×

zbMATH — the first resource for mathematics

Estimating non-simplified vine copulas using penalized splines. (English) Zbl 1384.62170
Summary: Vine copulas (or pair-copula constructions) have become an important tool for high-dimensional dependence modeling. Typically, so-called simplified vine copula models are estimated where bivariate conditional copulas are approximated by bivariate unconditional copulas. We present the first nonparametric estimator of a non-simplified vine copula that allows for varying conditional copulas using penalized hierarchical B-splines. Throughout the vine copula, we test for the simplifying assumption in each edge, establishing a data-driven non-simplified vine copula estimator. To overcome the curse of dimensionality, we approximate conditional copulas with more than one conditioning argument by a conditional copula with the first principal component as conditioning argument. An extensive simulation study is conducted, showing a substantial improvement in the out-of-sample Kullback-Leibler divergence if the null hypothesis of a simplified vine copula can be rejected. We apply our method to the famous uranium data and present a classification of an eye state data set, demonstrating the potential benefit that can be achieved when conditional copulas are modeled.

MSC:
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62G07 Density estimation
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aas, K; Berg, D, Models for construction of multivariate dependence: a comparison study, Eur. J. Finance, 15, 639-659, (2009)
[2] Aas, K; Czado, C; Frigessi, A; Bakken, H, Pair-copula constructions of multiple dependence, Insur. Math. Econ., 44, 182-198, (2009) · Zbl 1165.60009
[3] Abegaz, F; Gijbels, I; Veraverbeke, N, Semiparametric estimation of conditional copulas, J. Multivar. Anal., 110, 43-73, (2012) · Zbl 1244.62040
[4] Acar, EF; Craiu, RV; Yao, F, Dependence calibration in conditional copulas: a nonparametric approach, Biometrics, 67, 445-453, (2011) · Zbl 1217.62068
[5] Acar, EF; Genest, C; Nešlehová, J, Beyond simplified pair-copula constructions, J. Multivar. Anal., 110, 74-90, (2012) · Zbl 1243.62067
[6] Bedford, T; Cooke, RM, Vines: a new graphical model for dependent random variables, Ann. Stat., 30, 1031-1068, (2002) · Zbl 1101.62339
[7] Brechmann, EC; Czado, C; Aas, K, Truncated regular vines in high dimensions with application to financial data, Can. J Stat., 40, 68-85, (2012) · Zbl 1274.62381
[8] Dennis Cook, R; Johnson, ME, Generalized burr-Pareto-logistic distributions with applications to a uranium exploration data set, Technometrics, 28, 123-131, (1986)
[9] Dißmann, J; Brechmann, E; Czado, C; Kurowicka, D, Selecting and estimating regular vine copulae and application to financial returns, Comput. Statist. Data Anal., 59, 52-69, (2013) · Zbl 1400.62114
[10] Duong, T.: ks: Kernel Smoothing. R package version 1.10.4 (2016) · Zbl 1165.60009
[11] Efron, B, Selection criteria for scatterplot smoothers, Ann. Statist., 29, 470-504, (2001) · Zbl 1012.62040
[12] Eilers, PHC; Marx, BD, Flexible smoothing with B-splines and penalties, Statist. Sci., 11, 89-121, (1996) · Zbl 0955.62562
[13] Fermanian, J-D; Wegkamp, MH, Time-dependent copulas, J. Multivar. Anal., 110, 19-29, (2012) · Zbl 1352.62073
[14] Fischer, M; Köck, C; Schlüter, S; Weigert, F, An empirical analysis of multivariate copula models, Quant. Finance, 9, 839-854, (2009) · Zbl 1180.91314
[15] Forsey, D.R., Bartels, R.H.: Hierarchical B-spline refinement. In SIGGRAPH ’88: Proceedings of the 15th annual conference on Computer graphics and interactive techniques, New York, NY, pp. 205-212. ACM (1988)
[16] Forsey, DR; Bartels, RH, Surface Fitting with hierarchical splines, ACM Trans. Graph., 14, 134-161, (1995)
[17] Gijbels, I; Omelka, M; Veraverbeke, N, Multivariate and functional covariates and conditional copulas, Electron. J. Stat., 6, 1273-1306, (2012) · Zbl 1295.62031
[18] Gijbels, I; Veraverbeke, N; Omelka, M, Conditional copulas, association measures and their applications, Comput. Stat. Data Anal., 55, 1919-1932, (2011) · Zbl 1328.62366
[19] Hall, P., Yao, Q.: Approximating conditional distribution functions using dimension reduction. Ann. Statist. 33(3), 1404-1421 (2005, 06) · Zbl 1072.62008
[20] Hobæk Haff, I; Aas, K; Frigessi, A, On the simplified pair-copula construction: simply useful or too simplistic?, J. Multivar. Anal., 101, 1296-1310, (2010) · Zbl 1184.62079
