Gaussian approximation to the extreme value index estimator of a heavy-tailed distribution under random censoring

TL;DR

This paper develops a Gaussian approximation for the Hill estimator used in heavy-tailed distributions with censored data, establishing its asymptotic normality under relaxed assumptions.

Contribution

It introduces a Gaussian process approximation for the censored data Hill estimator and relaxes previous assumptions on distribution functions and sample fractions.

Findings

The estimator is asymptotically normal under second-order regular variation.

The methodology applies to a broader class of heavy-tailed distributions.

Provides a foundation for more flexible inference in censored heavy-tail analysis.

Abstract

We make use of the empirical process theory to approximate the adapted Hill estimator, for censored data, in terms of Gaussian processes. Then, we derive its asymptotic normality, only under the usual second-order condition of regular variation. Our methodology allows to relax the assumptions, made in Einmahl, Fils-Villetard and Guillou(2008), on the heavy-tailed distribution functions and the sample fraction of upper order statistics.