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 qu antitative risk analysis of an engineering system. One of the importan t questions raised is whether the tail is bounded or unbounded. Theref ore, 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 r aises 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 centra l values which are not informative for the tail. Too extreme threshold s will result in high estimation variances. In this paper we propose t o use a finite sample mean square error (MSE) to select such threshold s and to estimate tail characteristics. Estimators for the extreme val ue index, the end-point and extreme quantiles are based on the so-call ed generalized quantile plot. This plot is used to discern between bou nded and unbounded tail behaviour. A semi-parametric bootstrap techniq ue is used to estimate the MSE at each threshold and to select the opt imal threshold at which the MSE is minimized. Confidence limits are ob tained using the sampling distribution of estimators at the optimal th reshold. In a verification study and an application to wall thickness values of tubes, the MSE-criterion is applied to various extremal prop erties such as endpoints or extreme quantiles and to other parameters that are critically dependent on the tail behaviour of a random variab le such as reliability index. (C) 1998 Elsevier Science Ltd. All right s reserved.