A COMPARISON OF POISSON, NEGATIVE BINOMIAL, AND SEMIPARAMETRIC MIXED POISSON REGRESSION-MODELS - WITH EMPIRICAL APPLICATIONS TO CRIMINAL CAREERS DATA

Specifications and moment properties of the univariate Poisson and neg ative binomial distributions me briefly reviewed and illustrated. prop erties and limitations of the corresponding Poisson and negative binom ial (gamma mixtures of Poissons) regression models are described it is shown how a misspecification of the mixing distribution of a mixed Po isson model to accommodate hidden heterogeneity ascribable to unobserv ed variables-although not affecting the consistency of maximum likelih ood estimators of the Poisson mean rate parameter or its regression pa rameterization-can lead to inflated t ratios of regression coefficient s and associated incorrect inferences. Then the recently developed sem iparametric maximum likelihood estimator for regression models compose d of arbitrary mixtures of Poisson processes is specified and further developed It is concluded that the semiparametric mixed Poisson regres sion model adds considerable flexibility to Poisson-family regression models and provides opportunities for interpretation of empirical patt erns not available in the conventional approaches.