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

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