High Quantile Estimation and the Port Methodology

Abstract

In many areas of application, a typical requirement is to estimate a high quantile χ1−p of probability 1−p, a value, high enough, so that the chance of an exceedance of that value is equal to p, small. The semi-parametric estimation of high quantiles depends not only on the estimation of the tail index γ, the primary parameter of extreme events, but also on an adequate estimation of a scale first order parameter. The great majority of semi-parametric quantile estimators, in the literature, do not enjoy the adequate behaviour, in the sense that they do not suffer the appropriate linear shift in the presence of linear transformations of the data. Recently, and for heavy tails (γ > 0), a new class of quantile estimators was introduced with such a behaviour. They were named PORT-quantile estimators, with PORT standing for peaks over random threshold. In this paper, also for heavy tails, we introduce a new class of PORT-quantile estimators with the above mentioned behaviour, using the PORT methodology and incorporating Hill and moment PORT-classes of tail index estimators in one of the most recent classes of quantile estimators suggested in the literature. Under convenient restrictions on the underlying model, these classes of estimators are consistent and asymptotically normal for adequate k, the number of top order statistics used in the semi-parametric estimation of χ1−p.

Keywords

• statistics of extremes
• heavy tails
• high quantiles
• semi-parametric estimation
• PORT methodology
• asymptotic behaviour