A Hill Type Estimator of the Weibull Tail-Coefficient

Abstract

We present a new estimator of the Weibull tail-coefficient. The Weibull tail-coefficient is defined as the regular variation coefficient of the inverse cumulative hazard function. Our estimator is based on the log-spacings of the upper order statistics. Therefore, it is very similar to the Hill estimator for the extreme value index. We prove the weak consistency and the asymptotic normality of our estimator. Its asymptotic as well as its finite sample performances are compared to classical ones.

A weighted mean excess function approach to the estimation of Weibull-type tails

“…Several authors exploited the latter property of Weibull-type models to construct estimators for θ := 1/α. For instance Beirlant et al (1996) and Girard (2004) performed a Hill-type operation on the upper part of the QQ plot and introduced the estimator θ k,n := 1 k k j =1 log(X n−j +1,n ) − log(X n−k,n ) 1 k k j =1 log(log n+1 j ) − log(log n+1 k+1 )…”

Kernel regression with Weibull-type tails

“…In the analysis of univariate Weibull-type tails, the estimation of θ and the subsequent estimation of upper extreme quantiles assume a central position. We refer to Broniatowski (1993), Beirlant et al (1995), Girard (2004), Girard (2005, 2008a), Diebolt et al (2008), Dierckx et al (2009), , Goegebeur and Guillou (2011), and the references therein.…”

On the estimation of the functional Weibull tail-coefficient

“…Note that, since θ is defined through an asymptotic behavior of the tail, the estimator should only use the extreme-values of the sample and thus the extra condition k n /n → 0 is required. More specifically, most recent estimators are based on the log-spacings between the k n upper order statistics [6,16,[24][25][26][27].…”