Semiparametric Estimation Methods And Theoretical Research

In various fields of human activities,regression models often can be used to research the relationship between observation data variables.To deal with complex data in all kinds of practical problems,many important regression models have been proposed and developed.There are some statistical methods and theories to build for statistical inference on them.The dissertation is concerned with the problems of estimation of interesting parameters in some regression models,which include single index varying coefficient model,partially single index varying coefficient model and parametric regression model.The research works of this dissertation are summarized as follow.For the single index varying coefficient model,a two-stage estimation method has been proposed to estimate the parametric component and nonparametric com-ponent.The first stage involves the outer product of gradients estimation method to find the consistent initial estimators for index parameters,and the second stage adopts the refined estimation method to improve the efficiency of the index pa-rameters and the coefficient functions,which is obtained by minimizing the global loss function.Under some regularity conditions,we prove the consistency of the initial estimators of the index parameters,and the asymptotic normality of the estimators of the index parameters and the coefficient functions.Using the co-ordinate descent method,an iterative algorithm is proposed to implement the proposed two-stage estimation method.The results of simulation study and real data example show that the proposed method performs well.A method of estimating functions is introduced to study the single index varying coefficient model,which improve the existing estimating equation method.Based on the bias correction technique and the delete-one-component method,we propose a estimating function and the refined estimation equation is obtained.Under some regularity conditions,we establish the asymptotic properties of esti-mators of the parametric and nonparametric components.The specific algorithm for solving the above estimation equation is proposed by a fixed-point iterative scheme.The effectiveness of the proposed method is verified by numerical simu-lation and a real data example.For the partially single index varying coefficient model,a stepwise estimation method is proposed to estimate the parametric and nonparametric parts,such as the index parameters,the coefficient parameters and the coefficient functions.By means of the profile least squares method,we obtain the estimators of the coefficient functions and the coefficient parameters.At the same time the esti-mating equation method is presented to estimate the index parameters.In order to implement the proposed estimation method,we give an iterative algorithm.The asymptotic properties of the estimators of the parametric and nonparametric components are proved under some regularity conditions.Numerical simulation study shows that the proposed method performs well in finite samples.The estimation problem of the single index varying coefficient model with missing covariates at random is considered.By the inverse probability weighted method and the local estimating equation method,the weighted estimating equa-tion for the index parameters and the weighted local estimating equation for the coefficient functions are constructed.In addition,the estimators of the index parameters and the coefficient functions are derived,and their asymptotic nor-malities are also studied under some regularity conditions.Simulation studies are carried out to assess the performance of our method,and a real dataset is analyzed for illustration.To handle different missing data cases,a weighted semiparametric estimation method is proposed to estimate the coefficient parameters in parametric regres-sion model,when the observation variables of high dimension is allowing.For the continuous and discrete components in covariates,the selection probability func-tion is modeled respectively by two semiparametric models under MAR.By the estimators of the selection probability and the inverse probability weighted esti-mating equations,the estimators of the coefficient parameters in the parametric regression model are derived,and their asymptotic normalities are proved.Some simulated examples under different missing cases,suggest that our methods have good finite sample properties.