International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 70, Pages 3867-3875

Measures of concordance determined by D4-invariant copulas

H. H. Edwards, P. Mikusiński, and M. D. Taylor

Department of Mathematics, University of Central Florida, Orlando 32816-1364, FL, USA

Received 25 March 2004

Copyright © 2004 H. H. Edwards et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.