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In this paper, we study robust variable selection problem by combining the idea of quantile regression and random LASSO. A two-step algorithm is proposed to solve the proposed optimization problem. In the first step, we use bootstrap samples and variable subsets to estimate the importance of each variable. In the second step, the importance measures are used in generating variable subsets and then the adaptive quantile LASSO is applied to reduce the bias of estimators. The proposed method is robust and can handle the situation of highly-correlated variables. Meanwhile, the number of selected variables is no longer limited by the sample size. Simulation studies indicate that the proposed method has good robustness and better performance when the error term is heavy-tailed and there are highly correlated variables. Finally, we apply the proposed methodology to analyze a real data. The results reveal that the propose has better the predictive ability.

Robust variable selection based on the random quantile LASSO