Abstract
In this article, we consider the penalized LAD regression which deals with variable selection and estimation simultaneously for single-index models. The proposed estimator is robust and efficient with respect to heavy tailed errors or outliers in the response. Then, we introduce a practical algorithm where the unknown link function is estimated by local linear smoothing combined with LAD regression and the parametric index is estimated through linear LAD regression. Furthermore, we show that, under certain appropriate conditions, the penalized LAD estimator has desired large sample properties including -consistency, and oracle property. Simulation studies and a real data application are carried to illustrate the performances of the proposed method for finite-sample cases. Finally, based on the idea of sure independence screening procedure and the distance correlation (DC-SIS) proposed by Li et al. (2012), a robust two-step estimator is introduced to deal with ultra-high dimensional data.
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