Tail empirical process and weighted extreme value index estimator for randomly right-censored data

TL;DR

This paper introduces a new tail empirical process and a weighted Hill-type estimator for heavy-tailed, right-censored data, demonstrating improved performance and applications in survival analysis and goodness-of-fit testing.

Contribution

It develops a novel tail empirical process and a weighted estimator for the extreme value index under censorship, with proven consistency, asymptotic normality, and practical advantages.

Findings

Estimator outperforms existing methods in bias and MSE

Application to Australian AIDS patient survival data

Provides a goodness-of-fit test for Pareto-like models

Abstract

A tail empirical process for heavy-tailed and right-censored data is introduced and its Gaussian approximation is established. In this context, a (weighted) new Hill-type estimator for positive extreme value index is proposed and its consistency and asymptotic normality are proved by means of the aforementioned process in the framework of second-order conditions of regular variation. In a comparative simulation study, the newly defined estimator is seen to perform better than the already existing ones in terms of both bias and mean squared error. As a real data example, we apply our estimation procedure to evaluate the tail index of the survival time of Australian male Aids patients. It is noteworthy that our approach may also serve to develop other statistics related to the distribution tail such as second-order parameter and reduced-bias tail index estimators. Furthermore, the proposed tail empirical process provides a goodness-of-fit test for Pareto-like models under censorship.