[go: up one dir, main page]
IDEAS home Printed from https://ideas.repec.org/a/taf/jnlasa/v107y2012i497p214-222.html
   My bibliography  Save this article

Quantile Regression for Analyzing Heterogeneity in Ultra-High Dimension

Author

Listed:
  • Lan Wang
  • Yichao Wu
  • Runze Li
Abstract
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
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1080/01621459.2012.656014
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1080/01621459.2012.656014?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Zou, Hui, 2006. "The Adaptive Lasso and Its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1418-1429, December.
    2. Koenker,Roger, 2005. "Quantile Regression," Cambridge Books, Cambridge University Press, number 9780521845731.
    3. Kim, Yongdai & Choi, Hosik & Oh, Hee-Seok, 2008. "Smoothly Clipped Absolute Deviation on High Dimensions," Journal of the American Statistical Association, American Statistical Association, vol. 103(484), pages 1665-1673.
    4. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    5. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
    6. Jianqing Fan & Jinchi Lv, 2008. "Sure independence screening for ultrahigh dimensional feature space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(5), pages 849-911, November.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kean Ming Tan & Lan Wang & Wen‐Xin Zhou, 2022. "High‐dimensional quantile regression: Convolution smoothing and concave regularization," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(1), pages 205-233, February.
    2. Tang, Linjun & Zhou, Zhangong & Wu, Changchun, 2012. "Weighted composite quantile estimation and variable selection method for censored regression model," Statistics & Probability Letters, Elsevier, vol. 82(3), pages 653-663.
    3. Shan Luo & Zehua Chen, 2014. "Sequential Lasso Cum EBIC for Feature Selection With Ultra-High Dimensional Feature Space," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1229-1240, September.
    4. Tang, Yanlin & Song, Xinyuan & Wang, Huixia Judy & Zhu, Zhongyi, 2013. "Variable selection in high-dimensional quantile varying coefficient models," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 115-132.
    5. Loann David Denis Desboulets, 2018. "A Review on Variable Selection in Regression Analysis," Econometrics, MDPI, vol. 6(4), pages 1-27, November.
    6. Zhang, Ting & Wang, Lei, 2020. "Smoothed empirical likelihood inference and variable selection for quantile regression with nonignorable missing response," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    7. Jiang, Rong & Qian, Weimin & Zhou, Zhangong, 2012. "Variable selection and coefficient estimation via composite quantile regression with randomly censored data," Statistics & Probability Letters, Elsevier, vol. 82(2), pages 308-317.
    8. Canhong Wen & Xueqin Wang & Shaoli Wang, 2015. "Laplace Error Penalty-based Variable Selection in High Dimension," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(3), pages 685-700, September.
    9. Fan, Zengyan & Lian, Heng, 2018. "Quantile regression for additive coefficient models in high dimensions," Journal of Multivariate Analysis, Elsevier, vol. 164(C), pages 54-64.
    10. Ricardo P. Masini & Marcelo C. Medeiros & Eduardo F. Mendes, 2023. "Machine learning advances for time series forecasting," Journal of Economic Surveys, Wiley Blackwell, vol. 37(1), pages 76-111, February.
    11. Lina Liao & Cheolwoo Park & Hosik Choi, 2019. "Penalized expectile regression: an alternative to penalized quantile regression," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(2), pages 409-438, April.
    12. 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.
    13. Joel L. Horowitz, 2015. "Variable selection and estimation in high-dimensional models," CeMMAP working papers 35/15, Institute for Fiscal Studies.
    14. Bang, Sungwan & Jhun, Myoungshic, 2012. "Simultaneous estimation and factor selection in quantile regression via adaptive sup-norm regularization," Computational Statistics & Data Analysis, Elsevier, vol. 56(4), pages 813-826.
    15. Yazhao Lv & Riquan Zhang & Weihua Zhao & Jicai Liu, 2015. "Quantile regression and variable selection of partial linear single-index model," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(2), pages 375-409, April.
    16. Yao, Fang & Sue-Chee, Shivon & Wang, Fan, 2017. "Regularized partially functional quantile regression," Journal of Multivariate Analysis, Elsevier, vol. 156(C), pages 39-56.
    17. Tian, Yuzhu & Song, Xinyuan, 2020. "Bayesian bridge-randomized penalized quantile regression," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    18. Joel L. Horowitz, 2015. "Variable selection and estimation in high‐dimensional models," Canadian Journal of Economics/Revue canadienne d'économique, John Wiley & Sons, vol. 48(2), pages 389-407, May.
    19. Wang, Shangshan & Xiang, Liming, 2017. "Two-layer EM algorithm for ALD mixture regression models: A new solution to composite quantile regression," Computational Statistics & Data Analysis, Elsevier, vol. 115(C), pages 136-154.
    20. Yang, Hu & Yang, Jing, 2014. "A robust and efficient estimation and variable selection method for partially linear single-index models," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 227-242.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:taf:jnlasa:v:107:y:2012:i:497:p:214-222. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Longhurst (email available below). General contact details of provider: http://www.tandfonline.com/UASA20 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.