Asymptotic Confidence Bands for Copulas Based on the Local Linear Kernel Estimator

Diam Bâ¹, Cheikh Tidiane Seck², Gane Samb Lô^1,3

¹LERSTAD, Université Gaston Berger de Saint-Louis, Saint-Louis, Sénégal.
²LERSTAD, Université Alioune Diop, Bambey, Sénégal.
³LSTA, Université Pierre et Marie Curie, Paris, France.

DOI: 10.4236/am.2015.612183                

Abstract

In this paper, we establish asymptotically optimal simultaneous confidence bands for the copula function based on the local linear kernel estimator proposed by Chen and Huang [1]. For this, we prove under smoothness conditions on the derivatives of the copula a uniform in bandwidth law of the iterated logarithm for the maximal deviation of this estimator from its expectation. We also show that the bias term converges uniformly to zero with a precise rate. The performance of these bands is illustrated by a simulation study. An application based on pseudo-panel data is also provided for modeling the dependence structure of Senegalese households’ expense data in 2001 and 2006.

Keywords

Copula Function, Kernel Estimation, Local Linear Estimator, Uniform in Bandwidth Consistency, Simultaneous Confidence Bands

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Chen, S.X. and Huang, T.-M. (2007) Nonparametric Estimation of Copula Functions for Dependence Modeling. Canadian Journal of Statistics, 35, 265-282. http://dx.doi.org/10.1002/cjs.5550350205
[2]	Nelsen, R.B. (2006) An Introduction to Copulas. 2nd Edition, Springer, New York.
[3]	Mason, D.M. and Swanepoel, J.W.H. (2010) A General Result on the Uniform in Bandwidth Consistency of Kernel-Type Function Estimators. TEST, 20, 72-94. http://dx.doi.org/10.1007/s11749-010-0188-0
[4]	Tsukahara, H. (2005) Semiparametric Estimation in Copula Models. The Canadian Journal of Statistics, 33, 357-375. http://dx.doi.org/10.1002/cjs.5540330304
[5]	Scaillet, O. and Fermanian, J.-D. (2002) Nonparametric Estimation of Copulas for Time Series. FAME Research Paper No. 57. http://ssrn.com/abstract=372142 http://dx.doi.org/10.2139/ssrn.372142
[6]	Genest, C. and Rivest, L. (1993) Statistical Inference for Archimedean Copulas. Journal of the American Statistical Association, 88, 1034-1043. http://dx.doi.org/10.1080/01621459.1993.10476372
[7]	Deheuvels, P. (1979) La fonction de dépendence empirique et ses propriétés. Un test non paramétrique. d’indépendance. Bulletin Royal Belge de l’Académie des Sciences, 65, 274-292.
[8]	Fermanian, J., Radulovic, D. and Wegkamp, M. (2004) Weak Convergence of Empirical Copula Processes. International Statistical Institute (ISI) and Bernoulli Society for Mathematical Statistics and Probability, 10, 847-860. http://dx.doi.org/10.3150/bj/1099579158
[9]	Gijbels, I. and Mielniczuk, J. (1990) Estimation of the Density of a Copula Function. Communications in Statistics, Series A, 19, 445-464. http://dx.doi.org/10.1080/03610929008830212
[10]	Omelka, M., Gijbels, I. and Veraverbeke, N. (2009) Improved Kernel Estimators of Copulas: Weak Convergence and Goodness-of-Fit Testing. The Annals of Statistics, 37, 3023-3058. http://dx.doi.org/10.1214/08-AOS666
[11]	Dony, J. (2007) On the Uniform in Bandwidth Consistency of Kernel-Type Estimators and Conditional. Proceedings of the European Young Statisticians Meeting, Castro Urdiales, 10-14 September 2007.
[12]	Einmahl, U. and Mason, D.M. (2005) Uniform in Bandwidth Consistency of Kernel-Type Function Estimators. The Annals of Statistics, 33, 1380-1403. http://dx.doi.org/10.1214/009053605000000129
[13]	Deheuvels, P. and Mason, D.M. (2004) General Asymptotic Confidence Bands Based on Kernel-Type Function Estimators. Statistical Inference Stochastic Process, 7, 225-277. http://dx.doi.org/10.1023/B:SISP.0000049092.55534.af
[14]	Zari, T. (2010) Contribution à l’étude du processus empirique de copule. Thèse de doctorat, Université Paris, Paris, 6.
[15]	Wichura, M.J. (1973) Some Strassen-Type Laws of the Iterated Logarithm for Multiparameter Stochastic Processes with Independent Increments. The Annals of Probability, 1, 272-296. http://dx.doi.org/10.1214/aop/1176996980
[16]	van der Vaart, A.W. and Wellner, J.A. (1996) Weak Convergence and Empirical Processes. Springer, New York. http://dx.doi.org/10.1007/978-1-4757-2545-2
[17]	Lo, G.S., Sall, S.T. and Mergane, P.D. (2015) Functional Weak Laws for the Weighted Mean Losses or Gains and Applications. Applied Mathematics, 6, 847-863. http://dx.doi.org/10.4236/am.2015.65079