Constructing Copulas on the Parabolic Boundary of Kendall's Tau-Spearman's Rho Region

Copula is an effective method that describes the joint distribution of dependent random variables. Meanwhile, Kendall's tau and Spearman's rho are two common and important measures measuring the dependence for random variables. The expression form of Kendall's tau - Spearman's rho region has been given and every point on the linear portion of the boundary is attainable. However, it is not determined that other points could be attainable on the parabolic boundary, although countable points could be attainable because of a known family of Copulas at present. Based on constructing a family of Copulas for which Kendall's tau and Spearman's rho lie on the countable points of parabolic boundary by convex combination, and this paper obtains another new family of Copulas for which Kendall's tau and Spearman's rho lie on points of parabolic boundary by ordinal sum.