On the some properties of a symmetric Grubbs' copula

We investigate one-sided Grubbs’ statistics for a normal sample. Those statistics are standardized maximum and standardized minimum, i.e. studentized extreme deviation statistics. The two-parameter distribution of these statistics is considered, which arises when the one abnormal observation (outlier) differs from the other observations of its variance. We derive the formula for calculating the probability density function of studentized outlier deviation from sample average. A new two-parameter copula is extracted from the joint distribution of Grubbs’ statistics. The Grubbs’ copula is proved to be symmetric. As a result, one-sided Grubbs’ statistics have the property of exchangeability. Computer simulation of scatterplots from Grubbs’ copula is being performed. The scatterplot analysis shows that the Grubbs’ copula describes the negative statistical dependence. To study the effect of the copula’s parameters on the strength of this dependence, the estimation of the Kendall’s tau rank correlation coefficient is performed. The estimation algorithm uses computer simulation and it is realized in the R-package. We find that the copula’s parameters $n$ and $\nu>0$ have a multidirectional influence on the Kendall’s tau coefficient. If we do not change the parameter $\nu$ then the growth of the parameter $n$ leads to a decrease (in absolute value) of the Kendall’s tau coefficient, which reflects a decrease in the relationship’s strength between the marginals in Grubbs’ copula. If we do not change the parameter $n$, then growth of the parameter $\nu$ to 1 leads to a decrease in the Kendall’s tau coefficient (in absolute value), which reflects a decrease in the strength of the relationship. Further growth of the parameter $\nu$ leads to an increase in the Kendall’s tau coefficient (in absolute value), which reflects increased negative interdependence between the marginals.