Nonconcave penalized likelihood with a diverging number of parameters

Jianqing Fan, Heng Peng

Research output: Contribution to journalArticlepeer-review

706 Scopus citations


A class of variable selection procedures for parametric models via non-concave penalized likelihood was proposed by Fan and Li to simultaneously estimate parameters and select important variables. They demonstrated that this class of procedures has an oracle property when the number of parameters is finite. However, in most model selection problems the number of parameters should be large and grow with the sample size. In this paper some asymptotic properties of the nonconcave penalized likelihood are established for situations in which the number of parameters tends to ∞ as the sample size increases. Under regularity conditions we have established an oracle property and the asymptotic normality of the penalized likelihood estimators. Furthermore, the consistency of the sandwich formula of the covariance matrix is demonstrated. Nonconcave penalized likelihood ratio statistics are discussed, and their asymptotic distributions under the null hypothesis are obtained by imposing some mild conditions on the penalty functions. The asymptotic results are augmented by a simulation study, and the newly developed methodology is illustrated by an analysis of a court case on the sexual discrimination of salary.

Original languageEnglish (US)
Pages (from-to)928-961
Number of pages34
JournalAnnals of Statistics
Issue number3
StatePublished - Jun 2004

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • Asymptotic normality
  • Diverging parameters
  • Likelihood ratio statistic
  • Model selection
  • Nonconcave penalized likelihood
  • Oracle property
  • Standard errors


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