Extreme value analysis of actuarial risks: estimation and model validation

TL;DR

This paper reviews statistical methods for analyzing extreme risks in actuarial science, emphasizing empirical process theory for estimation and validation, and discusses multivariate tail dependence and real insurance data application.

Contribution

It introduces advanced empirical process techniques for extreme value estimation and model validation, and explores limitations of classical multivariate models with alternative approaches.

Findings

Empirical process theory effectively analyzes extreme risks.

Classical multivariate models have limitations in tail dependence.

Application to health insurance data demonstrates practical utility.

Abstract

We give an overview of several aspects arising in the statistical analysis of extreme risks with actuarial applications in view. In particular it is demonstrated that empirical process theory is a very powerful tool, both for the asymptotic analysis of extreme value estimators and to devise tools for the validation of the underlying model assumptions. While the focus of the paper is on univariate tail risk analysis, the basic ideas of the analysis of the extremal dependence between different risks are also outlined. Here we emphasize some of the limitation of classical multivariate extreme value theory and sketch how a different model proposed by Ledford and Tawn can help to avoid pitfalls. Finally, these theoretical results are used to analyze a data set of large claim sizes from health insurance.