Improvements in the estimation of the Weibull tail coefficient -- a comparative study

TL;DR

This paper introduces new classes of Weibull tail-coefficient estimators based on power transformations of log-excesses, evaluated through extensive simulations to improve estimation accuracy in extreme value analysis.

Contribution

It proposes novel WTC-estimators using power of log-excesses within a second-order framework, enhancing estimation performance over existing methods.

Findings

New estimators show reduced bias and RMSE in simulations.

Comparative analysis demonstrates improved accuracy over traditional estimators.

Performance varies with threshold choice and distribution type.

Abstract

The Weibull tail-coefficient (WTC) plays a crucial role in extreme value statistics when dealing with Weibull-type tails. Several distributions, such as normal, Gamma, Weibull, and Logistic distributions, exhibit this type of tail behaviour. The WTC, denoted by $\theta$, is a parameter in a right-tail function of the form $ \bar F(x) :=1-F(x) =: {\rm e}^{-H(x)}$, where $H(x)=-\ln(1-F(x))$ represents a regularly varying cumulative hazard function with an index of regular variation equal to 1/$\theta$, $\theta\in\mathbb{R}^{+}$. The commonly used WTC-estimators in literature are often defined as functions of the log-excesses, making them closely related to estimators of the extreme value index (EVI) for Pareto-type tails. For a positive EVI, the classical estimator is the Hill estimator. Generalized means have been successfully employed in estimating the EVI, leading to reduction of bias and of root mean square error for appropriate threshold values. In this study, we propose and investigate new classes of WTC-estimators based on power $p$ of the log-excesses within a second-order framework. The performance of these new estimators is evaluated through a large-scale Monte-Carlo simulation study, comparing them with existing WTC-estimators available in the literature.