Statistical Methodology 32 (2016) 147–160 Contents lists available at ScienceDirect Statistical Methodology journal homepage: www.elsevier.com/locate/stamet Going beyond oracle property: Selection consistency and uniqueness of local solution of the generalized linear model Chi Tim Ng a,∗ , Seungyoung Oh b , Youngjo Lee b a Department of Statistics, Chonnam National University, Gwangju 500-757, South Korea b Department of Statistics, Seoul National University, Seoul 151-747, South Korea article info Article history: Received 22 September 2015 Received in revised form 23 May 2016 Accepted 26 May 2016 Available online 8 June 2016 MSC: 62F12 62J12 Keywords: Generalized linear model Penalized likelihood estimation Oracle property SCAD penalty Selection consistency abstract 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. © 2016 Elsevier B.V. All rights reserved. 1. Introduction The penalized likelihood approach based on the least absolute shrinkage and selection operator (LASSO) penalty of Tibshirani [18] has been widely used in the simultaneous variable selection and estimation. One advantage of the LASSO estimation is the uniqueness of the local solution due to the ∗ Corresponding author. E-mail addresses: easterlyng@gmail.com (C.T. Ng), mnbv1689@naver.com (S. Oh), youngjo@snu.ac.kr (Y. Lee). http://dx.doi.org/10.1016/j.stamet.2016.05.006 1572-3127/© 2016 Elsevier B.V. All rights reserved.