Alternative modelling and inference methods for claim size distributions

TL;DR

This paper introduces novel methods for modeling claim size distributions, especially in the upper tail, by adjusting base distributions like the Pareto with mixtures and smoothing techniques, enhancing flexibility and interpretability.

Contribution

It extends traditional tail modeling by proposing mixture and smoothing approaches for better fit and introduces a modified estimation method and tests for model validation.

Findings

New mixture models improve tail fit for claim sizes.

Modified estimation method enhances parameter accuracy.

Proposed tests effectively distinguish Pareto from alternative models.

Abstract

The upper tail of a claim size distribution of a property line of business is frequently modelled by Pareto distribution. However, the upper tail does not need to be Pareto distributed, extraordinary shapes are possible. Here, the opportunities for the modelling of loss distributions are extended. The basic idea is the adjustment of a base distribution for their tails. The (generalised) Pareto distribution is used as base distribution for different reasons. The upper tail is in the focus and can be modelled well for special cases by a discrete mixture of the base distribution with a combination of the base distribution with an adapting distribution via the product of their survival functions. A kind of smoothed step is realised in this way in the original line function between logarithmic loss and logarithmic exceedance probability. The lower tail can also be adjusted. The new approaches offer the opportunity for stochastic interpretation and are applied to observed losses. For parameter estimation, a modification of the minimum Anderson Darling distance method is used. A new test is suggested to exclude that the observed upper tail is better modelled by a simple Pareto distribution. Q-Q plots are applied, and secondary results are also discussed.