On mixtures of extremal copulas and attainability of concordance signatures
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The concordance signature of a random vector or its distribution is defined to be the set of concordance probabilities for margins of all orders. It is proved that the concordance signature of a copula is always equal to the concordance signature of some unique mixture of the extremal copulas. Applications of the result include a characterization of the set of Kendall rank correlation matrices as the cut polytope as well as a method for determining whether sets of concordance probabilities are attainable. 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 shown 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 bivariate and multivariate Kendall's tau in the absence of ties
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