Weibull tail-distributions revisited: a new look at some tail estimators

Laurent Gardes (1) , Stéphane Girard (1) , Armelle Guillou (2)

1 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
2 IRMA - Institut de Recherche Mathématique Avancée

In this paper, we propose to include Weibull tail-distributions in a more general family of distributions. In particular, the considered model also encompasses the whole Fréchet maximum domain of attraction as well as log-Weibull tail-distributions. The asymptotic normality of some tail estimators based on the log-spacings between the largest order statistics is established in an unified way within the considered family. This result permits to understand the similarity between most estimators of the Weibull tail-coefficient and the Hill estimator. Some different asymptotic properties, in terms of bias, rate of convergence, are also highlighted.

Mots clés: Distribution tail; extreme quantile; maximum domain of attraction; asymptotic normality

Domaines: Théorie [stat.TH]; Statistiques [math.ST]

Laurent Gardes, Stéphane Girard, Armelle Guillou. Weibull tail-distributions revisited: a new look at some tail estimators. Journal of Statistical Planning and Inference, 2011, 141 (1), pp.429-444. ⟨10.1016/j.jspi.2010.06.018⟩. ⟨hal-00340661v4⟩