Robust Variable Selection With Exponential Squared Loss

Abstract

Robust variable selection procedures through penalized regression have been gaining increased attention in the literature. They can be used to perform variable selection and are expected to yield robust estimates. However, to the best of our knowledge, the robustness of those penalized regression procedures has not been well characterized. In this article, we propose a class of penalized robust regression estimators based on exponential squared loss. The motivation for this new procedure is that it enables us to characterize its robustness in a way that has not been done for the existing procedures, while its performance is near optimal and superior to some recently developed methods. Specifically, under defined regularity conditions, our estimators are -consistent and possess the oracle property. Importantly, we show that our estimators can achieve the highest asymptotic breakdown point of 1/2 and that their influence functions are bounded with respect to the outliers in either the response or the covariate domain. We performed simulation studies to compare our proposed method with some recent methods, using the oracle method as the benchmark. We consider common sources of influential points. Our simulation studies reveal that our proposed method performs similarly to the oracle method in terms of the model error and the positive selection rate even in the presence of influential points. In contrast, other existing procedures have a much lower noncausal selection rate. Furthermore, we reanalyze the Boston Housing Price Dataset and the Plasma Beta-Carotene Level Dataset that are commonly used examples for regression diagnostics of influential points. Our analysis unravels the discrepancies of using our robust method versus the other penalized regression method, underscoring the importance of developing and applying robust penalized regression methods.

KEY WORDS:

• Breakdown point
• Influence function
• Robust regression

Acknowledgments

Huang's research is partially supported by a funding through Projet 211 Phase 3 of Shanghai University of Finance and Economics and Shanghai Leading Academic Discipline Project, B803. Wang's research is partially supported by the National Natural Science Foundation of China (11271383), RFDP(20110171110037), the Scientific Research Foundation for Returned Overseas Chinese Scholars, State Education Ministry, and Fundamental Research Funds for the Central Universities. Zhang's research is partially supported by the National Institutes of Health grant R01DA016750-08 from the U.S. National Institute on Drug Abuse.