A semi-parameter estimator of a “scale” second order parameter based upon the log-excesses

In this paper we are interested in the derivation of the asymptotic and finite-sample distributional properties of a ldquoquasi-maximum likelihoodrdquo estimator of a ldquoscalerdquo second order parameter beta, directly based on the log-excesses of an available sample. Such estimation is of primordial importance for the adaptive selection of the optimal sample fraction to be used in the classical semi-parametric tail index estimation as well as for the reduced-bias estimation of the tail index, high quantiles and other parameters of extreme or even rare events.