Penalized empirical likelihood based variable selection for partially linear quantile regression models with missing responses

Abstract

In this paper, we consider variable selection for partially linear quantile regression models with missing response at random. We first propose a role penalized empirical likelihood based variable selection method, and show that such variable selection method is consistent and satisfies sparsity. Further more, to avoid the influence of nonparametric estimator on the variable selection for the parametric components, we also propose a double penalized empirical likelihood variable selection method. Some simulation studies and a real data application are undertaken to assess the finite sample performance of the proposed variable selection methods, and simulation results indicate that the proposed variable selection methods are workable.

Keywords

• Quantile regression,Partially linear model,Variable selection,Penalized empirical likelihood

References

1. Koenker, R. and Bassett, G.S. Regression quantiles, Econometrica 46, 33-50, 1978.
2. Buchinsky, M. Recent advances in quantile regression models: A practical guide for empir- ical research, Journal of Human Resources 33, 88-126, 1998.
3. Koenker, R. and Machado, J. Goodness of t and related inference processes for quantile regression, Journal of the American Statistical Association 94, 1296-1310, 1999.
4. Tang, C.Y. and Leng, C.L. An empirical likelihood approach to quantile regression with auxiliary information, Statistics & Probability Letters 82, 29-36, 2012.
5. Hendricks, W. and Koenker, R. Hierarchical spline models for conditional quantiles and the demand for electricity, Journal of the American Statistical Association 87, 58-68, 1992.
6. Yu, K. and Jones, M.C. Local linear quantile regression, Journal of The American Statistical Association 93, 228-237, 1998.
7. Dabrowska, D.M. Nonparametric quantile regression with censored data, The Indian Journal of Statistics, Series A 54, 252-259, 1992.
8. Lee, S. Ecient semiparametric estimation of a partially linear quantile regression model, Econometric Theory 19, 1-31, 2003.

9. Sun, Y. Semiparametric ecient estimation of partially linear quantile regression models, The Annals of Economics and Finance 6, 105-127, 2005.
10. He, X. and Liang, H. Quantile regression estimates for a class of linear and partially linear errors-in-variables models, Statistica Sinica 10, 129-140, 2000.
11. Chen, S. and Khan, S. Semiparametric estimation of a partially linear censored regression model, Economic Theory 17, 567-590, 2001.
12. Lv, X. and Li, R. Smoothed empirical likelihood analysis of partially linear quantile regres- sion models with missing response variables, Advances in Statistical Analysis 97, 317-347, 2013.
13. Tang, C.Y. and Leng, C. Penalized high-dimensional empirical likelihood, Biometrika 97, 905-920, 2010.
14. Ren, Y.W. and Zhang, X.S. Variable selection using penalized empirical likelihood, Science China Mathematics 54, 1829-1845, 2011.
15. Variyath, A.M., Chen, J. and Abraham, B. Empirical likelihood based variable selection, Journal of Statistical Planning and Inference 140, 971-981, 2010.
16. Wu, T.T., Li, G. and Tang, C. Empirical likelihood for censored linear regression and vari- able selection, Scandinavian Journal of Statistics 42, 798-812, 2015.
17. Zhao, P. X. Empirical likelihood based variable selection for varying coecient partially linear models with censored data, Journal of Mathematical Research with Applications 33, 493-504, 2013.
18. Xi, R., Li, Y. and Hu, Y. Bayesian Quantile Regression Based on the Empirical Likelihood with spike and slab priors, Bayesian Analysis, Online rst, DOI: 10.1214/15-BA975, 2016.
19. Hou, W., Song, L.X., Hou, X.Y. and Wang, X.G. Penalized empirical likelihood via bridge estimator in Cox's proportional hazard model, Communications in Statistics-Theory and Methods 43, 426-440, 2014.
20. Frank, I. and Friedman, J. A statistical view of some chemometrics regression tools, Tech- nometrics 35, 109-135, 1993.
21. Tibshirani, R. Regression shrinkage and selection via the Lasso, J R Stat Soc Ser B 58, 267-288, 1996.
22. Fan, J.Q. and Li, R.Z. Variable selection via nonconcave penalized likelihood and its oracle properties, Journal of the American Statistical Association 96, 1348-1360, 2001.
23. Zhang, C. Nearly unbiased variable selection underminimax concave penalty, Ann Stat 38, 894-942, 2010.
24. Schumaker, L.L. Spline Functions, Wiley, New York, 1981.
25. Kaslow, R.A., Ostrow, D.G., Detels, R., Phair, J.P., Polk, B.F. and Rinaldo Jr, C.R. The Multicenter AIDS Cohort Study: rationale, organization, and selected characteristics of the participants, American Journal of Epidemiology 126, 310-318, 1987.
26. Huang, J., Wu, C. and Zhou, L. Varying-coecient models and basis function approxima- tions for the analysis of repeated measurements, Biometrika 89, 111-128, 2002.
27. Tang, Y., Wang, H. and Zhu, Z. Variable selection in quantile varying coecient models with longitudinal data, Comput Stat Data Anal 57, 435-449, 2013.
28. Wang, H., Zhu, Z. and Zhou, J. Quantile regression in partially linear varying coecient models, Ann Stat. 37, 3841-3866, 2009.
29. Zhao, P.X. and Xue, L.G. Variable selection for semiparametric varying coecient partially linear errors-in-variables models, Journal of Multivariate Analysis 101, 1872-1883, 2010.
30. Fan, J.Q. and Li, R. New estimation and model selection procedures for semiparametric modeling in longitudinal data analysis, Journal of the American Statistical Association 99, 710-723, 2004.
31. Xue, L.G. and Zhu, L.X. Empirical likelihood semiparametric regression analysis for longi- tudinal data, Biometrika 94, 921-937, 2007.
32. Xue, L.G. Empirical likelihood for linear models with missing responses, Journal of Multi- variate Analysis 100, 1353-1366, 2009.
33. Zhao, P.X. and Xue, L.G. Empirical likelihood inferences for semiparametric varying coef- cient partially linear models with longitudinal data, Communications in Statistics-Theory and Methods 39, 1898-1914, 2010.