Peaks Over Random Threshold Methodology for Tail Index and High Quantile Estimation

Abstract

In this paper we present a class of semi-parametric high quantile estimators which enjoy a desirable property in the presence of linear transformations of the data. Such a feature is in accordance with the empirical counterpart of the theoretical linearity of a quantile χp: χp(δX + λ) = δχp(X) + λ, for any real λ and positive δ. This class of estimators is based on the sample of excesses over a random threshold, originating what we denominate PORT (Peaks Over Random Threshold) methodology. We prove consistency and asymptotic normality of two high quantile estimators in this class, associated with the PORT-estimators for the tail index. The exact performance of the new tail index and quantile PORT-estimators is compared with the original semiparametric estimators, through a simulation study.

Keywords

• heavy tails
• high quantiles
• semi-parametric estimation
• linear property
• sample of excesses