Double-Penalized Quantile Regression in Partially Linear Models
Author(s)
Yunlu Jiang
ABSTRACT
In this paper, we propose the double-penalized quantile regression estimators in partially linear models. An iterative algorithm is proposed for solving the proposed optimization problem. Some numerical examples illustrate that the finite sample performances of proposed method perform better than the least squares based method with regard to the non-causal selection rate (NSR) and the median of model error (MME) when the error distribution is heavy-tail. Finally, we apply the proposed methodology to analyze the ragweed pollen level dataset.
In this paper, we propose the double-penalized quantile regression estimators in partially linear models. An iterative algorithm is proposed for solving the proposed optimization problem. Some numerical examples illustrate that the finite sample performances of proposed method perform better than the least squares based method with regard to the non-causal selection rate (NSR) and the median of model error (MME) when the error distribution is heavy-tail. Finally, we apply the proposed methodology to analyze the ragweed pollen level dataset.
Cite this paper
Jiang, Y. (2015) Double-Penalized Quantile Regression in Partially Linear Models. Open Journal of Statistics, 5, 158-164. doi: 10.4236/ojs.2015.52019.
Jiang, Y. (2015) Double-Penalized Quantile Regression in Partially Linear Models. Open Journal of Statistics, 5, 158-164. doi: 10.4236/ojs.2015.52019.
References
[1] Engle, R.F., Granger, C.W., Rice, J. and Weiss, A. (1986) Semiparametric Estimates of the Relation between Weather and Electricity Sales. Journal of the American Statistical Association, 81, 310-320.
http://dx.doi.org/10.1080/01621459.1986.10478274 [2] Fan, J.Q. and Li, R.Z. (2004) New Estimation and Model Selection Procedures for Semiparametric Modeling in Longitudinal Data Analysis. Journal of the American Statistical Association, 99, 710-723.
http://dx.doi.org/10.1198/016214504000001060 [3] Bunea, F. (2004) Consistent Covariate Selection and Post Model Selection Inference in Semi-Parametric Regression. The Annals of Statistics, 32, 898-927.
http://dx.doi.org/10.1214/009053604000000247 [4] Li, R.Z. and Liang, H. (2008) Variable Selection in Semiparametric Regression Modeling. The Annals of Statistics, 36, 261-286.
http://dx.doi.org/10.1214/009053607000000604 [5] Liang, H. and Li., R.Z. (2009) Variable Selection for Partially Linear Models with Measurement Errors. Journal of the American Statistical Association, 104, 234-248.
http://dx.doi.org/10.1198/jasa.2009.0127 [6] Chen, B.C., Yu, Y., Zou, H. and Liang, H. (2012) Profiled Adaptive Elastic-Net Procedure for Partially Linear Models with High-Dimensional Covariates. Journal of Statistical Planning and Inference, 142, 1733-1745.
http://dx.doi.org/10.1016/j.jspi.2012.02.035 [7] Zou, H. and Zhang, H.H. (2009) On the Adaptive Elastic-Net with a Diverging Number of Parameters. The Annals of Statistics, 37, 1733-1751.
http://dx.doi.org/10.1214/08-AOS625 [8] Xie, H.L. and Huang, J. (2009) SCAD-Penalized Regression in High-Dimensional Partially Linear Models. The Annals of Statistics, 37, 673-696.
http://dx.doi.org/10.1214/07-AOS580 [9] Fan, J.Q. and Li, R.Z. (2001) Variable Selection via Nonconcave Penalized Likelihood and Its Oracle Properties. Journal of the American Statistical Association, 96, 1348-1360.
http://dx.doi.org/10.1198/016214501753382273 [10] Ni, X., Zhang, H.H. and Zhang, D.W. (2009) Automatic Model Selection for Partially Linear Models. Journal of multivariate Analysis, 100, 2100-2111.
http://dx.doi.org/10.1016/j.jmva.2009.06.009 [11] Koenker, R. (2005) Quantile Regression. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511754098 [12] Green, P.J. and Silverman, B.W. (1994) Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Chapman & Hall, London. [13] Hunter, D.R. and Lange, K. (2004) A Tutorial on MM Algorithms. The American Statistician, 58, 30-37.
http://dx.doi.org/10.1198/0003130042836 [14] Hunter, D.R. and Lange, K. (2000) Quantile Regression via an MM Algorithm. Journal of Computational and Graphical Statistics, 9, 60-77.
