Maximum likelihood estimation for N‐mixture models

The focus of this article is on the nature of the likelihood associated with N‐mixture models for repeated count data. It is shown that the infinite sum embedded in the likelihood associated with the Poisson mixing distribution can be expressed in terms of a hypergeometric function and, thence, in closed form. The resultant expression for the likelihood can be readily computed to a high degree of accuracy and is algebraically tractable. Specifically, the likelihood equations can be simplified to some advantage, the concentrated likelihood in the probability of detection formulated and problematic cases identified. The results are illustrated by means of a simulation study and a real world example. The study is extended to N‐mixture models with a negative binomial mixing distribution and results similar to those for the Poisson case obtained. N‐mixture models with mixing distributions which accommodate excess zeros and, separately, with a beta‐binomial distribution rather than a binomial used to model the intra‐site counts are also investigated. However the results for these settings, while computationally attractive, do not provide insight into the nature of the maximum likelihood estimates.