Penalized composite quasi-likelihood for ultrahigh dimensional variable selection

In high dimensional model selection problems, penalized least square approaches have been extensively used. The paper addresses the question of both robustness and efficiency of penalized model selection methods and proposes a data-driven weighted linear combination of convex loss functions, together with weighted L-1-penalty. It is completely data adaptive and does not require prior knowledge of the error distribution. The weighted L-1-penalty is used both to ensure the convexity of the penalty term and to ameliorate the bias that is caused by the L-1-penalty. In the setting with dimensionality much larger than the sample size, we establish a strong oracle property of the method proposed that has both the model selection consistency and estimation efficiency for the true non-zero coefficients. As specific examples, we introduce a robust method of composite L-1-L-2, and an optimal composite quantile method and evaluate their performance in both simulated and real data examples.