  Methods are developed for checking and completing systems of bivariate and
multivariate Kendall's tau concordance measures in applications where only
partial information about dependencies between variables is available. The
concept of a concordance signature of a multivariate continuous distribution is
introduced; this is the vector of concordance probabilities for margins of all
orders. 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 1's and
-1's. A method of estimating an attainable concordance signature from data is
derived and shown to correspond to using standard estimates of Kendall's tau in
the absence of ties. The set of attainable Kendall rank correlation matrices of
multivariate continuous distributions is proved to be identical to 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, as the degrees
of freedom tend to zero, to a mixture of extremal copulas sharing its
concordance signature with all elliptical distributions that have the same
correlation matrix. A characterization of the attainable signatures of
equiconcordant copulas is given.
