Optimal rates of convergence in the Weibull model based on kernel-type estimators

Cécile Mercadier and Philippe Soulier

Statistics & Probability Letters, 2012, vol. 82, issue 3, 548-556

Abstract: Let F be a distribution function in the maximal domain of attraction of the Gumbel distribution such that −log(1−F(x))=x1/θL(x) for a positive real number θ, called the Weibull tail index, and a slowly varying function L. It is well known that the estimators of θ have a very slow rate of convergence. We establish here a sharp optimality result in the minimax sense, that is when L is treated as an infinite dimensional nuisance parameter belonging to some functional class. We also establish the rate optimal asymptotic property of a data-driven choice of the sample fraction that is used for estimation.

Keywords: Weibull tail index; Rates of convergence; Kernel-type estimators; Optimal sample fraction; Sequential procedure (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1016/j.spl.2011.11.022

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