Estimation of the extreme value index in a censorship framework: asymptotic and finite sample behaviour

TL;DR

This paper introduces a new class of estimators for the extreme value index under heavy censoring, establishing their asymptotic properties and finite sample behavior, especially when censoring exceeds 50%.

Contribution

It derives the asymptotic normality of a broad class of estimators for censored heavy-tailed data, extending previous results and analyzing their finite sample performance.

Findings

New estimators with good bias properties.

Asymptotic normality established for heavy censoring cases.

Finite sample behavior analyzed, especially under high censoring.

Abstract

We revisit the estimation of the extreme value index for randomly censored data from a heavy tailed distribution. We introduce a new class of estimators which encompasses earlier proposals given in Worms and Worms (2014) and Beirlant et al. (2018), which were shown to have good bias properties compared with the pseudo maximum likelihood estimator proposed in Beirlant et al. (2007) and Einmahl et al. (2008). However the asymptotic normality of the type of estimators first proposed in Worms and Worms (2014) was still lacking, in the random threshold case. We derive an asymptotic representation and the asymptotic normality of the larger class of estimators and consider their finite sample behaviour. Special attention is paid to the case of heavy censoring, i.e. where the amount of censoring in the tail is at least 50\%. We obtain the asymptotic normality with a classical $\sqrt{k}$ rate where $k$ denotes the number of top data used in the estimation, depending on the degree of censoring.