[21] Hurvich, CM; Tsai, C-L, Regression and time series model selection in small samples, Biometrika, 76, 297-307, (1989) · Zbl 0669.62085
[22] Joe, H.: Dependence modeling with copulas. CRC Press, Boca Raton (2015) · Zbl 1346.62001
[23] Kauermann, G; Schellhase, C, Flexible pair-copula estimation in D-vines with penalized splines, Stat. Comput., 24, 1081-1100, (2014) · Zbl 1332.62117
[24] Kauermann, G; Schellhase, C; Ruppert, D, Flexible copula density estimation with penalized hierarchical B-splines, Scand. J. Stat., 40, 685-705, (2013) · Zbl 1364.62084
[25] Killiches, M., Kraus, D., Czado, C.: Examination and visualisation of the simplifying assumption for vine copulas in three dimensions. ArXiv e-prints (2016, February) · Zbl 1373.62227
[26] Kurowicka, D., Joe, H. (eds.): Dependence modeling. World Scientific, Singapore (2011)
[27] Kurz, M.S.: pacotest: Testing for Partial Copulas and the Simplifying Assumption in Vine Copulas, R package version 0.2 (2017) · Zbl 1012.62040
[28] Kurz, M.S., Spanhel, F.: Testing the simplifying assumption in high-dimensional vine copulas. In preparation (2016) · Zbl 0955.62562
[29] Lopez-Paz, D., Hernandez-Lobato, J., Ghahramani, Z.: Gaussian process vine copulas for multivariate dependence. In Proceedings of the 30th International Conference on Machine Learning, W&CP 28(2), pp. 10-18. JMLR (2013)
[30] Marx, B; Eilers, PHC, Multidimensional penalized signal regression, Technometrics, 47, 13-22, (2005)
[31] Nagler, T; Czado, C, Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas, J. Multivar. Anal., 151, 69-89, (2016) · Zbl 1346.62071
[32] Nelsen, R.: An introduction to copulas, 2nd edn. Springer, Berlin (2006) · Zbl 1152.62030
[33] Patton, A.J.: Modelling asymmetric exchange rate dependence. Int. Econ. Rev. 47(2), 527-556 (2006) · Zbl 0100.14202
[34] Rue, H., Martino, S., Chopin, N.: Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. J. Roy. Stat. Soc. B 71(2), 319-392 (2009) · Zbl 1248.62156
[35] Ruppert, D., Wand, M., Carroll, R.: Semiparametric Regression. Cambridge University Press, Cambridge (2003) · Zbl 1038.62042
[36] Schall, R, Estimation in generalized linear models with random effects, Biometrika, 78, 719-727, (1991) · Zbl 0850.62561
[37] Schepsmeier, U., Stoeber, J., Brechmann, E.C., Graeler, B., Nagler, T., Erhardt, T.: VineCopula: Statistical Inference of Vine Copulas. R package version 2.0.5 (2016) · Zbl 0763.65091
[38] Scott, D.W.: The Curse of Dimensionality and Dimension Reduction. Wiley, Hoboken (2008)
[39] Shi, X, A nondegenerate vuong test, Quant. Econ., 6, 85-121, (2015) · Zbl 1398.62050
[40] Sklar, A, Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, 8, 229-231, (1959) · Zbl 0100.14202
[41] Spanhel, F., Kurz, M.S.: Simplified vine copula models: Approximations based on the simplifying assumption. ArXiv e-prints (2015) · Zbl 1418.62225
[42] Spanhel, F; Kurz, MS, The partial copula: properties and associated dependence measures, Stat. Probab. Lett., 119, 76-83, (2016) · Zbl 1398.62150
[43] Stein, ML, A comparison of generalized cross validation and modified maximum likelihood for estimating the parameters of a stochastic process, Ann. Stat., 18, 1139-1157, (1990) · Zbl 0734.62091
[44] Stöber, J; Joe, H; Czado, C, Simplified pair copula constructions-limitations and extensions, J. Multivar. Anal., 119, 101-118, (2013) · Zbl 1277.62139
[45] Vatter, T; Chavez-Demoulin, V, Generalized additive models for conditional dependence structures, J. Multivar. Anal., 141, 147-167, (2015) · Zbl 1328.62390
[46] Veraverbeke, N., Omelka, M., Gijbels, I.: Estimation of a conditional copula and association measures. Scand. J. Stat. 38(4), 766-780 (2011) · Zbl 1246.62092
[47] Wahba, G, A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem, Ann. Stat., 13, 1378-1402, (1985) · Zbl 0596.65004
[48] Wand, M., Jones, M.C.: Kernel smoothing. Chapman and Hall, London (1995) · Zbl 0854.62043
[49] Wand, MP; Ormerod, JT, On semiparametric regression with O’sullivan penalised splines, Aust. N. Z. J. Stat., 50, 179-198, (2008) · Zbl 1146.62030
[50] Wood, S, Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models, J. Roy. Stat. Soc. B, 73, 3-36, (2011) · Zbl 1411.62089
[51] Zenger, C, Sparse grids, Notes Numer. Fluid Mech. Multidiscip. Des., 31, 241-251, (1991) · Zbl 0763.65091
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.