Threshold selection for extremal index estimation

We propose a new threshold selection method for nonparametric estimation of the extremal index of stochastic processes. The discrepancy method was proposed as a data-driven smoothing tool for estimation of a probability density function. Now it is modified to select a threshold parameter of an extremal index estimator. A modification of the discrepancy statistic based on the Cramér–von Mises–Smirnov statistic (Formula presented.) is calculated by k largest order statistics instead of an entire sample. Its asymptotic distribution as (Formula presented.) is proved to coincide with the (Formula presented.) -distribution. Its quantiles are used as discrepancy values. The convergence rate of an extremal index estimate coupled with the discrepancy method is derived. The discrepancy method is used as an automatic threshold selection for the intervals and K-gaps estimators. It may be applied to other estimators of the extremal index. The performance of our method is evaluated by simulated and real data examples.