Going beyond oracle property: Selection consistency and uniqueness of local solution of the generalized linear model

Recently, the selection consistency of penalized least square estimators has received a great deal of attention. For the penalized likelihood estimation with certain non-convex penalties, search space can be constructed within which there exists a unique local minimizer that exhibits selection consistency in high-dimensional generalized linear models under certain conditions. In particular, we prove that the SCAD penalty of Fan and Li (2001) and a new modified version of the unbounded penalty of Lee and Oh (2014) can be employed to achieve such a property. These results hold even for the non-sparse cases where the number of relevant covariates increases with the sample size. Simulation studies are provided to compare the performance of SCAD penalty and the newly proposed penalty.