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June 2004 Nonconcave penalized likelihood with a diverging number of parameters
Jianqing Fan, Heng Peng
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Ann. Statist. 32(3): 928-961 (June 2004). DOI: 10.1214/009053604000000256


A class of variable selection procedures for parametric models via nonconcave 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.


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Jianqing Fan. Heng Peng. "Nonconcave penalized likelihood with a diverging number of parameters." Ann. Statist. 32 (3) 928 - 961, June 2004.


Published: June 2004
First available in Project Euclid: 24 May 2004

zbMATH: 1092.62031
MathSciNet: MR2065194
Digital Object Identifier: 10.1214/009053604000000256

Primary: 62F12
Secondary: 62E20 , 62J02

Keywords: asymptotic normality , diverging parameters , likelihood ratio statistic , Model selection , nonconcave penalized likelihood , oracle property , standard errors

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 3 • June 2004
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