Robust Estimation Methods For Some Semiparametric Models

Semiparametric models have attracted much attention by scholars because of its advantages of flexible form,strong adaptability and easy interpretation.To a certain extent,it reduces the risk of false setting of specified parametric models,and can better expresses the potential structural relationship between response variables and predictors,and improve the prediction effect.In practical applications,we often encounter different types of complex data,such as functional data,outlier data and so on.It is of great theoretical and practical significance to make robust statistical analysis of these complex data.This paper mainly discusses the robust estimation and application of semiparametric models under functional data and non-functional data,the research content of this paper includes the following parts:In the second chapter,Walsh-average method is applied to the single index varying coefficient models to study the problem of estimation and variable selection.By using the step-by-step method and Walsh-average-based objective function,the estimates of parameters and nonparametric functions are obtained,and the theoretical properties of the proposed estimates are further discussed.In addition,based on SCAD penalty function,a variable selection method which can identify significant index parameters is proposed.Finally,the advantages of the method are illustrated by simulations and real practical application.In the third chapter,WLAD method and adaptive LASSO penalty estimation method are applied to the single index varying coefficient models to obtain the robust shrinkage estimates when the covariates and response of models may exist outliers.At the same time,the calculation of weights selection and the proofs of asymptotic properties for resulted estimates are given.The results of random experiment and case study show that the proposed method has good performance.In the fourth chapter,we use the exponential-squared loss function and the spline method to discuss the robust estimation problem of the single index partially functional linear models.Furthermore,we discuss the selection of the turning parameters and the specific implementation algorithm of the estimation problem.Under suitable conditions,the theoretical properties of the proposed estimators are proved.The feasibility of the proposed method is verified by numerical examples and real data analysis.In the fifth chapter,we study the estimation problem based on exponential-squared loss function for varying coefficient partially functional linear models to improve the efficiency loss of least squares estimation.We give the specific estimation approach based on B-spline basis,functional principal component analysis and exponential-squared loss function.In addition,we discuss the specific selection methods of turning parameters and truncation parameters,and further prove the theoretical properties of the proposed estimators under some given conditions.Finally,the excellent properties of the proposed method are verified by numerical simulation and real data analysis.In the sixth chapter,we study the robust estimation problem for varying coefficient partially functional linear models based on the rank regression method.First,we give the initial robust estimates for the models by combining the B-spline basis and the rank regression method.Then,we obtain the local rank regression estimation for the varying coefficients by using the back-fitting method.Under appropriate conditions,we obtain the consistency and asymptotic normality of the proposed estimators.In addition,the pseudo observation algorithm for solving the objective function and the selection of turning parameters are given.Finally,simulation studies and the application of real data show the usefulness of the proposed method.