One family, six distributions -- A flexible model for insurance claim severity

We propose a new class of claim severity distributions with six parameters, that has the standard two-parameter distributions, the log-normal, the log-Gamma, the Weibull, the Gamma and the Pareto, as special cases. This distribution is much more flexible than its special cases, and therefore more able to to capture important characteristics of claim severity data. Further, we have investigated how increased parameter uncertainty due to a larger number of parameters affects the estimate of the reserve. This is done in a large simulation study, where both the characteristics of the claim size distributions and the sample size are varied. We have also tried our model on a set of motor insurance claims from a Norwegian insurance company. The results from the study show that as long as the amount of data is reasonable, the five- and six-parameter versions of our model provide very good estimates of both the quantiles of the claim severity distribution and the reserves, for claim size distributions ranging from medium to very heavy tailed. However, when the sample size is small, our model appears to struggle with heavy-tailed data, but is still adequate for data with more moderate tails.