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 function al, 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 pn tends to infinity sufficiently slowly. Then we investigate the asymptotic variance and bias. In particular, those functionals T re 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