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

Let FF be a distribution function in the maximal domain of attraction of the Gumbel distribution such that −log(1−F(x))=x1/θL(x)−log(1−F(x))=x1/θL(x) for a positive real number θθ, called the Weibull tail index, and a slowly varying function LL. It is well known that the estimators of θθ have a very slow rate of convergence. We establish here a sharp optimality result in the minimax sense, that is when LL 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.