Kernel-type estimators for the extreme value index

A large part of the theory of extreme value index estimation is developed for positive extreme value indices. The best-known estimator of a positive extreme value index is probably the Hill estimator. This estimator belongs to the category of moment estimators, but can also be interpreted as a quasi-maximum likelihood estimator. It has been generalized to a kernel-type estimator, but this kernel-type estimator can, similarly to the Hill estimator, only be used for the estimation of positive extreme value indices. In the present paper, we introduce kernel-type estimators which can be used for estimating the extreme value index over the whole (positive and negative) range. We present a number of results on their distributional behavior and compare their performance with the performance of other estimators, such as moment-type estimators for the whole range and the quasi-maximum likelihood estimator, based on the generalized Pareto distribution. We also discuss an automatic bandwidth selection method and introduce a kernel estimator for a second-order parameter, controlling the speed of convergence.