GAMLSS for Longitudinal Multivariate Claim Count Models

By generalizing traditional regression frameworks, generalized additive models for location, scale, and shape (GAMLSSs) allow parametric or semiparametric modeling of one or more parameters of distributions that are not members of the linear exponential family. Consequently, these GAMLSS approaches offer an interesting theoretical framework to allow the use of several potentially helpful distributions in actuarial science. GAMLSS theory is coupled with longitudinal approaches for counting data because these approaches are essential to predictive pricing models. Indeed, they are mainly known for modeling the dependence between the number of claims from the contracts of the same insured over time. Considering that the models' cross-sectional counterparts have been successfully applied in actuarial work and the importance of longitudinal models, we show that the proposed approach allows one to quickly implement multivariate longitudinal models with nonparametric terms for ratemaking. This semiparametric modeling is illustrated using a dataset from a major insurance company in Canada. An analysis is then conducted on the improvement of predictive power that the use of historical data and nonparametric terms in the modeling allows. In addition, we found that the weight of past experience in bonus-malus predictive premiums analysis is reduced in comparison with a parametric model and that this method could help for continuous covariate segmentation. Our approach differs from previous studies because it does not use any simplifying assumptions as to the value of the a priori explanatory variables and because we have carried out a predictive pricing integrating nonparametric terms within the framework of the GAMLSS in an explicit way, which makes it possible to reproduce the same type of study using other distributions.