Abstract
In this article we present a class of mixed Poisson regression models with varying dispersion arising from non-conjugate to the Poisson mixing distributions for modelling overdispersed claim counts in non-life insurance. The proposed family of models combined with the adopted modelling framework can provide sufficient flexibility for dealing with different levels of overdispersion. For illustrative purposes, the Poisson-lognormal regression model with regression structures on both its mean and dispersion parameters is employed for modelling claim count data from a motor insurance portfolio. Maximum likelihood estimation is carried out via an expectation-maximization type algorithm, which is developed for the proposed family of models and is demonstrated to perform satisfactorily.
Highlights
During the last three decades, mixed Poisson regression models have been applied in various fields of studies, including non-life insurance for modelling overdispersed claim count data
The members of this family of models, which can be constructed based on a mixing distribution which is conjugate to the Poisson distribution, such as the negative binomial and the Poisson–inverse-Gaussian, have been the most popular choices due to the simplicity of their log-likelihood functions, which can be maximized using the standard maximum likelihood (ML) estimation approach
The ML estimates of the parameters for the negative binomial type I (NBI) and PLN regression models with varying dispersion and the zero-inflated Poisson (ZIP)
Summary
During the last three decades, mixed Poisson regression models have been applied in various fields of studies, including non-life insurance for modelling overdispersed claim count data. The class of mixed Poisson models we consider is very wide and the flexibility it provides in (i) the distributional choice for the mixing density and (ii) modelling jointly the mean and dispersion parameters as parametric functions of risk factors allows us to add the required amount of weight to the right tail area of the claim count distribution for accommodating different levels of overdispersion, resulting in an improved risk evaluation. At this point, it should be noted that, with the exception of very few articles, such as those by [8,9,10], modelling jointly all the parameters of mixed Poisson models in terms of explanatory variables has not been explored in depth.
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