Tail dependence estimation based on smooth estimation of diagonal section

Abstract

This paper is mainly developed around the diagonal section which is strongly related to tail dependence coefficients as defined in Nelsen [19]. Hence, we propose a flexible method for estimating tail dependence coefficients based on the new smooth estimation of the diagonal section based on the Bernstein polynomial approximation. To assess the performance of the new estimators we conduct the Monte-Carlo simulation study. As a result of the simulation study, both estimators perform satisfactory performance. Also, the estimation methods are illustrated by real data examples.

Keywords

• Copula
• tail dependence
• Bernstein polynomial
• diagonal section

References

1. Amblard, C., Girard, S., Estimation procedures for a semiparametric family of bivariate copulas, Journal of Computational and Graphical Statistics, 14(2) (2005), 363-377. DOI:10.1198/106186005X48722
2. Babu, G. J., Canty, A. J., Chaubey, Y. P., Application of Bernstein polynomials for smooth estimation of a distribution and density function, Journal of Statistical Planning and Inference, 105 (2002), 377-392. DOI:10.1016/S0378-3758(01)00265-8
3. Brown, B. M., Elliot, D., Paget, D. F., Lipschitz constants for the Bernstein polynomials of a Lipschitz continuous function, Journal of Approximation Theory, 49 (1987), 196-199. DOI:10.1016/0021-9045(87)90087-6
4. Caillault, C., Guegan, D., Empirical estimation of tail dependence using copulas: application to Asian markets, Quantitative Finance, 5 (2007), 489-501. DOI: 10.1080/14697680500147853
5. Deheuvels, P., La fonction de dependance empirique et ses proprietes un test non parametrique d independance, Acad. Roy. Belg. Bull. Cl. Sci., 65(5) (1979), 274-292.
6. Dimitrova, D., Kaishev, K., Penev, I., GeD spline estimation of multivariate Archimedean copulas, Computational Statistics and Data Analysis, 52 (2008), 3570-3582. DOI:10.1016/j.csda.2007.11.010
7. Duncan, M., Applied Geometry for Computer Graphics and CAD, Springer Verlag, London, 2005.
8. Durante, F., Okhrin, O., Estimation procedures for exchangeable Marshall copulas with hydrological application, Stoch. Environ. Res. Risk Assess, 29 (2015), 205-226. DOI:10.1007/s00477-014-0866-7

9. Durante, F., Fern´andez-S´anchez, J., Pappad´a, R., Copulas, diagonals, and tail dependence, Fuzzy Sets and Systems, 264 (2015), 22-41. DOI: 10.1016/j.fss.2014.03.014
10. Durante, F., Koles´arov´a, A., Mesiar, R., Sempi, C., Copulas with given diagonal sections: novel constructions and applications, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 15(4) (2007), 397-410. DOI: 10.1142/S0218488507004753
11. Embrechts, P., Mcneil, A., Straumann, D., Correlation and dependence in risk management: properties and pitfalls. In: Risk management: value at risk and beyond, Cambridge University Press, Cambridge, (2002), 176-223.
12. Erdely, A., Exact distribution under independence of the diagonal section of the empirical copula, Kybernetika, 44(6) (2008), 826-845.
13. Ferreira, M., Nonparametric estimation of the tail-dependence coefficient, REVSTAT Statistical Journal, 11(1) (2013), 1-16.
14. Ferreira, H., Ferreira, M., Tail dependence between order statistics, Journal of Multivariate Analysis, 105(1) (2012), 176-192. DOI:10.1016/j.jmva.2011.09.001
15. Feller, W., An Introduction to Probability Theory and its Applications, Wiley, New York, 1965.
16. Frahm, G., Junker, M., Schmidt, R., Estimating the tail-dependence coefficient: properties and pitfalls, Insur. Math. Econ., 37(1) (2005), 80-100. DOI:10.1016/j.insmatheco.2005.05.008
17. Goegebeur, Y., Guillou, A., Asymptotically unbiased estimation of the coefficient of tail dependence, Scandinavian Journal of Statistics, 40(1) (2012), 174-189. DOI:10.2307/23357259
18. Joe, H., Multivariate Models and Dependence Concepts, Chapman Hall, London, 1997.
19. Nelsen, R., An Introduction to Copulas, Springer Series in Statistics, Springer, New York, 2006.
20. Schmidt, R., Stadtmuller, U., Non-parametric estimation of tail dependence, Scand. J. Stat., 33(2) (2006), 307-335. DOI: 10.1111/j.1467-9469.2005.00483.x
21. Susam, S. O., Hudaverdi, B. U., Testing independence for Archimedean copula based on Bernstein estimate of Kendall distribution function, Journal of Statistical Computation and Simulation, 88(13) (2018), 2589-2599. DOI:10.1080/00949655.2018.1478978
22. Susam, S. O., Hudaverdi, B. U., A goodness-of-fit test based on Bezier curve estimation of Kendall distribution, Journal of Statistical Computation and Simulation, 90(13) (2020), 1194-1215. DOI:10.1080/00949655.2020.1720680
23. Susam, S. O., Erdogan, M., Plug-in estimation of dependence characteristics of Archimedean copula via Bezier curve, RevStat-Statistical Journal, (In Press).
24. Sklar, A., Fonctions de repartition a n dimensions et leurs marges, Publications de lInstitut de Statistique de lUniversite de Paris, 8 (1959), 229-231.