The concordance signature of a multivariate continuous distribution is the vector of concordance probabilities for margins of all orders; it underlies the bivariate and multivariate Kendalls tau measure of concordance. It is shown that every attainable concordance signature is equal to the concordance signature of a unique mixture of the extremal copulas, that is the copulas with extremal correlation matrices consisting exclusively of 1s and -1s. This result establishes that the set of attainable Kendall rank correlation matrices of multivariate continuous distributions in arbitrary dimension is the set of convex combinations of extremal correlation matrices, a set known as the cut polytope. A methodology for testing the attainability of concordance signatures using linear optimization and convex analysis is provided. The elliptical copulas are shown to yield a strict subset of the attainable concordance signatures as well as a strict subset of the attainable Kendall rank correlation matrices; the Student t copula is seen to converge to a mixture of extremal copulas sharing its concordance signature with all elliptical distributions that have the same correlation matrix. A method of estimating an attainable concordance signature from data is derived and shown to correspond to using standard estimates of Kendalls tau in the absence of ties. The methodology has application to Monte Carlo simulations of dependent random variables as well as expert elicitation of consistent systems of Kendalls tau dependence measures.
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