Measures of concordance determined by <i>D</i>₄‐invariant copulas

Abstract

A continuous random vector ( X , Y ) uniquely determines a copula C : [0, 1] 2 → [0, 1] such that when the distribution functions of X and Y are properly composed into C , the joint distribution function of ( X , Y ) results. A copula is said to be D 4 ‐invariant if its mass distribution is invariant with respect to the symmetries of the unit square. A D 4 ‐invariant copula leads naturally to a family of measures of concordance having a particular form, and all copulas generating this family are D 4 ‐invariant. The construction examined here includes Spearman’s rho and Gini’s measure of association as special cases.