Threshold selection for extremal index estimation

We propose a new threshold selection method for the nonparametric estimation of the extremal index of stochastic processes. The so-called 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. To this end, a specific normalization of the discrepancy statistic based on the Cram\'{e}r-von Mises-Smirnov statistic $\omega^2$ is calculated by the $k$ largest order statistics instead of an entire sample. Its asymptotic distribution as $k\to\infty$ is proved to be the same as the $\omega^2$-distribution. The quantiles of the latter distribution are used as discrepancy values. The rate of convergence 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 and it may be applied to other estimators of the extremal index.