Estimation of the Pareto tail index from extreme order statistics is an important problem in many settings such as income distributions (for inequality measurement), finance (for the evaluation of the value at risk), and insurance (determination of loss probabilities) among others. The upper tail of the distribution in which the data are sparse is typically fitted with a model such as the Pareto model from which quantities such as probabilities associated with extreme events are deduced. The success of this procedure relies heavily not only on the choice of the estimator for the Pareto tail index but also on the procedure used to determine the number k of extreme order statistics that are used for the estimation. For the choice of k most of the known procedures are based on the minimization of (an estimate of) the asymptotic mean square error of the maximum likelihood (or Hill) estimator (MLE) which is the traditional choice for the estimator of the Pareto tail index. In this paper we question the choice of the estimator and the resulting procedure for the determination of k, because we believe that the model chosen to describe the behaviour of the tail distribution can only be considered as approximate. If the data in the tail are not exactly but only approximately Pareto, then the MLE can be biased, i.e. it is not robust, and consequently the choice of k is also biased. We propose instead a weighted MLE for the Pareto tail index that downweights data "far" from the model, where "far" will be measured by the size of standardized residuals constructed by viewing the Pareto model as a regression model. The data that are downweighted this way do not systematically correspond to the largest quantiles. Based on this estimator and proceeding as in Ronchetti and Staudte (1994), we develop a robust prediction error criterion, called RC-criterion, to choose k. In simulation studies, we will compare our estimator and criterion to classical ones with exact and/or approximate Pareto data. Moreover, the analysis of real data sets will show that a robust procedure for selection, and not just for estimation, is needed.
Published March 2005 , 30 pages