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Title: On attainability of Kendall's tau matrices and concordance signatures Authors:  Johanna Neslehova - McGill University (Canada) [presenting]
Alexander Alexander John McNeil - University of York (United Kingdom)
Andrew Smith - University College Dublin (Ireland)
Abstract: The concordance signature of a random vector or its distribution is defined to be the set of concordance probabilities for margins of all orders. We will show that the concordance signature of a copula is always equal to the concordance signature of some unique mixture of so-called extremal copulas. This result has a number of interesting consequences, which we will explore, such as a characterization of the set of Kendall rank correlation matrices as the cut polytope, and a method for determining whether a set of concordance probabilities is attainable. We also use it to show that the widely-used elliptical distributions yield a strict subset of the attainable concordance signatures as well as a strict subset of the attainable Kendall rank correlation matrices, and prove that the Student $t$ copula converges to a mixture of extremal copulas sharing its concordance signature with all elliptical distributions that have the same correlation matrix. Finally, we will discuss a method of estimating an attainable concordance signature from data, and highlight applications to Monte Carlo simulations of dependent random variables as well as expert elicitation of consistent systems of Kendall's tau dependence measures.