On null and alternative distribution of statistics of two-side discordancy test for an extreme outlier

We find the joint distribution of Grubbs statistics for a normal sample. Those statistics are standardized maximum and standardized minimum. We note some properties of the joint distribution function. We apply the joint distribution function and find the exact distribution of the test statistics which uses in two-side discordancy test for an extreme outlier. We obtain recursive relationships for the distribution function of the statistics, which uses in two-side discordancy test. We obtain the region of critical values of the statistics, where the significance level of criteria equals the double significance level of the Grubbs criteria. We apply the joint distribution of Grubbs statistics and find the power function for the criteria in the case of a normal sample with a single outlier.