Identifying tails, bounds and end-points of random variables

The characterization of tails of random variables is of major concern in a safety analysis such as a structural reliability analysis or a quantitative risk analysis of an engineering system. One of the important questions raised is whether the tail is bounded or unbounded. Therefore, in a statistical analysis of a given data set, it makes sense to use only the extreme small or large data in the tail modelling. This raises the important issue of the selection of thresholds above which "tail behaviour" of the data can be justified. In general, thresholds close to the central data will bias the estimation towards the central values which are not informative for the tail. Too extreme thresholds will result in high estimation variances. In this paper we propose to use a finite sample mean square error (MSE) to select such thresholds and to estimate tail characteristics. Estimators for the extreme value index, the end-point and extreme quantiles are based on the so-called generalized quantile plot. This plot is used to discern between bounded and unbounded tail behaviour. A semi-parametric bootstrap technique is used to estimate the MSE at each threshold and to select the optimal threshold at which the MSE is minimized. Confidence limits are obtained using the sampling distribution of estimators at the optimal threshold. In a verification study and an application to wall thickness values of tubes, the MSE-criterion is applied to various extremal properties such as endpoints or extreme quantiles and to other parameters that are critically dependent on the tail behaviour of a random variable such as reliability index. (C) 1998 Elsevier Science Ltd. All rights reserved.