On three-parameter Grubbs copula-function

We study one-sided Grubbs's statistics for a normal sample, i. e. extreme studentized deviations of observations from the mean, computed from a normally distributed sample. 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 investigate the properties of the joint distribution of Grubbs's statistics. We prove the existence of a domain in which the joint distribution function of Grubbs's statistics is a linear function of their marginal distribution functions. We construct a three-parameter Grubbs's copula from the joint distribution of Grubbs's statistics. We prove the existence of a domain in which Grubbs's copula coincides with the Frechet–Hoeffding lower bound. We investigate the influence of the copulas parameters on the shape of this domain.