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**Nonconcave penalized likelihood with a diverging number of parameters.**
*(English)*
Zbl 1092.62031

Summary: A class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed by J. Fan and R. Li [ibid. 30, 74–99 (2002; Zbl 1012.62106); J. Am. Stat. Assoc. 96, No. 456, 1348–1360 (2001; Zbl 1073.62547)] 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 \(\infty\) 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.

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.

### MSC:

62F12 | Asymptotic properties of parametric estimators |

62E20 | Asymptotic distribution theory in statistics |

62F03 | Parametric hypothesis testing |

### Keywords:

model selection; nonconcave penalized likelihood; diverging parameters; oracle property; asymptotic normality; standard errors; likelihood ratio statistic### Software:

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\textit{J. Fan} and \textit{H. Peng}, Ann. Stat. 32, No. 3, 928--961 (2004; Zbl 1092.62031)

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