We consider the nonparametric estimation of the extremal index of
stochastic processes. The discrepancy method that was proposed by the author as a
data-driven smoothing tool for probability density function estimation is extended to
find a threshold parameter u for an extremal index estimator in case of heavy-tailed
distributions. To this end, the discrepancy statistics are based on the von Mises–
Smirnov statistic and the k largest order statistics instead of an entire sample. The
asymptotic chi-squared distribution of the discrepancy measure is derived. Its quan8
tiles may be used as discrepancy values. An algorithm to select u for an estimator of
the extremal index is proposed. The accuracy of the discrepancy method is checked
by a simulation study.
