A basic problem in the statistical analysis of scientific data consists of developing models for the analysis of so-called count data -- data consisting of the numbers of events of some type occurring over a given period of time. Examples include the number of incidents involving damage to ships of a specified type over a given time period, the number of crimes committed per year in the life course of a career criminal, and the number of firms of a given type founded per year in an economy. To account for factors that increase or decrease the rates of occurrence of such events, regression models of the Poisson or Poisson family (where the latter consists of mixtures of Poisson regressions) typically are formulated. The most commonly used mixed Poisson regression models -- which offer considerably more flexibility than the basic Poisson model -- are based on the specification of a parametric mixing distribution such as the gamma distribution. A new methodology for mixed Poisson regressions -- based on a nonparametric specification of the mixing distribution and the semiparametric maximum likelihood estimation of the model parameters -- has been defined and explored by the Principal Investigators. This project will systematically refine, reexamine, and more fully develop statistical procedures of estimation, testing, and model comparison for this new class of models. A computer simulation study will be conducted to examine the properties of the statistical estimators for relatively small samples. Four prototypical datasets from diverse subject matters will be analyzed to demonstrate the techniques and to make comparisons with existing analyses. Computer software for the empirical application of the techniques will be further developed. Because these new statistical models provide more flexibility in modeling event count data than conventional models, the methods to be developed could improve data analyses across a variety of scientific disciplines.
