Models of Insurance Claim Counts with Time Dependence Based on Generalization of Poisson and Negative Binomial Distributions

Longitudinal data (or panel data) consist of repeated observations of individual units that are observed over time. Each individual insured is assumed to be independent but correlation between contracts of the same individual is permitted. This paper presents an exhaustive overview of models for panel data that consist of generalizations of count distributions where the dependence between contracts of the same insureds can be modeled with Bayesian and frequentist models, based on generalization of Poisson and negative binomial distributions. This paper introduces some of those models to actuarial science and compares the fitting with specification tests for nested and non-nested models. It also shows why some intuitive models (past experience as regressors, multivariate distributions, or copula models) involving time dependence cannot be used to model the number of reported claims. We conclude that the random effects models have a better fit than the other models examined here because the fitting is improved and it allows for more flexibility in computing the next year's premium.