Advertisement

Copulas Based on Marshall–Olkin Machinery

  • Fabrizio Durante
  • Stéphane Girard
  • Gildas Mazo
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 141)

Abstract

We present a general construction principle for copulas that is inspired by the celebrated Marshall–Olkin exponential model. From this general construction method, we derive special subclasses of copulas that could be useful in different situations and recall their main properties. Moreover, we discuss possible estimation strategy for the proposed copulas. The presented results are expected to be useful in the construction of stochastic models for lifetimes (e.g., in reliability theory) or in credit risk models.

Notes

Acknowledgments

The first author acknowledges the support by Faculty of Economics and Management at Free University of Bozen-Bolzano via the project MUD.

References

  1. 1.
    Amblard, C., Girard, S.: A new extension of bivariate FGM copulas. Metrika 70, 1–17 (2009)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bernard, C., Liu, Y., MacGillivray, N., Zhang, J.: Bounds on capital requirements for bivariate risk with given marginals and partial information on the dependence. Depend. Model. 1, 37–53 (2013)zbMATHGoogle Scholar
  3. 3.
    Cuadras, C.M., Augé, J.: A continuous general multivariate distribution and its properties. Commun. Stat. A-Theory Methods 10(4), 339–353 (1981)CrossRefGoogle Scholar
  4. 4.
    Durante, F.: Generalized composition of binary aggregation operators. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 13(6), 567–577 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Durante, F.: A new class of symmetric bivariate copulas. J. Nonparametr. Stat. 18(7–8), 499–510 (2006/2007)Google Scholar
  6. 6.
    Durante, F., Fernández-Sánchez, J., Pappadà, R.: Copulas, diagonals and tail dependence. Fuzzy Sets Syst. 264, 22–41 (2015)Google Scholar
  7. 7.
    Durante, F., Foschi, R., Spizzichino, F.: Threshold copulas and positive dependence. Stat. Probab. Lett. 78(17), 2902–2909 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Durante, F., Kolesárová, A., Mesiar, R., Sempi, C.: Semilinear copulas. Fuzzy Sets Syst. 159(1), 63–76 (2008)CrossRefzbMATHGoogle Scholar
  9. 9.
    Durante, F., Okhrin, O.: Estimation procedures for exchangeable Marshall copulas with hydrological application. Stoch. Environ. Res Risk Assess 29(1), 205–226 (2015)CrossRefGoogle Scholar
  10. 10.
    Durante, F., Quesada-Molina, J.J., Úbeda-Flores, M.: On a family of multivariate copulas for aggregation processes. Inf. Sci. 177(24), 5715–5724 (2007)CrossRefzbMATHGoogle Scholar
  11. 11.
    Durante, F., Salvadori, G.: On the construction of multivariate extreme value models via copulas. Environmetrics 21(2), 143–161 (2010)MathSciNetGoogle Scholar
  12. 12.
    Durante, F., Sempi, C.: On the characterization of a class of binary operations on bivariate distribution functions. Publ. Math. Debrecen 69(1–2), 47–63 (2006)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Frahm, G.: On the extremal dependence coefficient of multivariate distributions. Stat. Probab. Lett. 76(14), 1470–1481 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Genest, C., Carabarím-Aguirre, A., Harvey, F.: Copula parameter estimation using Blomqvist’s beta. J. SFdS 154(1), 5–24 (2013)MathSciNetGoogle Scholar
  15. 15.
    Genest, C., Favre, A.C.: Everything you always wanted to know about copula modeling but were afraid to ask. J. Hydrol. Eng. 12(4), 347–368 (2007)CrossRefGoogle Scholar
  16. 16.
    Joe, H.: Dependence Modeling with Copulas. Chapman and Hall/CRC, London (2014)zbMATHGoogle Scholar
  17. 17.
    Kaas, R., Laeven, R.J.A., Nelsen, R.B.: Worst VaR scenarios with given marginals and measures of association. Insur. Math. Econom. 44(2), 146–158 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Krupskii, P., Joe, H.: Factor copula models for multivariate data. J. Multivar. Anal. 120, 85–101 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Li, X., Pellerey, F.: Generalized Marshall–Olkin distributions and related bivariate aging properties. J. Multivar. Anal. 102(10), 1399–1409 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Liebscher, E.: Construction of asymmetric multivariate copulas. J. Multivar. Anal. 99(10), 2234–2250 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Mai, J.F., Schenk, S., Scherer, M.: Exchangeable exogenous shock models. Bernoulli (2015). In pressGoogle Scholar
  22. 22.
    Mai, J.F., Scherer, M.: Lévy-Frailty copulas. J. Multivar. Anal. 100(7), 1567–1585 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Marshall, A.W.: Copulas, marginals, and joint distributions. In: Distributions with Fixed Marginals and Related Topics (Seattle, WA, 1993). IMS Lecture Notes Monograph Series, vol. 28, pp. 213–222. Institute of Mathematical Statistics, Hayward, CA (1996)Google Scholar
  24. 24.
    Marshall, A.W., Olkin, I.: A multivariate exponential distribution. J. Am. Stat. Assoc. 62, 30–44 (1967)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Marshall, A.W., Olkin, I., Arnold, B.C.: Inequalities: theory of majorization and its applications, 2nd edn. Springer Series in Statistics. Springer, New York (2011)CrossRefGoogle Scholar
  26. 26.
    Mazo, G., Girard, S., Forbes, F.: A flexible and tractable class of one-factor copulas. Statistics and Computing (2015), to appear. doi: 10.1007/s11222-015-9580-7
  27. 27.
    Mazo, G., Girard, S., Forbes, F.: Weighted least-squares inference based on dependence coefficients for multivariate copulas (2014). http://hal.archives-ouvertes.fr/hal-00979151
  28. 28.
    Muliere, P., Scarsini, M.: Characterization of a Marshall–Olkin type class of distributions. Ann. Inst. Stat. Math. 39(2), 429–441 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Pinto, J., Kolev, N.: Extended Marshall–Olkin model and its dual version. In: Cherubini, U., Durante, F., Mulinacci, S. (eds.) Marshall–Olkin Distributions—Advances in Theory and Applications, Springer Proceedings in Mathematics & Statistics. Springer International Publishing, Switzerland (2015)Google Scholar
  30. 30.
    Salvadori, G., De Michele, C., Durante, F.: On the return period and design in a multivariate framework. Hydrol. Earth Syst. Sci. 15, 3293–3305 (2011)CrossRefGoogle Scholar
  31. 31.
    Salvadori, G., Durante, F., De Michele, C.: Multivariate return period calculation via survival functions. Water Resour. Res. 49(4), 2308–2311 (2013)CrossRefGoogle Scholar
  32. 32.
    Sklar, A.: Fonctions de répartition à \(n\) dimensions et leurs marges. Publ. Inst. Stat. Univ. Paris 8, 229–231 (1959)MathSciNetGoogle Scholar
  33. 33.
    Tankov, P.: Improved Fréchet bounds and model-free pricing of multi-asset options. J. Appl. Probab. 48(2), 389–403 (2011)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Fabrizio Durante
    • 1
  • Stéphane Girard
    • 2
  • Gildas Mazo
    • 2
  1. 1.Faculty of Economics and ManagementFree University of Bozen-BolzanoBolzanoItaly
  2. 2.Laboratoire Jean Kuntzmann (LJK)INRIAGrenobleFrance

Personalised recommendations