Balancing stability and bias reduction in variable selection with the Mnet estimator

We propose a new penalized approach for variable selection using a combination of minimax concave and ridge penalties. The proposed method is designed to deal with p ≥ n problems with highly correlated predictors. We call the propose approach the Mnet method. Similar to the elastic net of Zou and Hastie (2005), the Mnet also tends to select or drop highly correlated predictors together. However, unlike the elastic net, the Mnet is selection consistent and equal to the oracle ridge estimator with high probability under reasonable conditions. We develop an efficient coordinate descent algorithm to compute the Mnet estimates. Simulation studies show that the Mnet has better performance in the presence of highly correlated predictors than either the elastic net or MCP. Finally, we illustrate the application of the Mnet to real data from a gene expression study in ophthalmology.