Ridge regression estimators for the extreme value index

Abstract

We consider bias reduced estimators of the extreme value index (EVI) in case of Pareto-type distributions and under all max-domains of attraction. To this purpose we revisit the regression approach started in Feuerverger and Hall (Ann. Stat. 27, 760–781, 1999) and Beirlant et al. (Extremes 2, 177–200, 1999) in the case of a positive EVI, and in Beirlant et al. (2005) for real-valued EVI. We generalize these approaches using ridge regression exploiting the mathematical fact that the bias tends to 0 when the number of top data points used in the estimation is decreased. The penalty parameter is selected by minimizing the asymptotic mean squared error of the proposed estimator. The accuracy and utility of the ridge regression estimators are studied using simulations and are illustrated with case studies on reinsurance claim size data as well as daily wind speed data.