We consider the nonparametric estimation of extremal index of stochastic processes.
There are nonparametric
methods like well-known blocks, runs and intervals estimators
of the extremal index which all require the selection of an appropriate threshold
u. Some modifications of blocks estimator (see, Drees (2011)) require the block size without u.
In order to estimate u
we develop the approach based on the discrepancy method. The latter was proposed first for a nonparametric estimation of probability density functions Vapnik et al. (1992).
The discrepancy statistics based on the von Mises-Smirnov (M-S) and the Kolmogorov-Smirnov (K-S) statistics were used as the discrepancy measures and some quantiles of limit distributions of M-S and K-S statistics were used as the discrepancy value $\delta$. The method was modified by the author for heavy-tailed densities Markovich (2016) and the extremal index Markovich (2015). To this end, the discrepancy statistics M-S and K-S were calculated not by entire sample but only by K largest order statistics.
The selection of K and $\delta$ is still an open problem. To overcome this problem
we obtain now the limit distribution of the modified M-S statistic regarding the value K. This allows us to select $\delta$ using quantiles of the latter distribution.
To this aim, we use the exponential limit distribution of the normalized inter-cluster size derived in Ferro, Segers (2003). The cluster means the number of consecutive observations exceeding threshold u between two consecutive nonexceedances. The exposition is accompanying by simulated examples.
