On distrubution of Grubbs' statistics in case of normal sample with outlier

We investigate one-sided Grubbs' statistics for a normal sample. Those statistics are standardized maximum and standardized minimum. i. e. studentized extreme deviation statistics. We consider the case of the sample when there is one abnormal observation (outlier), unknown to what number according. The outlier differs from other observations in values of population mean and dispersion. We obtain recursive relationships for the marginal distribution function of one-sided Grubbs's statistics. We find asymptotic formulas for marginal distribution functions. We obtain recursive relationships for the joint distribution function one-sided Grubbs' statistics. We investigate the properties of the joint distribution function.