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