Penalized Empirical Likelihood And Its Application For High-dimensional Semi-parametric Regression Models

With the continuous development of Internet big data,the amount of information in the data is more and more,and the corresponding dimension of the data is also higher and higher,how to effectively handle high-dimensional data has become a hot research in recent years.In some actual data analyses,one can obtain not only the basic sample information,but also information on the relevant regression coefficients.With these messages of regression coefficients,the estimation efficiency of the model can be improved.In reality,the true structure of the model is often unknown,and simply using only parametric regression models or non-parametric regression models to make statistical inferences,the predicted results will be highly biased and wrong results will be obtained.The semi-parametric regression model organically combines the simplicity of a parametric regression model with the flexibility of a non-parametric regression model,with the advantages of both.The study of semi-parametric regression models is very helpful for real life applications.However,in the semi-parametric regression model,the dimensionality of the covariate increases with the sample size,that is,when the covariate dimension is high,there is a "dimensional curse"problem.A large number of scholars have shown that combining the empirical likelihood method with the SCAD penalty function and applying it to the model can effectively solve the problem of variable selection in the case of high-dimensional data,thereby reducing the complexity of the model and solving the problem of model instability in making predictions.Therefore,the study of high-dimensional semi-parametric regression model for penalty empirical inference not only expands the scope of application of semi-parametric regression models,but also provides scientific reference for scholars who study such models.The first chapter mainly introduces the historical background and significance of the research on high-dimensional semi-parametric regression models for penalized empirical likelihood estimation,and the current state of domestic and international research on semi-parametric regression models and penalized empirical likelihood estimation,and concludes with a brief description of the content and structure of the article.In the second chapter,we introduces how to explore the empirical likelihood inference of the nonlinear semi-parametric measurement error model under the premise that the error is a martingale difference sequence.First,the effect of measurement error is eliminated by the method of inverse convolution,so as to obtain an unbiased estimate of the parameters.Then,the introduced auxiliary function is used to construct the empirical likelihood ratio statistic and its empirical likelihood confidence region.Finally,it is proved that it follows the progressive chi-square distribution.In the third chapter,we consider the penalized empirical likelihood inference for the high-dimensional partial linear semi-parametric measurement error model when the error is a martingale difference sequence.First of all,the effect of measurement error is eliminated by the method of inverse convolution,and an unbiased estimate is obtained.Secondly,the empirical likelihood method is used to construct the empirical likelihood ratio statistic.Then,for the discrete number of parameters,the parameter penalized empirical likelihood(PEL)method is proposed to construct the penalized empirical likelihood ratio statistic and its penalized empirical likelihood confidence region,and proves that the proposed penalized empirical likelihood estimator has Oracle property.Finally,according to the assumptions of the model,the numerical simulation of the punishment empirical likelihood estimation of the parameter components is carried out to verify the accuracy of the estimation results,and an example analysis is carried out.In the fourth chapter,for the high-dimensional partial linear varying coefficient panel data model with fixed effects.First of all,the B-spline method is used to obtain non-parametric estimates,and then the local linear dummy variable method is used to eliminate the fixed effects,and a suitable auxiliary function is introduced to construct the empirical log-likelihood ratio function.Finally,the idea of SCAD function and penalty function is combined,the penalized empirical likelihood method is introduced and applied to the model,the penalized empirical likelihood ratio statistic is constructed and the confidence region is obtained from this,which proves that its nature is subject to the progressive chi-square distribution.