Lígia Henriques-Rodrigues (School of Sciences and Technology and CIMA, University of Évora, Portugal)

Abstract: The Weibull tail-coefficient (WTC) is a pivotal parameter in extreme value statistics, particularly for distributions with Weibull-type tails. Various distributions, such as the normal, Gamma, Weibull, and Logistic distributions, exhibit such tail behavior. The WTC, represented by $\theta$, serves as a parameter in a right-tail function formulated as $\overline F(x) :=1-F(x) =: {\rm e}^{-H(x)}$, where $H(x)=-\ln(1-F(x))$ denotes a regularly varying cumulative hazard function with an index of regular variation equal to $1/\theta$, where $\theta\in\mathbb{R}^{+}$.
 Commonly used WTC estimators in the literature are typically defined as functions of the log-excesses, establishing a close connection with 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 shown promise in estimating the EVI, resulting in reduced bias and root mean square error for suitable threshold values. In this study, we propose and explore new classes of WTC estimators based on the power mean of the log-excesses and on the power $p$ of the log-excesses within a second-order framework. The performance of these novel estimators is assessed through an extensive Monte-Carlo simulation study.  Joint work with Frederico Caeiro and M. Ivette Gomes
Keywords: Power mean-of-order $p$, Semi-parametric estimation, Statistics of Extremes, Weibull tail-coefficient.
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