Electronic Journal of Statistics

Inference for elliptical copula multivariate response regression models

Yue Zhao and Christian Genest

Full-text: Open access

Abstract

The estimation of the coefficient matrix in a multivariate response linear regression model is considered in situations where we can observe only strictly increasing transformations of the continuous responses and covariates. It is further assumed that the joint dependence between all the observed variables is characterized by an elliptical copula. Penalized estimators of the coefficient matrix are obtained in a high-dimensional setting by assuming that the coefficient matrix is either element-wise sparse or row-sparse, and by incorporating the precision matrix of the error, which is also assumed to be sparse. Estimation of the copula parameters is achieved by inversion of Kendall’s tau. It is shown that when the true coefficient matrix is row-sparse, the estimator obtained via a group penalty outperforms the one obtained via a simple element-wise penalty. Simulation studies are used to illustrate this fact and the advantage of incorporating the precision matrix of the error when the correlation among the components of the error vector is strong. Moreover, the use of the normal-score rank correlation estimator is revisited in the context of high-dimensional Gaussian copula models. It is shown that this estimator remains as the optimal estimator of the copula correlation matrix in this setting.

Article information

Source
Electron. J. Statist., Volume 13, Number 1 (2019), 911-984.

Dates
Received: September 2017
First available in Project Euclid: 30 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1553911236

Digital Object Identifier
doi:10.1214/19-EJS1534

Mathematical Reviews number (MathSciNet)
MR3934620

Zentralblatt MATH identifier
07056144

Subjects
Primary: 62F12: Asymptotic properties of estimators 62J02: General nonlinear regression
Secondary: 62H20: Measures of association (correlation, canonical correlation, etc.) 62H99: None of the above, but in this section

Keywords
Copula Dantzig selector Kendall’s tau Lasso normal-score rank correlation regression sparsity $U$-statistic van der Waerden correlation

Rights
Creative Commons Attribution 4.0 International License.

Citation

Zhao, Yue; Genest, Christian. Inference for elliptical copula multivariate response regression models. Electron. J. Statist. 13 (2019), no. 1, 911--984. doi:10.1214/19-EJS1534. https://projecteuclid.org/euclid.ejs/1553911236


