Advertisement

Journal of Theoretical Probability

, Volume 28, Issue 4, pp 1311–1336 | Cite as

Conditioning-based metrics on the space of multivariate copulas and their interrelation with uniform and levelwise convergence and Iterated Function Systems

  • Juan Fernández Sánchez
  • Wolfgang Trutschnig
Article

Abstract

Using the one-to-one correspondence between copulas and special Markov kernels three strong metrics on the class \(\mathcal {C}_\rho \) of \(\rho \)-dimensional copulas with \(\rho \ge 3\) are studied. Being natural extensions of the two-dimensional versions introduced by Trutschnig (J Math Anal Appl 384:690–705, 2011), these metrics exhibit various good properties. In particular, it can be shown that the resulting metric spaces are separable and complete, which, as by-product, offers a simple separable and complete metrization of the so-called \(\partial \)-convergence studied by Mikusinski and Taylor (Ann Polon Math 96:75–95, 2009, Metrika 72:385–414, 2010). As an additional consequence of completeness, it is proved that the construction of singular copulas with fractal support via special Iterated Function Systems also converges with respect to any of the three introduced metrics. Moreover, the interrelation with the uniform metric \(d_\infty \) is studied and convergence with respect to \(d_\infty \) is characterized in terms of level-set and endograph convergence with respect to the Hausdorff metric.

Keywords

Copula Stochastic measure Markov kernel Iterated Function System Level set Endograph 

Mathematics Subject Classification (2010)

62H20 60E05 28A80 

Notes

Acknowledgments

The first author gratefully acknowledges the support of Grant MTM2011-22394 from the Spanish Ministry of Science and Innovation.

References

  1. 1.
    Barnsley, M.F.: Fractals Everywhere. Academic Press, Cambridge (1993)zbMATHGoogle Scholar
  2. 2.
    Bauer, H.: Wahrscheinlichkeitstheorie. de Gruyter, Berlin, New York (2002)zbMATHGoogle Scholar
  3. 3.
    Bhattacharya, R., Waymire, E.C.: A Basic Course in Probability Theory. Springer, New York (2007)zbMATHGoogle Scholar
  4. 4.
    Berger, A.: Chaos and Chance—An Introduction to Stochastic Aspects of Dynamics. de Gruyter, Berlin, New York (2001)zbMATHGoogle Scholar
  5. 5.
    Billingsley, P.: Convergence of Probability Measures. Wiley, Hoboken (1968)zbMATHGoogle Scholar
  6. 6.
    Chou, S.H., Nguyen, T.T.: On Fréchet theorem in the set of measure preserving functions over the unit interval. Int. J. Math. Math. Sci. 13(2), 373–378 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Darsow, W.F., Nguyen, B., Olsen, E.T.: Copulas and Markov processes. Ill. J. Math. 36(4), 600–642 (1992)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Di Bernardino, E., Laloë, T., Maume-Deschamps, V., Prieur, C.: Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory. ESAIM Probab. Stat. 17, 236–256 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Dudley, R.M.: Real Analysis and Probability. Cambridge University Press, Cambridge (2002)zbMATHCrossRefGoogle Scholar
  10. 10.
    Durante, F., Sarkoci, P., Sempi, C.: Shuffles of copulas. J. Math. Anal. Appl. 352, 914–921 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    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. 1–31. Springer, Berlin (2010)Google Scholar
  12. 12.
    Durante, F., Fernández Sánchez, J.: Multivariate shuffles and approximation of copulas. Stat. Probab. Lett. 80(23–24), 1827–1834 (2010)Google Scholar
  13. 13.
    Durante, F., Fernández Sánchez, J.: On the approximation of copulas via shuffles of Min. Stat. Probab. Lett. 82(10), 1761–1767 (2012)zbMATHCrossRefGoogle Scholar
  14. 14.
    Edgar, G.: Measure, Topology, and Fractal Geometry. Springer, New York (2008)zbMATHCrossRefGoogle Scholar
  15. 15.
    Elstrodt, J.: Maß-und Integrationstheorie. Springer, Berlin, Heidelberg, New York (1999)zbMATHCrossRefGoogle Scholar
  16. 16.
    Fredricks, G.A., Nelsen, R.B., Rodríguez-Lallena, J.A.: Copulas with fractal supports. Insur. Math. Econ. 37, 42–48 (2005)zbMATHCrossRefGoogle Scholar
  17. 17.
    Gruber, P.M.: Convex and Discrete Geometry. Springer, Berlin, Heidelberg (2007)zbMATHGoogle Scholar
  18. 18.
    Janssen, P., Swanepoel, J., Veraverbeke, N.: Large sample behavior of the Bernstein copula estimator. J. Stat. Plan. Infer. 142, 1189–1197 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Kallenberg, O.: Foundations of Modern Probability. Springer, New York, Berlin, Heidelberg (1997)zbMATHGoogle Scholar
  20. 20.
    Klenke, A.: Probability Theory—A Comprehensive Course. Springer, Berlin, Heidelberg (2007)Google Scholar
  21. 21.
    Lancaster, H.O.: Correlation and complete dependence of random variables. Ann. Math. Stat. 34, 1315–1321 (1963)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Lavric, B.: Continuity of monotone functions. Arch. Math. 29, 1–4 (1993)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Li, X., Mikusinski, P., Taylor, M.D.: Strong approximation of copulas. J. Math. Anal. Appl. 255, 608–623 (1998)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mikusinski, P., Taylor, M.D.: Markov operators and \(n\)-copulas. Ann. Polon. Math. 96, 75–95 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Mikusinski, P., Taylor, M.D.: Some approximations of \(n\)-copulas. Metrika 72, 385–414 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Molchanov, I.: Theory of Random Sets. Springer, London (2005)zbMATHGoogle Scholar
  27. 27.
    Nelsen, R.B.: An Introduction to Copulas. Springer, New York (2006)zbMATHGoogle Scholar
  28. 28.
    Olsen, E.T., Darsow, W.F., Nguyen, B.: Copulas and Markov operators, In: Proceedings of the Conference on Distributions with Fixed Marginals and Related Topics, IMS Lecture Notes. Monograph Series, vol. 28, pp. 244–259 (1996)Google Scholar
  29. 29.
    Rudin, W.: Real and Complex Analysis. McGraw-Hill International Editions, Singapore (1987)zbMATHGoogle Scholar
  30. 30.
    Taylor, M.D.: Multivariate measures of concordance. Inst. Stat. Math. 59, 789–806 (2006)CrossRefGoogle Scholar
  31. 31.
    Trutschnig, W.: Characterization of the sendograph-convergence of fuzzy sets by means of their \(L_p\)- and levelwise convergence. Fuzzy Set Syst. 161(8), 1064–1077 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Trutschnig, W.: On a strong metric on the space of copulas and its induced dependence measure. J. Math. Anal. Appl. 384, 690–705 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Trutschnig, W.: Some results on the convergence of (quasi-)copulas. Fuzzy Set Syst. 191, 113–121 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Trutschnig, W., Fernández-Sánchez, J.: Idempotent and multivariate copulas with fractal support. J. Stat. Plan. Infer. 142, 3086–3096 (2012)zbMATHCrossRefGoogle Scholar
  35. 35.
    van der Vaart, A.W.: Asymptotic Statistics. Cambridge University Press, Cambridge (2000)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Grupo de Investigación de Análisis MatemáticoUniversidad de AlmeríaAlmeríaSpain
  2. 2.Department for MathematicsUniversity SalzburgSalzburgAustria

Personalised recommendations