On Minimal Copulas under the Concordance Order

Jae Youn Ahn, Sebastian Fuchs

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3 Scopus citations

Abstract

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. 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. The concept of minimal copulas has already been proved to be useful in various continuous and concordance order preserving optimization problems including variance minimization and the detection of lower bounds for certain measures of concordance. We substantiate this key role of minimal copulas by showing that every continuous and concordance order preserving functional on copulas is minimized by some minimal copula, and, in the case the continuous functional is even strictly concordance order preserving, it is minimized by minimal copulas only. Applying the above results, we may conclude that every minimizer of Spearman’s rho is also a minimizer of Kendall’s tau.

Original languageEnglish
Pages (from-to)762-780
Number of pages19
JournalJournal of Optimization Theory and Applications
Volume184
Issue number3
DOIs
StatePublished - 1 Mar 2020

Bibliographical note

Funding Information:
Open access funding provided by Paris Lodron University of Salzburg. Jae Youn Ahn was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean Government (NRF-2017R1D1A1B03032318).

Funding Information:
Open access funding provided by Paris Lodron University of Salzburg. Jae Youn Ahn was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean Government (NRF-2017R1D1A1B03032318).

Publisher Copyright:
© 2019, The Author(s).

Keywords

  • Concordance order
  • Countermonotonicity
  • Extreme negative dependence
  • Kendall’s tau
  • Minimal copula
  • Optimization
  • Spearman’s rho

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