The adaptive robust and efficient variable selection method for the linear regression model

.The penalized robust variable selection procedure based on the exponential squared loss (ESL) proposed by Wang et al. (Citation2013) has shown that it not only has very good robustness for outliers but also is as asymptotically efficient as the least squares method without outliers under normal error. However, its finite-sample relative efficiency with respect to the penalized least squares estimator may be much smaller than one under normal error when the sample size n is not very large. To address this issue, we propose an adaptive robust variable selection method via a hybrid of the squared loss function and ESL function. By a data-driven procedure to select the weight, the proposed method is equivalent to the penalized least squares method, while it can maintain the robustness of the penalized ESL estimator. Under some conditions, the asymptotic properties of the proposed estimator are established. Furthermore, the promising performances of the proposed method are supported by extensive numerical simulations and a real data example.