The use of generalized linear models (GLM) to estimate claims reserves has become a standard method in insurance. Most frequently, the exponential dispersion family (EDF) is used; see e.g. England, Verrall. We study the so-called Tweedie EDF and test the sensitivity of the claims reserves and their mean square error of predictions (MSEP) over this family. Furthermore, we develop second order Taylor approximations for the claims reserves and the MSEPs for members of the Tweedie family that are difficult to obtain in practice, but are close enough to models for which claims reserves and MSEP estimations are easy to determine. As a result of multiple case studies, we find that claims reserves estimation is relatively insensitive to which distribution is chosen amongst the Tweedie family, in contrast to the MSEP, which varies widely.

We reconsider the problem of producing fair and accurate tariffs based on aggregated insurance data giving numbers of claims and total costs for the claims. Jørgensen and de Souza (Scand Actuarial J., 1994) assumed Poisson arrival of claims and gamma distributed costs for individual claims. Jørgensen and de Souza (1994) directly modelled the risk or expected cost of claims per insured unit, μ say. They observed that the dependence of the likelihood function on μ is as for a linear exponential family, so that modelling similar to that of generalized linear models is possible. In this paper we observe that, when modelling the cost of insurance claims, it is generally necessary to model the dispersion of the costs as well as their mean. In order to model the dispersion we use the framework of double generalized linear models. Modelling the dispersion increases the precision of the estimated tariffs. The use of double generalized linear models also allows us to handle the case where only the total cost of claims and not the number of claims has been recorded.

A practical method of allowing for covariates in compound Poisson modelling distributions is discussed.