ON SMOOTH STATISTICAL TAIL FUNCTIONALS

Many estimators of the extreme value index of a distribution function F that are based on a certain number k(n) of largest order statistics can be represented as a statistical tail functional, that is a functio nal T applied to the empirical tail quantile function Q(n). We study t he asymptotic behaviour of such estimators with a scale and location i nvariant functional T under weak second order conditions on F, For tha t purpose first a new approximation of the empirical tail quantile fun ction is established, As a consequence we obtain weak consistency and asymptotic normality of T(Q(n)) if T is continuous and Hadamard differ entiable, respectively, at the upper quantile function of a generalize d Pareto distribution and k(n) tends to infinity sufficiently slowly, Then we investigate the asymptotic variance and bias, In particular, t hose functionals T are characterized that lead to an estimator with mi nimal asymptotic variance, Finally, we introduce a method to construct estimators of the extreme value index with a made-to-order asymptotic behaviour.