Science China Mathematics

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

Variable selection in censored quantile regression with high dimensional data

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Abstract

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.

Keywords

adaptive LASSO censoring high dimensional quantile regression 

MSC(2010)

62G20 62J05 62J07 

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Notes

Acknowledgements

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.

References

  1. 1.
    Alhamzawi R, Yu K, Benoit D. Bayesian adaptive lasso quantile regression. Stat Model, 2012, 12: 279–297MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Belloni A, Chernozhukov V. ℓ1-Penalized quantile regression in high dimensional sparse models. Ann Statist, 2011, 39: 82–130MathSciNetCrossRefMATHGoogle 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: 2313–2351MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Carroll R, Fan J, Gijbels I, et al. Generalized partially linear single-index models. J Amer Statist Assoc, 1997, 92: 477–489MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen J, Chen Z. Extended Bayesian information criteria for model selection with large model space. Biometrika, 2008, 95: 759–771MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chernozhukov V, Hong H. Three-step censored quantile regression and extramarital affairs. J Amer Statist Assoc, 2002, 97: 872–882MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fan J, Li R. Variable selection via nonconcave penalized likelihood and its oracle properties. J Amer Statist Assoc, 2001, 96: 1348–1360MathSciNetCrossRefMATHGoogle 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: 3567–3604MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fitzenberger B, Winker P. Improving the computation of censored quantile regressions. Comput Statist Data Anal, 2007, 52: 88–108MathSciNetCrossRefMATHGoogle 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: 1075–1089MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Huang J, Ma S, Zhang C. Adaptive Lasso for sparse high dimensional regression models. Statist Sinica, 2008, 18: 1603–1618MathSciNetMATHGoogle Scholar
  14. 14.
    Koenker R. Quantile Regression. Cambridge: Cambridge University Press, 2005CrossRefMATHGoogle 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: 33–50MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Koenker R, Park B. An interior point algorithm for nonlinear quantile regression. J Econometrics, 1996, 71: 265–283MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Meinshausen N, Yu B. LASSO-type recovery of sparse representations for high dimensional data. Ann Statist, 2009, 37: 246–270MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Powell J. Censored regression quantiles. J Econometrics, 1986, 32: 143–155MathSciNetCrossRefMATHGoogle 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: 635–655MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Tibshirani R. Regression shrinkage and selection via the LASSO. J Roy Statist Soc Ser B, 1996, 58: 267–288MathSciNetMATHGoogle Scholar
  24. 24.
    Tobin J. Estimation of relationships for limited dependent variables. Econometrica, 1958, 26: 24–36MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Volgushev S, Wagener J, Dette H. Censored quantile regression processes under dependence and penalization. Electron J Stat, 2014, 8: 2405–2447MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Wang H, Fygenson M. Inference for censored quantile regression models in longitudinal studies. Ann Statist, 2009, 37: 756–781MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Wang H, Zhou J, Li Y. Variable selection for censored quantile regression. Statist Sinica, 2013, 23: 145–167MathSciNetMATHGoogle Scholar
  28. 28.
    Wang L, Wu Y, Li R. Quantile regression for analyzing heterogeneity in ultra-high dimension. J Amer Statist Assoc, 2012, 107: 214–222MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Wu Y, Liu Y. Variable selection in quantile regression. Statist Sinica, 2009, 19: 801–817MathSciNetMATHGoogle Scholar
  30. 30.
    Zhang C, Huang J. The sparsity and bias of the LASSO selection in high dimensional linear regression. Ann Statist, 2008, 36: 1567–1594MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Zou H. The adaptive LASSO and its oracle properties. J Amer Statist Assoc, 2006, 101: 1418–1429MathSciNetCrossRefMATHGoogle 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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