Measures of concordance determined by D4‐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 D4‐invariant if its mass distribution is invariant with respect to the symmetries of the unit square. A D4‐invariant copula leads naturally to a family of measures of concordance having a particular form, and all copulas generating this family are D4‐invariant. The construction examined here includes Spearman’s rho and Gini’s measure of association as special cases.

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