On Smooth Statistical Tail Functionals

Abstract

Many estimators of the extreme value index of a distribution functionF that are based on a certain numberkn of largest order statistics can be represented as a statistical tail function al, that is a functionalT applied to the empirical tail quantile functionQn. We study the asymptotic behaviour of such estimators with a scale and location invariant functionalT under weak second order conditions onF. 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 ofT(Qn) ifT is continuous and Hadamard differentiable, respectively, at the upper quantile function of a generalized Pareto distribution andkpn tends to infinity sufficiently slowly. Then we investigate the asymptotic variance and bias. In particular, those functionalsT 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