http://dx.doi.org/10.2307/1390613 [15] Zou, H. and Li, R.Z. (2008) One-Step Sparse Estimates in Nonconcave Penalized Likelihood Models. The Annals of Statistics, 36, 1509-1533.
http://dx.doi.org/10.1214/009053607000000802 [16] Chen, J.H. and Chen, Z.H. (2008) Extended Bayesian Information Criteria for Model Selection with Large Model Spaces. Biometrika, 95, 759-771.
http://dx.doi.org/10.1093/biomet/asn034 [17] Ruppert, D., Wand, M.P. and Carroll, R. J. (2003) Semiparametric Regression. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511755453
[1] Engle, R.F., Granger, C.W., Rice, J. and Weiss, A. (1986) Semiparametric Estimates of the Relation between Weather and Electricity Sales. Journal of the American Statistical Association, 81, 310-320.
http://dx.doi.org/10.1080/01621459.1986.10478274 [2] Fan, J.Q. and Li, R.Z. (2004) New Estimation and Model Selection Procedures for Semiparametric Modeling in Longitudinal Data Analysis. Journal of the American Statistical Association, 99, 710-723.
http://dx.doi.org/10.1198/016214504000001060 [3] Bunea, F. (2004) Consistent Covariate Selection and Post Model Selection Inference in Semi-Parametric Regression. The Annals of Statistics, 32, 898-927.
http://dx.doi.org/10.1214/009053604000000247 [4] Li, R.Z. and Liang, H. (2008) Variable Selection in Semiparametric Regression Modeling. The Annals of Statistics, 36, 261-286.
http://dx.doi.org/10.1214/009053607000000604 [5] Liang, H. and Li., R.Z. (2009) Variable Selection for Partially Linear Models with Measurement Errors. Journal of the American Statistical Association, 104, 234-248.
http://dx.doi.org/10.1198/jasa.2009.0127 [6] Chen, B.C., Yu, Y., Zou, H. and Liang, H. (2012) Profiled Adaptive Elastic-Net Procedure for Partially Linear Models with High-Dimensional Covariates. Journal of Statistical Planning and Inference, 142, 1733-1745.
http://dx.doi.org/10.1016/j.jspi.2012.02.035 [7] Zou, H. and Zhang, H.H. (2009) On the Adaptive Elastic-Net with a Diverging Number of Parameters. The Annals of Statistics, 37, 1733-1751.
http://dx.doi.org/10.1214/08-AOS625 [8] Xie, H.L. and Huang, J. (2009) SCAD-Penalized Regression in High-Dimensional Partially Linear Models. The Annals of Statistics, 37, 673-696.
http://dx.doi.org/10.1214/07-AOS580 [9] Fan, J.Q. and Li, R.Z. (2001) Variable Selection via Nonconcave Penalized Likelihood and Its Oracle Properties. Journal of the American Statistical Association, 96, 1348-1360.
http://dx.doi.org/10.1198/016214501753382273 [10] Ni, X., Zhang, H.H. and Zhang, D.W. (2009) Automatic Model Selection for Partially Linear Models. Journal of multivariate Analysis, 100, 2100-2111.
http://dx.doi.org/10.1016/j.jmva.2009.06.009 [11] Koenker, R. (2005) Quantile Regression. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511754098 [12] Green, P.J. and Silverman, B.W. (1994) Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Chapman & Hall, London. [13] Hunter, D.R. and Lange, K. (2004) A Tutorial on MM Algorithms. The American Statistician, 58, 30-37.
http://dx.doi.org/10.1198/0003130042836 [14] Hunter, D.R. and Lange, K. (2000) Quantile Regression via an MM Algorithm. Journal of Computational and Graphical Statistics, 9, 60-77.
http://dx.doi.org/10.2307/1390613 [15] Zou, H. and Li, R.Z. (2008) One-Step Sparse Estimates in Nonconcave Penalized Likelihood Models. The Annals of Statistics, 36, 1509-1533.
http://dx.doi.org/10.1214/009053607000000802 [16] Chen, J.H. and Chen, Z.H. (2008) Extended Bayesian Information Criteria for Model Selection with Large Model Spaces. Biometrika, 95, 759-771.
http://dx.doi.org/10.1093/biomet/asn034 [17] Ruppert, D., Wand, M.P. and Carroll, R. J. (2003) Semiparametric Regression. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511755453