On Three-Parameter Grubbs' Copula-Function

We study one-sided Grubbs' 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 when the sample has an abnormal observation (outlier) with unknown number. The outlier differs from other observations in mean value and dispersion. We investigate the properties of the joint distribution of Grubbs' statistics. We prove the existence of domain in which the joint distribution function of Grubbs' statistics is a linear function of their marginal distribution functions. We construct a three-parameter Grubbs' copula from the joint distribution of Grubbs' statistics. We prove the existence of a domain in which Grubbs' copula coincides with the Frechet-Hoeffding lower bound. We investigate the influence of the copulas parameters on the shape of this domain.