The paper presents asymptotic properties of nonconcave penalized likelihood estimators in the context of models with a diverging number of parameters. Fan and Peng propose a class of variable selection procedures using nonconcave penalized likelihood, which can simultaneously estimate parameters and select important variables. They demonstrate that this method has an oracle property when the number of parameters is finite. However, in many model selection problems, the number of parameters grows with the sample size. The authors establish asymptotic properties for situations where the number of parameters tends to infinity as the sample size increases. Under regularity conditions, they show that the penalized likelihood estimators have an oracle property and asymptotic normality. They also demonstrate the consistency of the sandwich formula for the covariance matrix and discuss the asymptotic distributions of nonconcave penalized likelihood ratio statistics under the null hypothesis. The results are supported by a simulation study and an analysis of a court case on sexual discrimination of salary. The paper also discusses the sparsity, asymptotic normality, consistency of the sandwich formula, and likelihood ratio theory of the nonconcave penalized likelihood method. The authors show that the method can be applied to nonparametric estimation and that the oracle property holds for the SCAD and hard thresholding penalty functions. The paper concludes with numerical examples and a simulation study to illustrate the effectiveness of the method.The paper presents asymptotic properties of nonconcave penalized likelihood estimators in the context of models with a diverging number of parameters. Fan and Peng propose a class of variable selection procedures using nonconcave penalized likelihood, which can simultaneously estimate parameters and select important variables. They demonstrate that this method has an oracle property when the number of parameters is finite. However, in many model selection problems, the number of parameters grows with the sample size. The authors establish asymptotic properties for situations where the number of parameters tends to infinity as the sample size increases. Under regularity conditions, they show that the penalized likelihood estimators have an oracle property and asymptotic normality. They also demonstrate the consistency of the sandwich formula for the covariance matrix and discuss the asymptotic distributions of nonconcave penalized likelihood ratio statistics under the null hypothesis. The results are supported by a simulation study and an analysis of a court case on sexual discrimination of salary. The paper also discusses the sparsity, asymptotic normality, consistency of the sandwich formula, and likelihood ratio theory of the nonconcave penalized likelihood method. The authors show that the method can be applied to nonparametric estimation and that the oracle property holds for the SCAD and hard thresholding penalty functions. The paper concludes with numerical examples and a simulation study to illustrate the effectiveness of the method.