On estimation of loss distributions and risk measures

The estimation of certain loss distribution and analyzing its properties is a key issue in several finance mathematical and actuarial applications. It is common to apply the tools of extreme value theory and generalized Pareto distribution in problems related to heavy-tailed data. Our main goal is to study third party liability claims data obtained from Estonian Traffic Insurance Fund (ETIF). The data is quite typical for insurance claims containing very many observations and being heavytailed. In our approach the fitting consists of two parts: for main part of the distribution we use lognormal fit (which was the most suitable based on our previous studies) and a generalized Pareto distribution is used for the tail. Main emphasis of the fitting techniques is on the proper threshold selection. We seek for stability of parameter estimates and study the behaviour of risk measures at a wide range of thresholds. Two related lemmas will be proved.