Tail index and second-order parameters’ semi-parametric estimation based on the log-excesses

Abstract

In this paper we are interested in the derivation of the asymptotic and finite-sample distributional properties of a ‘quasi-maximum likelihood’ estimator of a ‘scale’ second-order parameter β, 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. An application in the area of survival analysis is provided, on the basis of a data set on males diagnosed with cancer of the tongue.

Keywords:

• statistics of extremes
• heavy tails
• log-excesses
• semi-parametric estimation
• maximum likelihood
• survival analysis
• Monte Carlo simulation

AMS Subject Classification: :

• Primary: 62G32
• Secondary: 65C05