Semiparametric Regression Models for Claims Reserving and Credibility: the Mixed Model Approach

Verrall (1996) and England & Verrall (2001) considered the use of smoothing methods in the context of claims reserving, by applying two smoothing procedures in a likelihood-based way, namely the locally weighted regression smoother (‘loess’) and the cubic smoothing spline smoother. Using the statistical methodology of semiparametric regression and its connection with mixed models (see e.g. Ruppert et al., 2003), this paper revisits smoothing models for loss reserving and considers their use in an example from credibility. Next to the flexibility of a semiparametric regression model, advantages of the presented approach are threefold. Firstly, because the constructed semiparametric models have an interpretation as (generalized) linear mixed models ((G)LMMs), standard statistical theory and software for (G)LMMs can be used. Secondly, a Bayesian implementation of these smoothing models is relatively straightforward and allows simulation from the full predictive distribution of quantities of interest. Since actuaries are interested in predictions, this is a major advantage. Thirdly, more complicated statistical models, dealing for example with semicontinuous data or extensive longitudinal data, can be handled within the same framework. Throughout this work, data examples illustrate these different aspects. Evidently, the methodology is not restricted to the problems discussed in this paper, but is relevant for other kinds of actuarial regression problems.