IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v184y2020i3d10.1007_s10957-019-01618-4.html
   My bibliography  Save this article

On Minimal Copulas under the Concordance Order

Author

Listed:
  • Jae Youn Ahn

    (Ewha Womans University)

  • Sebastian Fuchs

    (Universität Salzburg)

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 $$d\ge 3$$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.

Suggested Citation

  • Jae Youn Ahn & Sebastian Fuchs, 2020. "On Minimal Copulas under the Concordance Order," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 762-780, March.
    • Handle: RePEc:spr:joptap:v:184:y:2020:i:3:d:10.1007_s10957-019-01618-4
    DOI: 10.1007/s10957-019-01618-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-019-01618-4
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-019-01618-4?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Sebastian Fuchs & Yann McCord & Klaus D. Schmidt, 2018. "Characterizations of Copulas Attaining the Bounds of Multivariate Kendall’s Tau," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 424-438, August.
    2. Thibaut Lux & Antonis Papapantoleon, 2016. "Improved Fr\'echet$-$Hoeffding bounds on $d$-copulas and applications in model-free finance," Papers 1602.08894, arXiv.org, revised Jun 2017.
    3. Marco Scarsini, 1984. "Strong measures of concordance and convergence in probability," Post-Print hal-00542387, HAL.
    4. Marco Scarsini & Alfred Muller, 2006. "Stochastic order relations and lattices of probability measures," Post-Print hal-00539119, HAL.
    5. M. Taylor, 2007. "Multivariate measures of concordance," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(4), pages 789-806, December.
    6. Müller, Alfred & Scarsini, Marco, 2000. "Some Remarks on the Supermodular Order," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 107-119, April.
    7. Jae Youn Ahn, 2015. "Negative Dependence Concept in Copulas and the Marginal Free Herd Behavior Index," Papers 1503.03180, arXiv.org.
    8. Joe, Harry, 1990. "Multivariate concordance," Journal of Multivariate Analysis, Elsevier, vol. 35(1), pages 12-30, October.
    9. Dhaene, Jan & Denuit, Michel, 1999. "The safest dependence structure among risks," Insurance: Mathematics and Economics, Elsevier, vol. 25(1), pages 11-21, September.
    10. Wang, Bin & Wang, Ruodu, 2011. "The complete mixability and convex minimization problems with monotone marginal densities," Journal of Multivariate Analysis, Elsevier, vol. 102(10), pages 1344-1360, November.
    11. Dhaene, J. & Denuit, M. & Goovaerts, M. J. & Kaas, R. & Vyncke, D., 2002. "The concept of comonotonicity in actuarial science and finance: theory," Insurance: Mathematics and Economics, Elsevier, vol. 31(1), pages 3-33, August.
    12. Manuel Úbeda-Flores, 2005. "Multivariate versions of Blomqvist’s beta and Spearman’s footrule," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(4), pages 781-788, December.
    13. Dhaene, J. & Denuit, M. & Goovaerts, M. J. & Kaas, R. & Vyncke, D., 2002. "The concept of comonotonicity in actuarial science and finance: applications," Insurance: Mathematics and Economics, Elsevier, vol. 31(2), pages 133-161, October.
    14. Roger Nelsen & Manuel Úbeda-Flores, 2012. "Directional dependence in multivariate distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(3), pages 677-685, June.
    15. Peter Tankov, 2010. "Improved Frechet bounds and model-free pricing of multi-asset options," Papers 1004.4153, arXiv.org, revised Mar 2011.
    16. Lee, Woojoo & Ahn, Jae Youn, 2014. "On the multidimensional extension of countermonotonicity and its applications," Insurance: Mathematics and Economics, Elsevier, vol. 56(C), pages 68-79.
    17. Durante, Fabrizio & Fernández-Sánchez, Juan, 2010. "Multivariate shuffles and approximation of copulas," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1827-1834, December.
    18. Marco Scarsini, 1984. "On measures of concordance," Post-Print hal-00542380, HAL.
    19. Taylor M. D., 2016. "Multivariate measures of concordance for copulas and their marginals," Dependence Modeling, De Gruyter, vol. 4(1), pages 1-13, October.
