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Quantile Regression for Analyzing Heterogeneity in Ultra-High Dimension


  • Lan Wang
  • Yichao Wu
  • Runze Li


Ultra-high dimensional data often display heterogeneity due to either heteroscedastic variance or other forms of non-location-scale covariate effects. To accommodate heterogeneity, we advocate a more general interpretation of sparsity, which assumes that only a small number of covariates influence the conditional distribution of the response variable, given all candidate covariates; however, the sets of relevant covariates may differ when we consider different segments of the conditional distribution. In this framework, we investigate the methodology and theory of nonconvex, penalized quantile regression in ultra-high dimension. The proposed approach has two distinctive features: (1) It enables us to explore the entire conditional distribution of the response variable, given the ultra-high-dimensional covariates, and provides a more realistic picture of the sparsity pattern; (2) it requires substantially weaker conditions compared with alternative methods in the literature; thus, it greatly alleviates the difficulty of model checking in the ultra-high dimension. In theoretic development, it is challenging to deal with both the nonsmooth loss function and the nonconvex penalty function in ultra-high-dimensional parameter space. We introduce a novel, sufficient optimality condition that relies on a convex differencing representation of the penalized loss function and the subdifferential calculus. Exploring this optimality condition enables us to establish the oracle property for sparse quantile regression in the ultra-high dimension under relaxed conditions. The proposed method greatly enhances existing tools for ultra-high-dimensional data analysis. Monte Carlo simulations demonstrate the usefulness of the proposed procedure. The real data example we analyzed demonstrates that the new approach reveals substantially more information as compared with alternative methods. This article has online supplementary material.

Suggested Citation

  • Lan Wang & Yichao Wu & Runze Li, 2012. "Quantile Regression for Analyzing Heterogeneity in Ultra-High Dimension," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(497), pages 214-222, March.
  • Handle: RePEc:taf:jnlasa:v:107:y:2012:i:497:p:214-222
    DOI: 10.1080/01621459.2012.656014

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    Cited by:

    1. repec:eee:jmvana:v:164:y:2018:i:c:p:54-64 is not listed on IDEAS
    2. Sherwood, Ben, 2016. "Variable selection for additive partial linear quantile regression with missing covariates," Journal of Multivariate Analysis, Elsevier, vol. 152(C), pages 206-223.
    3. Sokbae Lee & Myung Hwan Seo & Youngki Shin, 2016. "The lasso for high dimensional regression with a possible change point," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 193-210, January.
    4. repec:eee:csdana:v:115:y:2017:i:c:p:136-154 is not listed on IDEAS
    5. Demian Pouzo, 2015. "On the Non-Asymptotic Properties of Regularized M-estimators," Papers 1512.06290,, revised Oct 2016.
    6. repec:spr:aistmt:v:69:y:2017:i:4:d:10.1007_s10463-016-0566-9 is not listed on IDEAS
    7. repec:spr:aistmt:v:70:y:2018:i:3:d:10.1007_s10463-017-0599-8 is not listed on IDEAS
    8. repec:eee:jmvana:v:165:y:2018:i:c:p:1-13 is not listed on IDEAS
    9. He, Qianchuan & Kong, Linglong & Wang, Yanhua & Wang, Sijian & Chan, Timothy A. & Holland, Eric, 2016. "Regularized quantile regression under heterogeneous sparsity with application to quantitative genetic traits," Computational Statistics & Data Analysis, Elsevier, vol. 95(C), pages 222-239.
    10. Xiang Zhang & Yichao Wu & Lan Wang & Runze Li, 2016. "Variable selection for support vector machines in moderately high dimensions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 53-76, January.
    11. repec:spr:coopap:v:70:y:2018:i:1:d:10.1007_s10589-017-9977-7 is not listed on IDEAS
    12. Tang, Yanlin & Wang, Huixia Judy & Zhu, Zhongyi, 2013. "Variable selection in quantile varying coefficient models with longitudinal data," Computational Statistics & Data Analysis, Elsevier, vol. 57(1), pages 435-449.
    13. repec:eee:stapro:v:137:y:2018:i:c:p:304-311 is not listed on IDEAS
    14. Lian, Heng & Meng, Jie & Fan, Zengyan, 2015. "Simultaneous estimation of linear conditional quantiles with penalized splines," Journal of Multivariate Analysis, Elsevier, vol. 141(C), pages 1-21.
    15. Zhaoping Hong & Yuao Hu & Heng Lian, 2013. "Variable selection for high-dimensional varying coefficient partially linear models via nonconcave penalty," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(7), pages 887-908, October.
    16. repec:bla:istatr:v:85:y:2017:i:3:p:494-518 is not listed on IDEAS
    17. Yao, Fang & Sue-Chee, Shivon & Wang, Fan, 2017. "Regularized partially functional quantile regression," Journal of Multivariate Analysis, Elsevier, vol. 156(C), pages 39-56.
    18. Kaul, Abhishek & Koul, Hira L., 2015. "Weighted ℓ1-penalized corrected quantile regression for high dimensional measurement error models," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 72-91.
    19. Sokbae Lee & Yuan Liao & Myung Hwan Seo & Youngki Shin, 2016. "Oracle Estimation of a Change Point in High Dimensional Quantile Regression," Papers 1603.00235,, revised Dec 2016.
    20. Xia, Xiaochao & Liu, Zhi & Yang, Hu, 2016. "Regularized estimation for the least absolute relative error models with a diverging number of covariates," Computational Statistics & Data Analysis, Elsevier, vol. 96(C), pages 104-119.

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