Characterizations of degree one bivariate measures of concordance

We call a measure of concordance κ of an ordered pair ( X , Y ) of two continuous random variables a bivariate measure of concordance. This κ may be considered to be a function κ ( C ) of the copula C associated with ( X , Y ) . κ is considered to be of degree n if, given any two copulas A and B , the value of their convex sum, κ ( t A + ( 1 − t ) B ) , is a polynomial in t of degree n . Examples of bivariate measures of concordance are Spearman’s rho, Blomqvist’s beta, Gini’s measure of association, and Kendall’s tau. The first three of these are of degree one, but Kendall’s tau is of degree two. We exhibit three characterizations of bivariate measures of concordance of degree one.