Variable selection for high dimensional partially linear varying coefficient errors-in-variables models

Abstract

In this paper,  we consider variable selection procedure for the high dimensional  partially linear varying coefficient models where  the parametric part covariates are measured with additive errors.  The penalized bias-corrected profile least squares estimators are conducted, and their asymptotic properties are also studied under some regularity conditions. The rate of convergence and the asymptotic normality of the resulting estimates are established. We further demonstrate that, with proper choices of the penalty functions and the regularization parameter, the resulting estimates perform asymptotically as well as an oracle property. Choice of smoothing parameters is also discussed. Finite sample performance of the proposed variable selection procedures is assessed by Monte Carlo simulation studies.

Keywords

• High dimensionality,Measurement error,Semiparametric models,Local linear regression,Variable selection

References

1. Ahmad, I., Leelahanon, S. and Li, Q. Efficient estimation of a semiparametric partially linear varying coefficient model, The Annals of Statistics 33 (1), 258-283, 2005.
2. Carroll, R. J., Ruppert, D., Stefanski, L. A. and Crainiceanu, C. M. Measurement Error in Nonlinear Models (2nd ed), New York: Chapman and Hall, 2006.
3. Fan, J.Q. and Huang, T. Profile likelihood inferences on semiparametric varying coefficient partially linear models, Bernoulli 11 (6), 1031-1057, 2005.
4. Fan, J.Q. and Li, R.Z. Variable selection via nonconcave penalized likelihood and its oracle properties, Journal of the American Statistical Association 96 (456), 1348-1360, 2001.
5. Fan, J.Q. and Peng, H. Nonconcave penalized likelihood with a diverging number of parameters, The Annals of Statistics 32 (3), 928-961, 2004.
6. Feng, S.Y. and Xue, L.G. Bias-corrected statistical inference for partially linear varying coefficient errors-in-variables models with restricted condition, Annals of the Institute of Statistical Mathematics 66 (1), 121-140, 2014.
7. Fuller, W. A. Measurement Error Models, New York: John Wiley, 1987.
8. Hu, X.M., Wang, Z.Z. and Zhao, Z.Z. Empirical likelihood for semiparametric varying-coefficient partially linear errors-in-variables models, Statistics and Probability Letters 79 (8), 1044-1052, 2009.

9. Hall, P. and Heyde, C. (1980). Martingale limit theory and its application. Academic Press.
10. Huang, Z. and Zhang, R.Q. Empirical likelihood for nonparametric parts in semiparametric varying-coefficient partially linear models, Statistics and Probability Letters 79(16), 1798-1808, 2009.
11. Kai, B.,Li, R. Z. and Zou, H. New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models, The Annals of Statistics 39 (1), 305-332, 2011.
12. Li, G.R., Feng, S.Y. and Peng, H. Profile-type smoothed score function for a varying coefficient partially linear model, Journal of Multivariate Analysis 102 (2), 372-385, 2011.
13. Li, G.R., Lin, L. and Zhu, L.X. Empirical likelihood for varying coefficient partially linear model with diverging number of parameters, Journal of Multivariate Analysis 105 (1), 85-111, 2012.
14. Li, R.Z. and Liang, H. Variable selection in semiparametric regression modeling, The Annals of Statistics 36 (1), 261-286, 2008.
15. Liang, H., Hadle, W. and Carroll, R.J. Estimation in a semiparametric partially linear errors-in-variables model, The Annals of Statistics 27 (5), 1519-1535, 1999.
16. Ruppert, D., Wand, M. and Carroll, R. Semiparametric Regression. Cambridge University Press, 2003.
17. Tibshirani, R. J. Regression shrinkage and selection via the Lasso, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 58 (1), 267-288, 1996.
18. Wang, K. and Lin, L. Simultaneous structure estimation and variable selection in partial linear varying coefficient models for longitudinal data, Journal of Statistical Computation and Simulation 85 (7), 1459-1473, 2015.
19. Wang, X. L., Li, G. R. and Lin, L. Empirical likelihood inference for semiparametric varying-coefficient partially linear EV models, Metrika 73 (2), 171-185, 2011.
20. Wei, C. H. Statistical inference for restricted partially linear varying coefficient errors-in-variables models, Journal of Statistical Planning and Inference 142 (8), 2464-2472, 2012.
21. You, J. H. and Chen, G. M. Estimation of a semiparametric varying-coefficient partially linear errors-in-variables model, Journal of Multivariate Analysis 97 (2), 324-341, 2006.
22. Zhang, C.H. Nearly unbiased variable selection under minimax concave penalty, The Annals of Statistics 38 (2), 894-942, 2010.
23. Zhang, W., Lee, S., Y. and Song, X. Local polynomial fitting in semivarying coefficient models, Journal of Multivariate Analysis 82 (1), 166-188, 2002.
24. Zhao, P.X. and Xue, L.G. Variable selection for semiparametric varying coefficient partially linear models, Statistics and Probability Letters 79 (20), 2148-2157, 2009.
25. Zhao, P.X. and Xue, L.G. Variable selection for semiparametric varying coefficient partially linear errors-in-variables models, Journal of Multivariate Analysis 101 (8), 1872-1883, 2010.
26. Zhou, Y. and Liang, H. Statistical inference for semiparametric varying-coefficient partially linear models with generated regressors, The Annals of Statistics 37 (1), 427-458, 2009.
27. Zou, H. The adaptive lasso and its oracle properties, Journal of the American Statistical Association 101 (476), 1418-1429, 2006.
28. Zou, H. and Hastie, T. Regularization and variable selection via the elastic net, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 67 (2), 301-320, 2005.