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The L1 penalized LAD estimator for high dimensional linear regression

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  • Wang, Lie

Abstract

In this paper, the high-dimensional sparse linear regression model is considered, where the overall number of variables is larger than the number of observations. We investigate the L1 penalized least absolute deviation method. Different from most of the other methods, the L1 penalized LAD method does not need any knowledge of standard deviation of the noises or any moment assumptions of the noises. Our analysis shows that the method achieves near oracle performance, i.e. with large probability, the L2 norm of the estimation error is of order O(klogp/n). The result is true for a wide range of noise distributions, even for the Cauchy distribution. Numerical results are also presented.

Suggested Citation

  • Wang, Lie, 2013. "The L1 penalized LAD estimator for high dimensional linear regression," Journal of Multivariate Analysis, Elsevier, vol. 120(C), pages 135-151.
    • Handle: RePEc:eee:jmvana:v:120:y:2013:i:c:p:135-151
    DOI: 10.1016/j.jmva.2013.04.001
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    References listed on IDEAS

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    1. A. Belloni & V. Chernozhukov & L. Wang, 2011. "Square-root lasso: pivotal recovery of sparse signals via conic programming," Biometrika, Biometrika Trust, vol. 98(4), pages 791-806.
    2. Wang, Hansheng & Li, Guodong & Jiang, Guohua, 2007. "Robust Regression Shrinkage and Consistent Variable Selection Through the LAD-Lasso," Journal of Business & Economic Statistics, American Statistical Association, vol. 25, pages 347-355, July.
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    Cited by:

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    6. Alexandre Belloni & Victor Chernozhukov & Lie Wang, 2013. "Pivotal estimation via square-root lasso in nonparametric regression," CeMMAP working papers CWP62/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
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    12. Alexandre Belloni & Victor Chernozhukov & Kengo Kato, 2013. "Uniform post selection inference for LAD regression and other z-estimation problems," CeMMAP working papers CWP74/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    13. Han, Dongxiao & Huang, Jian & Lin, Yuanyuan & Shen, Guohao, 2022. "Robust post-selection inference of high-dimensional mean regression with heavy-tailed asymmetric or heteroskedastic errors," Journal of Econometrics, Elsevier, vol. 230(2), pages 416-431.
    14. Umberto Amato & Anestis Antoniadis & Italia De Feis & Irene Gijbels, 2021. "Penalised robust estimators for sparse and high-dimensional linear models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(1), pages 1-48, March.
    15. Wu, Xiaofei & Ming, Hao & Zhang, Zhimin & Cui, Zhenyu, 2024. "Multi-block alternating direction method of multipliers for ultrahigh dimensional quantile fused regression," Computational Statistics & Data Analysis, Elsevier, vol. 192(C).
    16. Xiaofei Wu & Rongmei Liang & Hu Yang, 2022. "Penalized and constrained LAD estimation in fixed and high dimension," Statistical Papers, Springer, vol. 63(1), pages 53-95, February.
    17. Chen, Le-Yu & Lee, Sokbae, 2023. "Sparse quantile regression," Journal of Econometrics, Elsevier, vol. 235(2), pages 2195-2217.
    18. Xianchao Xiu & Lingchen Kong & Yan Li & Houduo Qi, 2018. "Iterative reweighted methods for $$\ell _1-\ell _p$$ ℓ 1 - ℓ p minimization," Computational Optimization and Applications, Springer, vol. 70(1), pages 201-219, May.
    19. Jianqing Fan & Quefeng Li & Yuyan Wang, 2017. "Estimation of high dimensional mean regression in the absence of symmetry and light tail assumptions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(1), pages 247-265, January.
    20. Li, Mei & Kong, Lingchen, 2019. "Double fused Lasso penalized LAD for matrix regression," Applied Mathematics and Computation, Elsevier, vol. 357(C), pages 119-138.
    21. Fan, Jianqing & Feng, Yang & Xia, Lucy, 2020. "A projection-based conditional dependence measure with applications to high-dimensional undirected graphical models," Journal of Econometrics, Elsevier, vol. 218(1), pages 119-139.
    22. Wang, Yibo & Karunamuni, Rohana J., 2022. "High-dimensional robust regression with Lq-loss functions," Computational Statistics & Data Analysis, Elsevier, vol. 176(C).
    23. Meira, Erick & Lila, Maurício Franca & Cyrino Oliveira, Fernando Luiz, 2023. "A novel reconciliation approach for hierarchical electricity consumption forecasting based on resistant regression," Energy, Elsevier, vol. 269(C).
    24. Yuyang Liu & Pengfei Pi & Shan Luo, 2023. "A semi-parametric approach to feature selection in high-dimensional linear regression models," Computational Statistics, Springer, vol. 38(2), pages 979-1000, June.

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    More about this item

    Keywords

    High dimensional regression; LAD estimator; L1 penalization; Variable selection;
    All these keywords.

    JEL classification:

    • L1 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance

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