Robust heavy-tailed versions of generalized linear models with applications in actuarial science

Generalized linear models (GLMs) form one of the most popular classes of models in statistics. The gamma variant is used, for instance, in actuarial science for the modelling of claim amount in insurance. A flaw of GLMs is that they are not robust against outliers (i.e., against erroneous or extreme data points). A difference in trends in the bulk of the data and the outliers thus yields skewed inference and prediction. Cantoni and Ronchetti (2001) proposed a robust frequentist approach which is now the most commonly applied. It consists in an estimator which is derived from a modification of the derivative of the log-likelihood. We propose an approach which is modelling-based and thus fundamentally different. It allows for an understanding and interpretation of the modelling, and it can be applied for both frequentist and Bayesian statistical analyses. We show that the approach possesses appealing theoretical and empirical properties. In particular, we show through a simulation study that it offers an advantage in terms of estimation performance.