On Tail Dependence for Three-parameter Grubbs' Copula

We consider one-sided Grubbs's statistics for a normal sample of the size n. These statistics are extreme studentized deviations of the observations from the sample mean. One abnormal observation (outlier) is assumed in the sample, its number is unknown. We consider the case when the outlier differs from other observations in values of population mean and dispersion, i. e., shift and scale parameters. We construct a copula-function by an inversion method from the joint distribution of Grubbs's statistics, it depends on three parameters: shift and scale parameters and n. It is proved that for Grubbs's copula-function, the coefficients of the upper-left and lower-right tail dependencies are equal each other. Moreover, their value is independent of the shift and scale parameters but it depends on parameter n. The dependence in the tails of the distribution of the three-parameter Grubbs's copula coincides with the dependence in the tails of the joint distribution of one-sided Grubbs's statistics calculated from the normal sample without outlier.