Modelling heavy-tailed insurance claim data using the hyper-erlang distribution with common scale parameter

Abstract

When modelling positively skewed insurance claim data, traditional distributions such as lognormal and Weibull often fail to accurately estimate the tail. Several methods have been developed to improve tail estimation without compromising the body fitting, including the transformed kernel density and the generalised lambda distribution. In this study, we investigate the robustness of a promising method, fitting of the hyper-Erlang distribution with a common scale parameter. A modified version of the Expectation Maximisation (EM) algorithm is used for distribution fitting, with some changes we proposed to improve the efficiency of the algorithm. Results from a preliminary study we conducted suggest that different initial estimates of the common scale parameter affect the performance of the modified EM algorithm. For fitting medical claim data provided by the Society of Actuaries, bootstrap samples are taken to determine an optimal initial estimate for the scale parameter. With this estimate, the hyper-Erlang distribution is able to provide a satisfactory fit to the data. The result is comparable to those produced by transformed kernel density and generalised lambda distribution.