2008/07/16 by Cun-Hui Zhang, Cun‐Hui Zhang · 1 citation
Mathematics · #Advanced Statistical Methods and Models #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #math.ST #stat.TH
paper · pdf · doi:10.1214/07-aos0316c
published as Annals of Statistics 2008, Vol. 36, No. 4, 1553-1560 · Published in at http://dx.doi.org/10.1214/07-AOS0316C the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2008/07/16 · arxiv created 2008/08/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Penalized methods are commonly used for selecting variables and fitting high-dimensional data. It is well known that the LASSO is biased and thus cannot attain the estimation efficiency of the oracle selector. Recent studies [6, 7, 8, 10, 11] showed that due to the interference of the bias, the LASSO requires quite strong conditions for consistent variable selection. Since the ℓ1 penalty has the smallest bias among all convex penalty functions with selection features, these studies naturally draw our attention to methodologies based on concave penalties, or equivalently, nonconcave penalized likelihood. Frank and Friedman [5] considered the ℓα penalty for general α ≥ 0, which is strictly concave for α < 1. Their main interest was to use α as a hyperparameter to “bridge ” between the subset selection with α = 0 and the ridge regression with α = 2. Important progresses were made by Fan and Li [3], who advocated the unbiasedness and continuity as essential for variable selectors and carefully developed the SCAD method. In the theoretical front, Fan and Peng [4] proved that the SCAD has the oracle property when the