Bivariate Mixed Poisson Regression Models with Varying Dispersion

George Tzougas*, Alice Pignatelli di Cerchiara

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)
49 Downloads (Pure)


The main purpose of this article is to present a new class of bivariate mixed Poisson regression models with varying dispersion that offers sufficient flexibility for accommodating overdispersion and accounting for the positive correlation between the number of claims from third-party liability bodily injury and property damage. Maximum likelihood estimation for this family of models is achieved through an expectation-maximization algorithm that is shown to have a satisfactory performance when three members of this family, namely, the bivariate negative binomial, bivariate Poisson–inverse Gaussian, and bivariate Poisson–Lognormal distributions with regression specifications on every parameter are fitted on two-dimensional motor insurance data from a European motor insurer. The a posteriori, or bonus-malus, premium rates that are determined by these models are calculated via the expected value and variance principles and are compared to those based only on the a posteriori criteria. Finally, we present an extension of the proposed approach with varying dispersion by developing a bivariate Normal copula-based mixed Poisson regression model with varying dispersion and dependence parameters. This approach allows us to consider the influence of individual and coverage-specific risk factors on the mean, dispersion, and copula parameters when modeling different types of claims from different types of coverage. For expository purposes, the Normal copula paired with negative binomial distributions for marginals and regressors on the mean, dispersion, and copula parameters is fitted on a simulated dataset via maximum likelihood.

Original languageEnglish
Pages (from-to)211-241
Number of pages31
JournalNorth American Actuarial Journal
Issue number2
Early online date30 Oct 2021
Publication statusPublished - 3 Apr 2023


  • Statistics, Probability and Uncertainty
  • Economics and Econometrics
  • Statistics and Probability

ASJC Scopus subject areas

  • Economics and Econometrics
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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