Integer-valued autoregressive processes with dynamic heterogeneity and their applications in automobile insurance

Bonus-malus systems in automobile insurance describe how the past claim frequencies determine the future insurance premiums. The potential risks of the policyholders vary due to differences in driving behavior, which leads to the unobserved heterogeneity in individual average claim counts. While the Poisson distribution has been used as a simple model for discrete count data, the negative binomial distribution is suggested for modeling the claim counts with unobserved heterogeneity by letting the mean parameter of the Poisson distribution follow a Gamma distribution. In this project, we introduce an integer-valued autoregressive process with dynamic heterogeneity to model the random fluctuations and correlations of the heterogeneity from year to year. Some properties of the model are studied, and a bonus-malus system is built and illustrated using the Gibb's Sampler algorithm. Finally, comparisons with other existing models are provided in terms of the extent to which they use the claim history.