Abstract:
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 $180^{\circ}$ 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 $90^{\circ}$ and $270^{\circ}$ partially coincide with the Frechet–Hoeffding upper bound (in the case of $n > 3$) and become the Frechet–Hoeffding upper bound (in the case $n = 3$).
Keywords:extreme studentized deviations of observations from the mean, copula, support of a copula, nonsingular copula, Frechet–Hoeffding lower bound, Frechet–Hoeffding upper bound, rotated version of a copula.
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