×

zbMATH — the first resource for mathematics

Nonparametric estimation of simplified vine copula models: comparison of methods. (English) Zbl 1404.62034
Summary: In the last decade, simplified vine copula models have been an active area of research. They build a high dimensional probability density from the product of marginals densities and bivariate copula densities. Besides parametric models, several approaches to nonparametric estimation of vine copulas have been proposed. In this article, we extend these approaches and compare them in an extensive simulation study and a real data application. We identify several factors driving the relative performance of the estimators. The most important one is the strength of dependence. No method was found to be uniformly better than all others. Overall, the kernel estimators performed best, but do worse than penalized B-spline estimators when there is weak dependence and no tail dependence.

MSC:
62G07 Density estimation
62H05 Characterization and structure theory for multivariate probability distributions; copulas
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aas, K., C. Czado, A. Frigessi, and H. Bakken (2009). Pair-copula constructions of multiple dependence. Insurance Math. Econom. 44(2), 182-198. · Zbl 1165.60009
[2] Bedford, T. and R. M. Cooke (2001). Probability density decomposition for conditionally dependent random variables modeled by vines. Ann. Math. Artif. Intell. 32(1-4), 245-268. · Zbl 1314.62040
[3] Bedford, T. and R. M. Cooke (2002). Vines - A new graphical model for dependent random variables. Ann. Statist. 30(4), 1031-1068. · Zbl 1101.62339
[4] Brechmann, E. C., C. Czado, and K. Aas (2012). Truncated regular vines in high dimensions with application to financial data. Canad. J. Statist. 40(1), 68-85. · Zbl 1274.62381
[5] Brechmann, E. C. and U. Schepsmeier (2013). Modeling dependence with C- and D-vine copulas: The R package CDVine. J. Stat. Softw. 52(3), 1-27.
[6] Charpentier, A., J.-D. Fermanian, and O. Scaillet (2006). The estimation of copulas: Theory and practice. In J. Rank (Ed.), Copulas: From Theory to Application in Finance, pp. 35-62. Risk Books, London.
[7] Chen, S. X. (1999). Beta kernel estimators for density functions. Comput. Statist. Data Anal. 31(2), 131-145. · Zbl 0935.62042
[8] Czado, C. (2010). Pair-copula constructions of multivariate copulas. In P. Jaworski, F. Durante, W. K. Härdle, and T. Rychlik (Eds.), Copula Theory and its Applications, pp. 93-109. Springer, Heidelberg.
[9] Czado, C., S. Jeske, and M. Hofmann (2013). Selection strategies for regular vine copulae. J. SFdS 154(1), 174-191. · Zbl 1316.62030
[10] Dißmann, J., E. C. Brechmann, C. Czado, and D. Kurowicka (2013). Selecting and estimating regular vine copulae and application to financial returns. Comput. Statist. Data Anal. 59(1), 52-69. · Zbl 1400.62114
[11] Eilers, P. H. C. and B. D. Marx (1996). Flexible smoothing with B-splines and penalties. Statist. Sci. 11(2), 89-121. With comments and a rejoinder by the authors. · Zbl 0955.62562
[12] Fischer, M., C. Köck, S. Schlüter, and F. Weigert (2009). An empirical analysis of multivariate copula models. Quant. Finance 9(7), 839-854. · Zbl 1180.91314
[13] Geenens, G., A. Charpentier, and D. Paindaveine (2017). Probit transformation for nonparametric kernel estimation of the copula density. Bernoulli 23(3), 1848-1873. · Zbl 1392.62101
[14] Genest, C., K. Ghoudi, and L.-P. Rivest (1995). A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82(3), 543-552. · Zbl 0831.62030
[15] Gijbels, I. and J. Mielniczuk (1990). Estimating the density of a copula function. Comm. Statist. - Theory and Methods 19(2), 445-464. · Zbl 0900.62188
[16] Hobæk Haff, I. and J. Segers (2015). Nonparametric estimation of pair-copula constructions with the empirical pair-copula. Comput. Statist. Data Anal. 84, 1-13. · Zbl 06984141
