Asymptotic properties for combined L-1 and concave regularization

作者：Fan Yingying^*; Lv Jinchi

来源：Biometrika, 2014, 101(1): 57-70.

DOI：10.1093/biomet/ast047

• Concave penalty
• Global optimum
• Lasso penalty
• Prediction
• Variable selection

Two important goals of high-dimensional modelling are prediction and variable selection. In this article, we consider regularization with combined L-1 and concave penalties, and study the sampling properties of the global optimum of the suggested method in ultrahigh-dimensional settings. The L-1 penalty provides the minimum regularization needed for removing noise variables in order to achieve oracle prediction risk, while a concave penalty imposes additional regularization to control model sparsity. In the linear model setting, we prove that the global optimum of our method enjoys the same oracle inequalities as the lasso estimator and admits an explicit bound on the false sign rate, which can be asymptotically vanishing. Moreover, we establish oracle risk inequalities for the method and the sampling properties of computable solutions. Numerical studies suggest that our method yields more stable estimates than using a concave penalty alone.

• 出版日期2014-3