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, \cite{Drees-2011}) require the block size without u.
In order to estimate u %a unknown parameter
we develop the approach based on the discrepancy method. The latter was proposed first for a nonparametric estimation of probability density functions %\cite{Markovich-89},
\cite{Vapnik}. The discrepancy statistics based on the von Mises-Smirnov's (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 \cite{Markovich-2016} and the extremal index \cite{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 \cite{Ferro}. The cluster means the number of consecutive observations exceeding threshold u between two consecutive nonexceedances. The exposition is accompanying by simulated examples.
