimators for essentially any number of parameters. We also show how the GIPSD exponential family can be extended to model deflated mass points. Our results provide easy access to likelihood-based inference and automated model selection procedures for GIPSDs that only involve one-dimensional numerical root-finding problems that are easily solved with simple routines. We consider four real-data examples which illustrate the utility and scope of our results.