Goodness-of-fit tests and applications for left-truncated Weibull distributions to non-life insurance

In risk theory with application to insurance, the identification of the relevant distributions for both the counting and the claim size processes from given observations is of major importance. In some situations left-truncated distributions can be used to model, not only the single claim severity, but also the inter-arrival times between two consecutive claims. We show that left-truncated Weibull distributions are particularly relevant, especially for the claim severity distribution. For that, we first demonstrate how the parameters can be estimated consistently from the data, and then show how a Kolmogorov-Smirnov goodness-of-fit test can be set up using modified critical values. These critical values are universal to all left-truncated Weibull distributions, independent of the actual Weibull parameters. To illustrate our findings we analyse three applications using real insurance data, one from a Swiss excess of loss treaty over automobile insurance, another from an American private passenger automobile insurance and a third from earthquake inter-arrival times in California.