Excess functions and estimation of the extreme-value index

Abstract

A general class of estimators of the extreme-value index is generated using estimates of mean, median and trimmed excess functions. Special cases yield earlier proposals in the literature, such as Pickands' (1975) estimator. A particular restatement of the mean excess function yields an estimator which can be derived from the slope at the right upper tail from a generalized quantile plot. From this viewpoint algorithms can be constructed to search for the number of extremes needed to minimize the mean square error of the estimator. Basic asymptotic properties of this estimator are derived. The method is applied in case studies of size distributions for alluvial diamonds and of wind speeds.