Estimating the Extreme Value Index and High Quantiles with Exponential Regression Models

In this paper we present exponential regression models for spacings, or ordered excesses over a given threshold, and for log-ratios of such spacings under maximum domain of attraction conditions. From these we derive estimators for the extreme value index (EVI) and for high quantiles, which share many attractive properties of the maximum likelihood estimators from the peaks-over-thresholds method, but offer the extra advantage of being generally applicable without restriction on the value of the EVI. Further, the exponential regression models can be refined with parameters of second order regular variation, which reduces the bias of the resulting estimators. The refined models also give rise to insightful and practical techniques to select the threshold in the estimation of the EVI and of high quantiles. We demonstrate asymptotic normality of the newly proposed estimators and compare their small sample behaviour to some classical methods in a simulation study.