Estimation of the Weibull tail coefficient through the power-mean-of-order p

Abstract

The Weibull tail coefficient (WTC) is the parameter θ θ in a right-tail function of the type F¯:=1−F F¯:=1−F, such that H:=−ln F¯ is a regularly varying function at infinity with an index of regular variation equal to θ∈R+. In a context of extreme value theory for maxima, it is possible to prove that we have an extreme value index (EVI) ξ=0, but usually a very slow rate of convergence. Most of the recent WTC-estimators are proportional to the class of Hill EVI-estimators, the average of the log-excesses associated with the k upper order statistics, 1≤k<n. The interesting performance of EVI-estimators based on generalized means leads us to base the WTC-estimation on the power mean-of-order-p (MOp) EVI-estimators. Consistency of the WTC-estimators is discussed and their performance, for finite samples, is illustrated through a small-scale Monte Carlo simulation study.