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

2024/08/11 by Martin Bladt, Bladt, Martin, Christoffer Øhlenschlæger +1
Computer Science · #Computability, Logic, AI Algorithms #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2408.05862

openalex publication_date 2024/08/11 · openalex created_date 2024/12/16 · openalex updated_date 2026/07/28

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.

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