Censored and extreme losses: functional convergence and applications to tail goodness-of-fit

TL;DR

This paper develops new theoretical tools to analyze the tail behavior of censored losses, enabling improved tail goodness-of-fit testing and parameter estimation in heavy-tailed insurance data.

Contribution

It establishes the functional convergence of extreme tail estimators for censored data and introduces new rules for selecting order statistics based on these results.

Findings

Convergence of Extreme Nelson–Aalen and Kaplan–Meier estimators established.

New rules for selecting order statistics improve tail analysis.

Application to real insurance data demonstrates practical utility.

Abstract

This paper establishes the functional convergence of the Extreme Nelson--Aalen and Extreme Kaplan--Meier estimators, which are designed to capture the heavy-tailed behaviour of censored losses. The resulting limit representations can be used to obtain the distributions of pathwise functionals with respect to the so-called tail process. For instance, we may recover the convergence of a censored Hill estimator, and we further investigate two goodness-of-fit statistics for the tail of the loss distribution. Using the the latter limit theorems, we propose two rules for selecting a suitable number of order statistics, both based on test statistics derived from the functional convergence results. The effectiveness of these selection rules is investigated through simulations and an application to a real dataset comprised of French motor insurance claim sizes.