Penalized Composite Quasi-Likelihood for Ultrahigh-Dimensional Variable Selection

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In high-dimensional model selection problems, penalized simple least-square approaches have been extensively used. This 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 L₁-penalty. It is completely data-adaptive and does not require prior knowledge of the error distribution. The weighted L₁-penalty is used both to ensure the convexity of the penalty term and to ameliorate the bias caused by the L₁-penalty. In the setting with dimensionality much larger than the sample size, we establish a strong oracle property of the proposed method that possesses both the model selection consistency and estimation efficiency for the true non-zero coefficients. As specific examples, we introduce a robust method of composite L1-L2, and optimal composite quantile method and evaluate their performance in both simulated and real data examples.