Sobolev Convergence of Empirical Bernstein Copulas

Abstract

In this work, we prove that Bernstein estimator always converges to the true copula under Sobolev distances. The rate of convergences is provided in case the true copula has bounded second order derivatives. Simulation study has also been done for Clayton copulas. We then use this estimator to estimate measures of complete dependence for weather data. The result suggests a nonlinear relationship between the dust density in Chiang Mai, Thailand and the temperature and the humidity level.

Keywords

• Bernstein estimator,copula,empirical,Sobolev convergence

References

1. [1] J. Behboodian, A. Dolati, and M. Úbeda-Flores. A multivariate version of gini’s rank association coefficient. Statist. Papers, 48(2):295–304, 2007.
2. [2] N. Blomqvist. On a measure of dependence between two random variables. Ann. Math. Statist., 21(4):593–600, 1950.
3. [3] H. Dette, K. F. Siburg, and P. A. Stoimenov. A copula-based non-parametric measure of regression dependence. Scand. J. Stat., 40(1):21–41, 2013.
4. [4] S. Gaißer, M. Ruppert, and F. Schmid. A multivariate version of hoeffding’s phisquare. J. Multivariate Anal., 101(10):2571–2586, 2010.
5. [5] W. Hoeffding. The collected works of Wassily Hoeffding. Springer-Verlag, 1994.
6. [6] P. Janssen, J. Swanepoel, and N. Veraverbeke. Large sample behavior of the bernstein copula estimator. J. Statist. Plann. Inference, 142(5):1189 – 1197, 2012.
7. [7] H. Joe. Multivariate concordance. J. Multivariate Anal., 35(1):12–30, 1990.
8. [8] M. G. Kendall. A new measure of rank correlation. Biometrika, 30(1/2):81–93, 1938.

9. [9] H. O. Lancaster. Measures and indeces of dependence. In M. Kotz and N. L. Johnson, editors, Encyclopedia of Statistical Sciences, volume 2, pages 334–339. Wiley, New York, 1982.
10. [10] R. B. Nelsen. Nonparametric measures of multivariate association. Lecture Notes- Monograph Series, 28:223–232, 1996.
11. [11] A. Rényi. On measures of dependence. Acta Math. Hungar., 10(3):441–451, 1959.
12. [12] F. Schmid, R. Schmidt, T. Blumentritt, S. Gaißer, and M. Ruppert. Copula-based measures of multivariate association. In P. Jaworski, F. Durante, W. K. Härdle, and T. Rychlik, editors, Copula Theory and Its Applications, volume 198 of Lecture Notes in Statistics – Proceedings, pages 209–236. Springer Berlin Heidelberg, 2010.
13. [13] B. Schweizer and E. F. Wolff. On nonparametric measures of dependence for random variables. Ann. Statist., 9(4):879–885, 1981.
14. [14] K. F. Siburg and P. A. Stoimenov. A measure of mutual complete dependence. Metrika, 71:239–251, 2010.
15. [15] A. Sklar. Fonctions de répartition á n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris, 8:229–231, 1959.
16. [16] M. Taylor. Bernstein Polynomials and n-Copulas. ArXiv e-prints, March 2009.
17. [17] M. D. Taylor. Multivariate measures of concordance. Ann. Inst. Statist. Math., 59(4):789–806, 2006.
18. [18] W. Trutsching. On a strong metric on the space of copulas and its induced dependence measure. J. Math. Anal. Appl., 384:690–705, 2011.
19. [19] M. Úbeda-Flores. Multivariate versions of blomqvist’s beta and spearman’s footrule. Ann. Inst. Statist. Math., 57(4):781–788, 2005.
20. [20] E. F. Wolff. N-dimensional measures of dependence. Stochastica, 4(3):175–188, 1980.