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

Let F be a distribution function in the maximal domain of attraction of the Gumbel distribution and such that − log(1− F (x)) = xL(x) for a positive real number θ, called the Weibul 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.