[17] Hurvich, C. M., J. S. Simonoff, and C.-L. Tsai (1998). Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion. J. R. Stat. Soc. Ser. B Stat. Methodol. 60(2), 271-293. · Zbl 0909.62039
[18] Janssen, P., J. Swanepoel, and N. Veraverbeke (2014). A note on the asymptotic behavior of the Bernstein estimator of the copula density. J. Multivariate Anal. 124, 480-487. · Zbl 1359.62103
[19] Joe, H. (1996). Families of m-variate distributions with given margins and m(m − 1)/2 bivariate dependence parameters. In L. Rüschendorf, B. Schweizer, and M. D. Taylor (Eds.), Distributions with Fixed Marginals and Related Topics, pp. 120-141. Inst. Math. Statist., Hayward CA.
[20] Joe, H. (2015). Dependence Modeling with Copulas. CRC Press, Boca Raton FL. · Zbl 1346.62001
[21] Kauermann, G. and C. Schellhase (2014). Flexible pair-copula estimation in D-vines using bivariate penalized splines. Stat. Comput. 24(6), 1081-1100. · Zbl 1332.62117
[22] Kim, G., M. J. Silvapulle, and P. Silvapulle (2007). Comparison of semiparametric and parametric methods for estimating copulas. Comput. Statist. Data Anal. 51(6), 2836-2850. · Zbl 1161.62364
[23] Loader, C. (1999). Local Regression and Likelihood. Springer-Verlag, New York. · Zbl 0929.62046
[24] Lorentz, G. G. (1953). Bernstein Polynomials. Univ. of Toronto Press, Toronto.
[25] Morales-Nápoles, O., R. Cooke, and D. Kurowicka (2011). Counting vines. In D. Kurowicka and H. Joe (Eds.), Dependence Modeling: Vine Copula Handbook, pp. 189-218. World Scientific Publishing, Singapore.
[26] Nagler, T. (2016). kdecopula: Kernel Smoothing for Bivariate Copula Densities. R package version 0.8.0. Available on CRAN.
[27] Nagler, T. (2017). kdecopula: An R Package for the Kernel Estimation of Bivariate Copula Densities. Available at https://arxiv.org/abs/1603.04229.
[28] Nagler, T. and C. Czado (2016). Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas. J. Multivariate Anal. 151, 69-89. · Zbl 1346.62071
[29] Ripley, B. D. (1987). Stochastic Simulation. John Wiley & Sons, Chichester. · Zbl 0613.65006
[30] Rose, D. (2015). Modeling and Estimating Multivariate Dependence Structures with the Bernstein Copula. Ph. D. thesis, Ludwig-Maximilians Universität München. · Zbl 1332.62005
[31] Ruppert, D., M. P. Wand, and R. J. Carroll (2003). Semiparametric Regression. Cambridge University Press, Cambridge. · Zbl 1038.62042
[32] Sancetta, A. and S. Satchell (2004). The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Econometric Theory 20(3), 535-562. · Zbl 1061.62080
[33] Scheffer, M. and G. N. F. Weiß(2017). Smooth nonparametric Bernstein vine copulas. Quant. Finance 17(1), 139-156. · Zbl 1402.91731
[34] Schellhase, C. (2016). penRvine: Pair-Copula Estimation in R-Vines using Bivariate Penalized Splines. R package version 0.2. Available on CRAN.
[35] Schepsmeier, U., J. Stoeber, E. C. Brechmann, B. Graeler, T. Nagler, and T. Erhardt et al. (2017). VineCopula: Statistical Inference of Vine Copulas. R package version 2.1.2. Available on CRAN.
[36] Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229-231. · Zbl 0100.14202
[37] Spanhel, F. and M. S. Kurz (2015). Simplified vine copula models: Approximations based on the simplifying assumption. Available at https://arxiv.org/abs/1510.06971. · Zbl 1418.62225
[38] Stöber, J. and C. Czado (2012). Sampling pair copula constructions with applications to mathematical finance. In J.-F. Mai and M. Scherer (Eds.), Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications. World Scientific Publishing, Singapore.
[39] Wahba, G. (1990). Spline Models for Observational Data. SIAM, Philadelphia, PA. · Zbl 0813.62001
[40] Weingessel, A. (2013). quadprog: Functions to solve Quadratic Programming Problems. R package version 1.5-5. Available on CRAN.
[41] Wood, S. N. (2006). Generalized Additive Models: An Introduction with R. Chapman and Hall/CRC, Boca Raton FL.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.