On tail dependence for three-parameter Grubbs' copula

L. K. Shiryaeva

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2020), pp. 51-73

Résumé

We consider one-sided Grubbs's statistics for a normal sample of the size $n$. Those statistics are extreme studentized deviations of the observations from the sample mean. A one abnormal observation (outlier) is proposed in the sample, it is unknown what according to number. 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 which is depended on three parameters: shift and scale parameters and $n$. It is proved that for Grubbs's copula-function coefficients of the upper-left and lower-right tail dependence are equal each other. Moreover, their value does not depend on the parameters shift and scale parameters but it is dependent 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.

Détail

Mots-clés : copula, normal distribution
Keywords: tail dependence, upper-left and lower-right tail dependences coefficients, joint distribution function of standardized minimum and maximum, outlier.