We study the properties of the new threshold selection method for non-parametric estimation of the extremal index of a stationary sequence proposed in [1]. The method is to apply the so-called discrepancy method based on the Cramer-von Mises-Smirnov's statistic calculated by the largest order statistics of a sample. The limit distribution of this statistic is derived if the proportion of the largest order statistics used tends to some nonzero constant. We also use the non-standard modification of the Cramer-von Mises-Smirnov's statistic to propose the goodness-of-fit test procedure of omega-squared type for distribution tails.