Penalized Composite Quasi-Likelihood for Ultrahigh-Dimensional Variable Selection

TL;DR

This paper introduces a robust, data-driven penalized composite quasi-likelihood method for ultrahigh-dimensional variable selection, achieving model selection consistency and estimation efficiency without prior error distribution knowledge.

Contribution

It proposes a novel, fully data-adaptive weighted composite quasi-likelihood approach with weighted L1-penalty, ensuring robustness and efficiency in ultrahigh-dimensional settings.

Findings

Achieves strong oracle property with model selection consistency

Demonstrates robustness through composite L1-L2 and quantile methods

Performs well in simulated and real data applications

Abstract

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 $L_1$-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 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 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.