Science China Mathematics

, Volume 61, Issue 4, pp 641–658 | Cite as

Variable selection in censored quantile regression with high dimensional data

  • Yali Fan
  • Yanlin Tang
  • Zhongyi Zhu


We propose a two-step variable selection procedure for censored quantile regression with high dimensional predictors. To account for censoring data in high dimensional case, we employ effective dimension reduction and the ideas of informative subset idea. Under some regularity conditions, we show that our procedure enjoys the model selection consistency. Simulation study and real data analysis are conducted to evaluate the finite sample performance of the proposed approach.


adaptive LASSO censoring high dimensional quantile regression 


62G20 62J05 62J07 


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This work was supported by National Natural Science Foundation of China (Grant Nos. 11401383, 11301391 and 11271080). The authors thank two anonymous reviewers for helpful comments.


  1. 1.
    Alhamzawi R, Yu K, Benoit D. Bayesian adaptive lasso quantile regression. Stat Model, 2012, 12: 279–297MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Belloni A, Chernozhukov V. ℓ1-Penalized quantile regression in high dimensional sparse models. Ann Statist, 2011, 39: 82–130MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bertsimas D, Tsitsiklis J. Introduction to Linear Optimization. Belmont: Athena Scientific, 1997Google Scholar
  4. 4.
    Candes E, Tao T. The Dantzig selector: Statistical estimation when p is much larger than n. Ann Statist, 2007, 35: 2313–2351MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Carroll R, Fan J, Gijbels I, et al. Generalized partially linear single-index models. J Amer Statist Assoc, 1997, 92: 477–489MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen J, Chen Z. Extended Bayesian information criteria for model selection with large model space. Biometrika, 2008, 95: 759–771MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chernozhukov V, Hong H. Three-step censored quantile regression and extramarital affairs. J Amer Statist Assoc, 2002, 97: 872–882MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fan J, Li R. Variable selection via nonconcave penalized likelihood and its oracle properties. J Amer Statist Assoc, 2001, 96: 1348–1360MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fan J, Lv J. Sure independence screening for ultrahigh dimensional feature space. J Roy Statist Soc Ser B, 2008, 70: 849–911MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fan J, Song R. Sure independence screening in generalized linear models with NP-dimensionality. Ann Statist, 2010, 38: 3567–3604MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fitzenberger B, Winker P. Improving the computation of censored quantile regressions. Comput Statist Data Anal, 2007, 52: 88–108MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Galvao A, Lamarche C, Lima L. Estimation of censored quantile regression for panel data with fixed effects. J Amer Statist Assoc, 2013, 108: 1075–1089MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Huang J, Ma S, Zhang C. Adaptive Lasso for sparse high dimensional regression models. Statist Sinica, 2008, 18: 1603–1618MathSciNetzbMATHGoogle Scholar
  14. 14.
    Koenker R. Quantile Regression. Cambridge: Cambridge University Press, 2005CrossRefzbMATHGoogle Scholar
  15. 15.
    Koenker R. Censored quantile regression redux. J Statist Software, 2008, 27: 1–25CrossRefGoogle Scholar
  16. 16.
    Koenker R, Bassett G. Regression quantiles. Econometrica, 1978, 46: 33–50MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Koenker R, Park B. An interior point algorithm for nonlinear quantile regression. J Econometrics, 1996, 71: 265–283MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Meinshausen N, Yu B. LASSO-type recovery of sparse representations for high dimensional data. Ann Statist, 2009, 37: 246–270MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Powell J. Censored regression quantiles. J Econometrics, 1986, 32: 143–155MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Scheetz T, Kim K, Swiderski R, et al. Regulation of gene expression in the mammalian eye and its relevance to eye disease. Proc Natl Acad Sci USA, 2006, 103: 14429–14434CrossRefGoogle Scholar
  21. 21.
    Tang Y, Song X,Wang H, et al. Variable selection in high dimensional quantile varying coeffcient models. J Multivariate Anal, 2013, 122: 115–132MathSciNetCrossRefGoogle Scholar
  22. 22.
    Tang Y, Wang H, He X, et al. An informative subset-based estimator for censored quantile regression. TEST, 2012, 21: 635–655MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Tibshirani R. Regression shrinkage and selection via the LASSO. J Roy Statist Soc Ser B, 1996, 58: 267–288MathSciNetzbMATHGoogle Scholar
  24. 24.
    Tobin J. Estimation of relationships for limited dependent variables. Econometrica, 1958, 26: 24–36MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Volgushev S, Wagener J, Dette H. Censored quantile regression processes under dependence and penalization. Electron J Stat, 2014, 8: 2405–2447MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wang H, Fygenson M. Inference for censored quantile regression models in longitudinal studies. Ann Statist, 2009, 37: 756–781MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wang H, Zhou J, Li Y. Variable selection for censored quantile regression. Statist Sinica, 2013, 23: 145–167MathSciNetzbMATHGoogle Scholar
  28. 28.
    Wang L, Wu Y, Li R. Quantile regression for analyzing heterogeneity in ultra-high dimension. J Amer Statist Assoc, 2012, 107: 214–222MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wu Y, Liu Y. Variable selection in quantile regression. Statist Sinica, 2009, 19: 801–817MathSciNetzbMATHGoogle Scholar
  30. 30.
    Zhang C, Huang J. The sparsity and bias of the LASSO selection in high dimensional linear regression. Ann Statist, 2008, 36: 1567–1594MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Zou H. The adaptive LASSO and its oracle properties. J Amer Statist Assoc, 2006, 101: 1418–1429MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.College of ScienceUniversity of Shanghai for Science and TechnologyShanghaiChina
  2. 2.School of Mathematical SciencesTongji UniversityShanghaiChina
  3. 3.Department of StatisticsFudan UniversityShanghaiChina

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