We study the properties of the new threshold selection method for non-parametric estimation of the extremal index of a stationary sequence proposed in [15]. The method is to apply the so-called A1 discrepancy method based on the Cram´er–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 nonstandard modification of the Cram´er–von Mises–Smirnov’s statistic to propose the goodness-of-fit test procedure of ω2 type for distribution tails.