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Maximum likelihood estimation for N ‐mixture models
Author(s) -
Haines Linda M.
Publication year - 2016
Publication title -
biometrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.298
H-Index - 130
eISSN - 1541-0420
pISSN - 0006-341X
DOI - 10.1111/biom.12521
Subject(s) - mixing (physics) , mathematics , poisson distribution , likelihood function , negative binomial distribution , mixture model , binomial (polynomial) , hypergeometric distribution , maximum likelihood , binomial distribution , statistics , beta binomial distribution , hypergeometric function , marginal likelihood , restricted maximum likelihood , zero inflated model , count data , poisson regression , mathematical analysis , population , physics , demography , quantum mechanics , sociology
Summary 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.