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High-dimensional robust regression with Lq-loss functions

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  • Wang, Yibo
  • Karunamuni, Rohana J.

Abstract

Robust procedures in high-dimensional regression are important because outliers are often present in data. For data with heavy-tailed errors, quantile regression and least absolute deviation regression methods have been widely used with great success. Some interesting Huber-loss-based and robust M-type regularized estimators have also been developed. However, high-dimensional regression estimation under Lq-loss functions (1≤q<2) has not been fully studied in the literature. A lack of smoothness of these loss functions near the origin makes the regularized optimization problems computationally challenging. Robust sparse regression estimation under the Lq-loss functions (1≤q<2) is investigated. A regularized estimator under the Lq-loss combined with a weighted penalty function is proposed and its properties, such as the model-selection oracle property and asymptotic normality, are studied. The l1 and l2 estimation error bounds of the proposed estimator are also obtained. A novel computational algorithm is also proposed. Monte Carlo studies are conducted to compare the finite-sample and robustness properties of the proposed procedure with some existing regularized robust methods. The methods are also compared using two real data examples. The numerical studies show the satisfactory finite-sample performance of our procedure.

Suggested Citation

  • Wang, Yibo & Karunamuni, Rohana J., 2022. "High-dimensional robust regression with Lq-loss functions," Computational Statistics & Data Analysis, Elsevier, vol. 176(C).
    • Handle: RePEc:eee:csdana:v:176:y:2022:i:c:s0167947322001475
    DOI: 10.1016/j.csda.2022.107567
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    References listed on IDEAS

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