On a family of copulas partially coinciding with the Frechet–Hoeffding boundaries

The Grubbs's test statistics are studied, i.e. absolute values of extreme studentized deviations of n random observations from the mean. We consider the case when random observations have arbitrary continuous marginal distributions. The existence of two regions is proved; in one of them, the joint distribution function of these statistics is a linear function of their marginal distribution functions, and in the other, the joint distribution function is zero. We construct a Grubbs’s copula from the joint distribution of Grubbs’s statistics. For the case n>3, the existence of two domains within the unit square in which the Grubbs's copula coincides with the lower Frechet-Hoeffding boundary is proved. In the case of n=3, the Grubbs's copula is the Frechet-Hoeffding lower bound. The Grubbs's copula rotated by 180∘ also partially coincides with the Frechet-Hoeffding lower bound (in the case of n>3) and is the Frechet-Hoeffding lower bound (in the case of n=3). We prove that Grubbs's copulas rotated by 90∘ and 270∘ partially coincide with the Frechet-Hoeffding upper bound (in the case of n>3) and become the Frechet-Hoeffding upper bound (in the case n=3).