Pairwise and Global Dependence in Trivariate Copula Models

  • Fabrizio Durante
  • Roger B. Nelsen
  • José Juan Quesada-Molina
  • Manuel Úbeda-Flores
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 444)


We investigate whether pairwise dependence properties related to all the bivariate margins of a trivariate copula imply the corresponding trivariate dependence property. The main finding is that, in general, information about the pairwise dependence is not sufficient to infer some aspects of global dependence. In essence, dependence is a multi-facet property that cannot be easily reduced to simplest cases.


Copula Dependence Stochastic model 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chakak, A., Koehler, K.J.: A strategy for constructing multivariate distributions. Comm. Statist. Simulation Comput. 24(3), 537–550 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Deheuvels, P.: Indépendance multivariée partielle et inégalités de Fréchet. In: Studies in Probability and Related Topics, Nagard, Rome, pp. 145–155 (1983)Google Scholar
  3. 3.
    Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D.: The concept of comonotonicity in actuarial science and finance: theory. Insurance Math. Econom. 31(1), 3–33 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Durante, F., Klement, E., Quesada-Molina, J.J.: Bounds for trivariate copulas with given bivariate marginals. J. Inequal. Appl. 2008, 1–9 (2008), article ID 161537Google Scholar
  5. 5.
    Durante, F., Sempi, C.: Copula theory: an introduction. In: Jaworski, P., Durante, F., Härdle, W., Rychlik, T. (eds.) Copula Theory and its Applications. Lecture Notes in Statistics - Proceedings, vol. 198, pp. 3–31. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  6. 6.
    Genest, C., Nešlehová, J.: A primer on copulas for count data. Astin Bull. 37(2), 475–515 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Hofert, M.: A stochastic representation and sampling algorithm for nested Archimedean copulas. J. Stat. Comput. Simul. 82(9), 1239–1255 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Hofert, M., Scherer, M.: CDO pricing with nested Archimedean copulas. Quant. Finance 11(5), 775–787 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Jaworski, P., Durante, F., Härdle, W.K. (eds.): Copulae in Mathematical and Quantitative Finance. Lecture Notes in Statistics - Proceedings, vol. 213. Springer, Heidelberg (2013)zbMATHGoogle Scholar
  10. 10.
    Jaworski, P., Durante, F., Härdle, W.K., Rychlik, T. (eds.): Copula Theory and its Applications. Lecture Notes in Statistics - Proceedings, vol. 198. Springer, Heidelberg (2010)zbMATHGoogle Scholar
  11. 11.
    Joe, H.: Multivariate models and dependence concepts. Monographs on Statistics and Applied Probability, vol. 73. Chapman & Hall, London (1997)zbMATHCrossRefGoogle Scholar
  12. 12.
    Loisel, S.: A trivariate non-Gaussian copula having 2-dimensional Gaussian copulas as margins. Tech. rep., Cahiers de recherche de l’Isfa (2009)Google Scholar
  13. 13.
    Mai, J.F., Scherer, M.: What makes dependence modeling challenging? Pitfalls and ways to circumvent them. Stat. Risk. Model. 30(4), 287–306 (2013)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Nelsen, R.B.: Nonparametric measures of multivariate association. In: Distributions with Fixed Marginals and Related Topics (Seattle, WA, 1993). IMS Lecture Notes Monogr. Ser., vol. 28, pp. 223–232, Inst. Math. Statist., Hayward, CA (1996)Google Scholar
  15. 15.
    Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Springer Series in Statistics. Springer, New York (2006)zbMATHGoogle Scholar
  16. 16.
    Nelsen, R.B., Úbeda-Flores, M.: How close are pairwise and mutual independence? Statist. Probab. Lett. 82(10), 1823–1828 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Okhrin, O., Okhrin, Y., Schmid, W.: On the structure and estimation of hierarchical Archimedean copulas. J. Econometrics 173(2), 189–204 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Quesada-Molina, J.J., Rodríguez-Lallena, J.A.: Some advances in the study of the compatibility of three bivariate copulas. J. Ital. Stat. Soc. 3(3), 397–417 (1994)zbMATHCrossRefGoogle Scholar
  19. 19.
    Rodríguez-Lallena, J.A., Úbeda-Flores, M.: Compatibility of three bivariate quasi-copulas: applications to copulas. In: Soft Methodology and Random Information Systems. Adv. Soft Comput., vol. 26, pp. 173–180. Springer, Berlin (2004)CrossRefGoogle Scholar
  20. 20.
    Salvadori, G., De Michele, C.: Statistical characterization of temporal structure of storms. Adv. Water Resour. 29(6), 827–842 (2006)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Fabrizio Durante
    • 1
  • Roger B. Nelsen
    • 2
  • José Juan Quesada-Molina
    • 3
  • Manuel Úbeda-Flores
    • 4
  1. 1.School of Economics and ManagementFree University of Bozen-BolzanoBolzanoItaly
  2. 2.Department of Mathematical SciencesLewis and Clark CollegePortlandUSA
  3. 3.Department of Applied MathematicsUniversity of GranadaGranadaSpain
  4. 4.Department of MathematicsUniversity of AlmeríaAlmeríaSpain

Personalised recommendations