Mean-of-Order-p Location-Invariant Extreme Value Index Estimation

Abstract

A simple generalisation of the classical Hill estimator of a positive extreme value index (EVI) has been recently introduced in the literature. Indeed, the Hill estimator can be regarded as the logarithm of the mean of order p = 0 of a certain set of statistics. Instead of such a geometric mean, we can more generally consider the mean of order p (MOP) of those statistics, with p real, and even an optimal MOP (OMOP) class of EVI-estimators. These estimators are scale invariant but not location invariant. With PORT standing for peaks over random threshold, new classes of PORT-MOP and PORT-OMOP EVI-estimators are now introduced. These classes are dependent on an extra tuning parameter q, 0 ≤ q < 1, and they are both location and scale invariant, a property also played by the EVI. The asymptotic normal behaviour of those PORT classes is derived. These EVI-estimators are further studied for finite samples, through a Monte-Carlo simulation study. An adequate choice of the tuning parameters under play is put forward, and some concluding remarks are provided.

Keywords

• bootstrap and/or heuristic threshold selection
• heavy tails
• location/scale invariant semi-parametric estimation
• Monte-Carlo simulation
• optimal levels
• statistics of extremes