Using of Special Hermite Functions for Investigation of Power Properties of Grubbs’ Criterion

We consider a normal sample with a single upper outlier. A distribution of studentized form of outlier's deviation from the sample mean is obtained. This distribution uses Hermite special functions with negative integer-valued index. The integral relationships for David's power measures of Grubbs criteria are obtained. We discuss the case, when Grubbs statistic is the likelihood-ratio statistic. We find the maximal deviation of power function for Grubbs criteria from the probability that the contaminant is the outlier and it is identified as discordant. We receive the region of critical values of Grubbs statistic, where the second power measure of David equals to the third and forth power measures of David. We make calculations of power function for Grubbs criteria in the case of normal samples with a single upper outlier with the right shift. The results of calculations are similar to the theoretically expected facts.