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 behavior of such estimators with scale and location invariant functional T under weak second order conditions on F. For that purpose rst 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 diierentiable, respectively, at the upper quantile function of a generalized Pareto distribution and k n tends to innnity suuciently 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 behavior.