The Use of Generalized Means in the Estimation of the Weibull Tail Coefficient

Abstract

Due to the specificity of the Weibull tail coefficient, most of the estimators available in the literature are based on the log excesses and are consequently quite similar to the estimators used for the estimation of a positive extreme value index. The interesting performance of estimators based on generalized means leads us to base the estimation of the Weibull tail coefficient on the power mean-of-order-$p$. Consistency and asymptotic normality of the estimators under study are put forward. Their performance for finite samples is illustrated through a Monte Carlo simulation. It is always possible to find a negative value of$p$(contrarily to what happens with the mean-of-order-$p$estimator for the extreme value index), such that, for adequate values of the threshold, there is a reduction in both bias and root mean square error.