On Minimal Copulas under the Concordance Orde

09/17/2018

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by   Jae Youn Ahn, et al.

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In the present paper, we study extreme negative dependence focussing on the concordance order for copulas. With the absence of a least element for dimensions d > 3, the set of all minimal elements in the collection of all copulas turns out to be a natural and quite important extreme negative dependence concept. This concept has already been proved to be useful in various optimization problems including variance minimization and the detection of lower bounds for certain measures of concordance. We investigate several sufficient conditions and we provide a necessary condition for a copula to be minimal: The sufficient conditions are related to the extreme negative dependence concept of d-countermonotonicity and the necessary condition is related to the collection of all copulas minimizing multivariate Kendall's tau.