Kybernetika 49 no. 1, 73-95, 2013

Construction of multivariate copulas in $n$-boxes

José M. González-Barrios and María M. Hernández-Cedillo


In this paper we give an alternative proof of the construction of $n$-dimensional ordinal sums given in Mesiar and Sempi \cite{mesiar}, we also provide a new methodology to construct $n$-copulas extending the patchwork methodology of Durante, Saminger-Platz and Sarkoci in \cite{durante08} and \cite{durante09}. Finally, we use the gluing method of Siburg and Stoimenov \cite{siburg} and its generalization in Mesiar {et al.} \cite{mesiarjagr} to give an alternative method of patchwork construction of $n$-copulas, which can be also used in composition with our patchwork method.


$n$-copulas, modular functions, rectangular patchwork


60A10, 60E05


  1. J. Aczél and J. Dhombres: Functional Equations in Several Variables. Cambridge University Press, Cambridge 1989.   CrossRef
  2. B. De Baets and H. De Meyer: Orthogonal grid constructions of copulas. IEEE Trans. Fuzzy Systems 15 (2007), 6, 1053-1062.   CrossRef
  3. U. Cherubini, E. Luciano and W. Vecchiato: Copula Methods in Finance. John Wiley and Sons, Chichester 2004.   CrossRef
  4. M. Dorey and P. Joubert: Modelling copulas: An overview. The Staple Inn Actuarial Society, London 2005.   CrossRef
  5. F. Durante, A. Kolesárová, R. Mesiar and C. Sempi: Copulas with given diagonal sections: novel constructions and applications. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 15 (2007), 397-410.   CrossRef
  6. F. Durante, S. Saminger-Platz and P. Sarkoci: On representations of 2-increasing binary aggregation functions. Inform. Sci. 178 (2008), 4534-4541.   CrossRef
  7. F. Durante, S. Saminger-Platz and P. Sarkoci: Rectangular patchwork for bivariate copulas and tail dependence. Comm. Statist. Theory Methods 38 (2009), 2515-2527.   CrossRef
  8. F. Durante and J. Fernández-Sánchez: Multivariate shuffles and approximation of copulas. Statist. Probab. Lett. 80 (2010), 1827-1834.   CrossRef
  9. P. Embrechts, A. Höing and A. Juri: Using copulae to bound the value-at-risk for functions of dependent risks. Finance Stochast. 7 (2003), 145-167.   CrossRef
  10. A. Erdely and M. Díaz-Vieira: Nonparametric and semiparametric bivariate modeling of petrophysical porosity-permeability dependence from well log data. In: Copula Theory and Its Applications (P. Jaworski, F. Durante, W. K. Härdle and T. Rychlik, eds.), Lect. Notes in Statistics, Springer (2010), pp. 267-278.   CrossRef
  11. M. M. Hernández-Cedillo: Topics on multivariate copulas and applications. Ph. D. Thesis, Universidad Nacional Autónoma de México 2013.   CrossRef
  12. H. Joe: Multivariate Models and Dependence Concepts. Chapman and Hall, London 1997.   CrossRef
  13. E. P. Klement, A. Kolesárová, R. Mesiar and C. Sempi: Copulas constructed from the horizontal section. Comm. Statist. Theory Methods 36 (2007), 2901-2911.   CrossRef
  14. Y. Malevergne and D. Sornette: Extreme Financial Risk. From Dependence to Risk Management. Springer-Verlag, Berlin 2006.   CrossRef
  15. R. Mesiar, V. Jágr, M. Jurá\u nová and M. Komorníková: Univariate conditioning of copulas. Kybernetika 44 (2008), 6, 807-816.   CrossRef
  16. A. J. McNeil, R. Frey and P. Embrechts: \it Quantitative Risk Management. Princeton University Press, Princeton 2005.   CrossRef
  17. R. Mesiar and C. Sempi: Ordinal sums and idempotent of copulas. Aequationes Math. 79 (2010), 1-2, 39-52.   CrossRef
  18. R. B. Nelsen: An Introduction to Copulas. Second edition. Springer, New York 2006.   CrossRef
  19. G. Salvadori, C. De Michele, N. T. Kottegoda and R. Rosso: Extremes in Nature. An Approach Using Copulas. Series: Water Science and Technology Library 56, Springer, Amsterdam 2007.   CrossRef
  20. K. F. Siburg and P. A. Stoimenov: Gluing copulas. Comm. Statist. Theory Methods 37 (2008), 3124-3134.   CrossRef
  21. M. H. Zhang: Modelling total tail dependence along diagonals. Insur. Math. Econ. 42 (2008), 73-80.   CrossRef