Export citation

References

  • [1] Barber, R. F., and Kolar, M. ROCKET: Robust confidence intervals via Kendall’s tau for transelliptical graphical models., Ann. Statist. 46 (2018), 3422–3450.
  • [2] Boucheron, S., Lugosi, G., and Massart, P., Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, Oxford, 2013.
  • [3] Cai, T. T., Liu, W., and Luo, X. A constrained $\ell _1$ minimization approach to sparse precision matrix estimation., J. Amer. Statist. Assoc. 106 (2011), 594–607.
  • [4] Cai, T. T., and Zhang, L. High-dimensional Gaussian copula regression: Adaptive estimation and statistical inference., Stat. Sinica 28 (2018), 963–993.
  • [5] Cambanis, S., Huang, S., and Simons, G. On the theory of elliptically contoured distributions., J. Multivariate Anal. 11 (1981), 368–385.
  • [6] Datta, A., and Zou, H. CoCoLasso for high-dimensional error-in-variables regression., Ann. Statist. 45 (2017), 2400–2426.
  • [7] de la Peña, V., and Giné, E., Decoupling: From Dependence to Independence. Randomly Stopped Processes. $U$-statistics and Processes. Martingales and Beyond. Springer-Verlag, New York, 1999.
  • [8] Devlin, S. J., Gnanadseikan, R., and Kettenring, J. R. Robust estimation and outlier detection with correlation coefficients., Biometrika 62 (1975), 531–545.
  • [9] Donoho, D. L., and Huo, X. Uncertainty principles and ideal atomic decomposition., IEEE Trans. Inform. Theory 47 (2001), 2845–2862.
  • [10] El Maache, H., and Lepage, Y. Spearman’s rho and Kendall’s tau for multivariate data sets. In, Mathematical statistics and applications: Festschrift for Constance van Eeden, vol. 42 of IMS Lecture Notes Monogr. Ser. Inst. Math. Statist., Beachwood, OH, 2003, pp. 113–130.
  • [11] Fan, J., Xue, L., and Zou, H. Multitask quantile regression under the transnormal model., J. Amer. Statist. Assoc. 111 (2016), 1726–1735.
  • [12] Fang, K., Kotz, S., and Ng, K., Symmetric Multivariate and Related Distributions. Chapman & Hall, London, 1990.
  • [13] Genest, C., Favre, A.-C., Béliveau, J., and Jacques, C. Metaelliptical copulas and their use in frequency analysis of multivariate hydrological data., Water Resources Research 43 (2007), https://doi.org/10.1029/2006WR005275.
  • [14] Giné, E., Latała, R., and Zinn, J. Exponential and moment inequalities for $U$-statistics. In, High Dimensional Probability, II (Seattle, WA, 1999), vol. 47. Birkhäuser Boston, Boston, MA, 2000, pp. 13–38.
  • [15] Hájek, J., and Šidák, Z., Theory of Rank Tests. Prague: Academia, 1967.
  • [16] Huang, J., and Zhang, T. The benefit of group sparsity., Ann. Statist. 38 (2010), 1978–2004.
  • [17] Hult, H., and Lindskog, F. Multivariate extremes, aggregation and dependence in elliptical distributions., Adv. Appl. Probab. 34 (2002), 587–608.
  • [18] Kendall, M. G., and Gibbons, J. D., Rank Correlation Methods, 5th ed. London: Edward Arnold, 1990.
  • [19] Klaassen, C. A. J., and Wellner, J. A. Efficient estimation in the bivariate normal copula model: Normal margins are least favourable., Bernoulli 3 (1997), 55–77.
  • [20] Kruskal, W. H. Ordinal measures of association., J. Amer. Statist. Assoc. 53 (1958), 814–861.
  • [21] Li, X., Xu, Y., Zhao, T., and Liu, H. Statistical and computational tradeoffs of regularized Dantzig-type estimators., e-prints (2015).
  • [22] Lin, J., Basu, S., Banerjee, M., and Michailidis, G. Penalized maximum likelihood estimation of multi-layered Gaussian graphical models., J. Mach. Learn. Res. 17 (2016), 1–51.
  • [23] Lindskog, F., McNeil, A. J., and Schmock, U. Kendall’s tau for elliptical distributions. In, Credit Risk: Measurement, Evaluation and Management, G. Bol, G. N. Akhaeizadeh, S. T. Rachev, T. Ridder, and K.-H. Vollmer, Eds. Physica-Verlag, 2003, pp. 149–156.
  • [24] Liu, H., Han, F., Yuan, M., Lafferty, J., and Wasserman, L. A. High dimensional semiparametric Gaussian copula graphical models., Ann. Statist. 40 (2012), 2293–2326.
  • [25] Liu, H., Lafferty, J., and Wasserman, L. A. The nonparanormal: Semiparametric estimation of high dimensional undirected graphs., J. Mach. Learn. Res. 10 (2009), 2295–2328.
  • [26] Liu, H., Zhang, J., Jiang, X., and Liu, J. The group Dantzig selector. In, Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics (Chia Laguna Resort, Sardinia, Italy, 13–15 May 2010), Y. W. Teh and M. Titterington, Eds., vol. 9 of Proceedings of Machine Learning Research, PMLR, pp. 461–468.
  • [27] Loh, P.-K., and Wainwright, M. J. High-dimension regression with noisy and missing data: Provable guarantees with non-convexity., Ann. Statist. 40 (2012), 1637–1664.
  • [28] Loh, P.-K., and Wainwright, M. J. Regularized $M$-estimators with nonconvexity: Statistical and algorithmic theory for local optima., J. Mach. Learn. Res. 16 (2015), 559–616.
  • [29] Lounici, K., Pontil, M., van de Geer, S., and Tsybakov, A. B. Oracle inequalities and optimal inference under group sparsity., Ann. Statist. 39 (2011), 2164–2204.
  • [30] Negahban, S. N., Ravikumar, P., Wainwright, M. J., and Yu, B. A unified framework for high-dimensional analysis of $M$-estimators with decomposable regularizers., Statist. Sci. 27 (2012), 538–557.
  • [31] Nelsen, R. B., An Introduction to Copulas, 2nd ed. Springer, New York, 2006.
  • [32] Obozinski, G., Wainwright, M. J., and Jordan, M. I. Support union recovery in high-dimensional multivariate regression., Ann. Statist. 39 (2011), 1–47.
  • [33] Ravikumar, P., Wainwright, M. J., Raskutti, G., and Yu, B. High-dimensional covariance estimation by minimizing $\ell _1$-penalized log-determinant divergence., Electron. J. Statist. 5 (2011), 935–980.
  • [34] Rothman, A. J., Bickel, P., Levina, E., and Zhu, J. Sparse permutation invariant covariance estimation., Electron. J. Statist. 2 (2008), 494–515.
  • [35] Rothman, A. J., Levina, E., and Zhu, J. Sparse multivariate regression with covariance estimation., J. Comput. Graph. Stat. 19 (2010), 947–962.
  • [36] Ruymgaart, F. H. Asymptotic normality of nonparametric tests for independence., Ann. Statist. 2 (1974), 892–910.
  • [37] Segers, J., van den Akker, R., and Werker, B. J. M. Semiparametric Gaussian copula models: Geometry and efficient rank-based estimation., Ann. Statist. 42 (2014), 1911–1940.
  • [38] Shorack, G. R., and Wellner, J. A., Empirical Processes with Applications to Statistics. Wiley, New York, 1986.
  • [39] van der Vaart, A. W., Asymptotic Statistics. Cambridge University Press, Cambridge, 1998.
  • [40] Vershynin, R. Introduction to the non-asymptotic analysis of random matrices. In, Compressed Sensing, Theory and Application, Y. Eldar and G. Kutyniok, Eds. Cambridge University Press, 2012, pp. 210–268.
  • [41] Wegkamp, M., and Zhao, Y. Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas., Bernoulli 25 (2016), 1184–1226.
  • [42] Xue, L., and Zou, H. Regularized rank-based estimation of high-dimensional nonparanormal graphical models., Ann. Statist. 40 (2012), 2541–2571.
  • [43] Yin, J., and Li, H. A sparse conditional Gaussian graphical model for analysis of genetical genomics data., Ann. Appl. Stat. 5 (2011), 831–851.
  • [44] Yuan, M., and Lin, Y. Model selection and estimation in regression with grouped variables., J. R. Statist. Soc. B 68 (2006), 49–67.
  • [45] Yuan, M., and Lin, Y. Model selection and estimation in the Gaussian graphical model., Biometrika 94 (2007), 19–35.
  • [46] Zhang, T., and Zou, H. Sparse precision matrix estimation via lasso penalized $D$-trace loss., Biometrika 101 (2014), 103–120.