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 functional T applied to the empirical tail quantile function Q(n). We study the asymptotic behaviour of such estimators with a scale and location invariant functional T under weak second order conditions on F, For that purpose first a new approximation of the empirical tail quantile function is established, As a consequence we obtain weak consistency and asymptotic normality of T(Q(n)) if T is continuous and Hadamard differentiable, respectively, at the upper quantile function of a generalized Pareto distribution and k(n) tends to infinity sufficiently slowly, Then we investigate the asymptotic variance and bias, In particular, those functionals T are characterized that lead to an estimator with minimal asymptotic variance, Finally, we introduce a method to construct estimators of the extreme value index with a made-to-order asymptotic behaviour.