Advanced dependence metrics and tail coefficients in a bounded copula construction

In this work, we study some properties of the family of copulas introduced previously by Bagré et al. We discuss from dependence metrics such as Spearman's rho, the medial correlation coefficient and distances separating this family of copulas from the Fréchet bounds (lower and upper bounds) and the product copula, respectively. The tail coefficients are also described. We show that the ranges of the Spearman rho and the medial correlation of this class are respectively -0.4784 to 0.4784 and $-0.3333$ to 0.3333. Furthermore, we prove that this class of copulas is bounded between Gumbel's bivariate logistic copula and modified Ali-Mikhail-Haq copula. In particular, we study the properties of two examples of this new family. In addition to theoretical properties, we present graphical illustrations such as level curves and scatter plots of observations from each of the examples.