Asymptotic distribution of a Pickands-type estimator of the extreme-value index

One of the main goals of extreme-value analysis is to estimate the probability of rare events given a sample from an unknown distribution. The upper tail behavior of this distribution is described by the extreme-value index. The aim of this Note is to establish the asymptotic distribution of the estimator of c R introduced in Gardes and Girard [A Pickands-type estimator of the extreme-value index, Technical Report LMC-IMAG, RR-1063, 2004]. We also give its rate of convergence in some typical situations.