    20. Cheung, Ka Chun & Lo, Ambrose, 2014. "Characterizing mutual exclusivity as the strongest negative multivariate dependence structure," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 180-190.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Fuchs, Sebastian & Schmidt, Klaus D., 2021. "On order statistics and Kendall’s tau," Statistics & Probability Letters, Elsevier, vol. 169(C).
    2. Fuchs, Sebastian & Di Lascio, F. Marta L. & Durante, Fabrizio, 2021. "Dissimilarity functions for rank-invariant hierarchical clustering of continuous variables," Computational Statistics & Data Analysis, Elsevier, vol. 159(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Fuchs, Sebastian & Di Lascio, F. Marta L. & Durante, Fabrizio, 2021. "Dissimilarity functions for rank-invariant hierarchical clustering of continuous variables," Computational Statistics & Data Analysis, Elsevier, vol. 159(C).
    2. Takaaki Koike & Liyuan Lin & Ruodu Wang, 2022. "Joint mixability and notions of negative dependence," Papers 2204.11438, arXiv.org, revised Jan 2024.
    3. Lee, Woojoo & Ahn, Jae Youn, 2014. "On the multidimensional extension of countermonotonicity and its applications," Insurance: Mathematics and Economics, Elsevier, vol. 56(C), pages 68-79.
    4. Ferreira Helena & Ferreira Marta, 2020. "Multivariate medial correlation with applications," Dependence Modeling, De Gruyter, vol. 8(1), pages 361-372, January.
    5. Ferreira Helena & Ferreira Marta, 2020. "Multivariate medial correlation with applications," Dependence Modeling, De Gruyter, vol. 8(1), pages 361-372, January.
    6. Lauzier, Jean-Gabriel & Lin, Liyuan & Wang, Ruodu, 2023. "Pairwise counter-monotonicity," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 279-287.
    7. Chaoubi, Ihsan & Cossette, Hélène & Gadoury, Simon-Pierre & Marceau, Etienne, 2020. "On sums of two counter-monotonic risks," Insurance: Mathematics and Economics, Elsevier, vol. 92(C), pages 47-60.
    8. Jean-Gabriel Lauzier & Liyuan Lin & Ruodu Wang, 2023. "Pairwise counter-monotonicity," Papers 2302.11701, arXiv.org, revised May 2023.
    9. Gaißer, Sandra & Ruppert, Martin & Schmid, Friedrich, 2010. "A multivariate version of Hoeffding's Phi-Square," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2571-2586, November.
    10. Gijbels, Irène & Kika, Vojtěch & Omelka, Marek, 2021. "On the specification of multivariate association measures and their behaviour with increasing dimension," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
    11. Jae Youn Ahn, 2015. "Negative Dependence Concept in Copulas and the Marginal Free Herd Behavior Index," Papers 1503.03180, arXiv.org.
    12. Chuancun Yin & Dan Zhu, 2016. "Sharp Convex Bounds on the Aggregate Sums–An Alternative Proof," Risks, MDPI, vol. 4(4), pages 1-8, September.
    13. Martynas Manstavičius, 2022. "Diversity of Bivariate Concordance Measures," Mathematics, MDPI, vol. 10(7), pages 1-18, March.
    14. Liebscher Eckhard, 2014. "Copula-based dependence measures," Dependence Modeling, De Gruyter, vol. 2(1), pages 1-16, October.
    15. Fuchs Sebastian & McCord Yann, 2019. "On the lower bound of Spearman’s footrule," Dependence Modeling, De Gruyter, vol. 7(1), pages 126-132, January.
    16. Ariel Neufeld & Antonis Papapantoleon & Qikun Xiang, 2020. "Model-free bounds for multi-asset options using option-implied information and their exact computation," Papers 2006.14288, arXiv.org, revised Jan 2022.
    17. Tavin, Bertrand, 2015. "Detection of arbitrage in a market with multi-asset derivatives and known risk-neutral marginals," Journal of Banking & Finance, Elsevier, vol. 53(C), pages 158-178.
    18. Chuancun Yin & Dan Zhu, 2016. "Sharp convex bounds on the aggregate sums--An alternative proof," Papers 1603.05373, arXiv.org, revised May 2016.
    19. Lee Woojoo & Ahn Jae Youn, 2017. "Measuring herd behavior: properties and pitfalls," Dependence Modeling, De Gruyter, vol. 5(1), pages 316-329, December.
    20. Ying Zhang & Chuancun Yin, 2014. "A new multivariate dependence measure based on comonotonicity," Papers 1410.7845, arXiv.org.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:184:y:2020:i:3:d:10.1007_s10957-019-01618-4. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.