Stationary birth-death processes generating inflation-deflation distributions: Avoiding the issue of dominance

A mixture of two or more count distributions has become deeply embedded in the analysis of excess counts, often relative to the stationary (equilibrium) distributions of birth-death processes such as the geometric, Poisson, Poisson-Lindley (PL), negative binomial (NB), hyper-Poisson (HP), and Conway-Maxwell-Poisson (CMP) distributions. However, the mechanism by which excess counts arise–namely, through modifications of the birth and death rates in the base distributions–has not yet been directly examined in the research literature. All well-known inflation mixture distributions are, in fact, parameterizations of the stationary distributions of birth-death processes. Thus, although the resulting distributions share the same shapes, they arise from distinct mechanisms and are not equivalent in regression analyses. This paper focuses on inflation-deflation stationary distributions arising from modified birth-death processes that form an exponential family and introduces two types of such distributions.