Extreme value index and tail probability estimates for mixed distributions

Abstract

For many practical applications, the mixed probability distribution provides a more realistic parent model from which extreme values are generated. Some fundamental asymptotic distributional results are given for the maximum of independent samples, and bounds are derived for the extreme value index that characterizes the limiting distribution. Based on these bounds, the peaks-over-threshold method is used to compute a probability weight estimator of the extreme value index. A simplified error measure is introduced to select the threshold level that optimizes the estimation of the variate with a given probability of exceedance. The peaks-over-threshold approach permits the determination and calibration of the distribution of the error in the estimated extreme value index. Similar results are obtained for the probability of exceedance as the number of data becomes asymptotically large and the probability of exceedance is much less than the inverse of the number of data. The mean and variance of the error in this estimator are used to find a simplified measure for the selection of the best threshold level. The results are tested by simulation using seasonal statistics of recorded meteorological winds, and then actual wind records from several sites are used to demonstrate the practical application of the methods.