On a Family of Copulas Partially Coinciding with the Fréchet–Hoeffding Boundaries

Grubbs’s test statistics are studied, that is, 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 Grubbs’s copula coincides with the lower Fréchet–Hoeffding bound is proved. In the case of n = 3, Grubbs’s copula is the Fréchet–Hoeffding lower bound. Grubbs’s copula rotated by 180° also partially coincides with the Fréchet–Hoeffding lower bound (in the case of n > 3) and is the Fréchet–Hoeffding lower bound (in the case of n = 3). We prove that Grubbs’s copulas rotated by 90° and 270° partially coincide with the Fréchet–Hoeffding upper bound (in the case of n > 3) and become the Fréchet–Hoeffding upper bound (in the case n = 3).