Nonconcave penalized likelihood with NP-dimensionality

Jianqing Fan, Jinchi Lv

Research output: Contribution to journalReview articlepeer-review

238 Scopus citations


Penalized likelihood methods are fundamental to ultrahigh dimensional variable selection. How high dimensionality such methods can handle remains largely unknown. In this paper, we show that in the context of generalized linear models, such methods possess model selection consistency with oracle properties even for dimensionality of nonpolynomial (NP) order of sample size, for a class of penalized likelihood approaches using folded-concave penalty functions, which were introduced to ameliorate the bias problems of convex penalty functions. This fills a long-standing gap in the literature where the dimensionality is allowed to grow slowly with the sample size. Our results are also applicable to penalized likelihood with the L1-penalty, which is a convex function at the boundary of the class of folded-concave penalty functions under consideration. The coordinate optimization is implemented for finding the solution paths, whose performance is evaluated by a few simulation examples and the real data analysis.

Original languageEnglish (US)
Article number5961830
Pages (from-to)5467-5484
Number of pages18
JournalIEEE Transactions on Information Theory
Issue number8
StatePublished - Aug 2011

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Library and Information Sciences
  • Computer Science Applications


  • Coordinate optimization
  • Lasso
  • SCAD
  • folded-concave penalty
  • high dimensionality
  • nonconcave penalized likelihood
  • oracle property
  • variable selection
  • weak oracle property


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