Several Studies On Single-index Varying-coefficient Model

Single-index varying coefficient model is a widely used semi-parametric model.It combines the advantages of varying coefficient model and single index model.It not only solves the possible ”dimension disaster” and the problems of difficult prediction and extension in the variable coefficient model,but also enriches the relationship between explanatory variables and response variables in the single index model.It is widely used in economic,financial,biomedical and other fields.With the improvement of data storage capacity,whether in social science or natural science,the data are becoming more and more complex.Functional data,panel data and longitudinal data are increasingly appearing in various research fields.At the same time,the research of regression model on new data types has also made great progress and produced fruitful results.However,because the single-index variable coefficient model is a complex semi-parametric model,it has little research in the fields of functional data,panel data and longitudinal data.Therefore,this paper will focus on three different data types based on the single-index varying coefficient model.This paper mainly introduces three models based on single-index varying coefficient structure.In the second chapter,we propose a single-index varying coefficient panel data model,which reflects the heterogeneity of the model through the change of link functions with different individuals.Combining RMAVE method with local linear regression,we estimate the parameters and link functions,and explain the steps of iterative algorithm.Under certain regular conditions,the large sample properties of parameters and estimators of connection function are proved.Finally,the results of numerical simulation verify our theoretical properties,and our model also shows better fitting ability than the single-index panel data model in a real example.In Chapter 3,we propose a functional single-index varying coefficient model,in which the response variables,individual curves and coefficient are functional data.The estimation in this chapter is mainly divided into two parts.In the first part,we weighted average of all grid points on the basis of the algorithm in Chapter 2,so as to obtain the estimated values of parameter curve and connection function.In the second part,we first estimate the individual curve by the least square method,and then perform spectral decomposition on the covariance matrix to obtain the estimation of eigenvalues and eigenfunctions,so as to obtain the estimation of covariance matrix.In this chapter,we also use the method of wild bootstrap to construct the simultaneous Confidence band of parameter function and connection function.The parameter curve,the link function,the eigenvalue of the covariance matrix and the asymptotic properties of the eigenfunction are given in the theoretical proof.In the numerical simulation,the parameter curve and connection function also perform well.In Chapter 4,we propose a functional single-index varying coefficient model based on longitudinal data.The covariates and response variables in this model are longitudinal data,and the coefficients are functional parameters.Considering the correlation between error terms will affect the estimation of parameters.After obtaining the estimated values of the index term parameters and the connection function,we estimate the covariance matrix with Cholesky decomposition,and use the estimated values of the covariance matrix to improve the estimation of the parameter curve and the function curve.Large sample properties of parameter and function before and after improvement are proved.The models and estimation methods in this paper enrich the related research of single-index varying coefficient model,and expand on panel data,functional data and longitudinal data,which provides new solutions to practical problems in the fields of economy,finance,biomedicine and so on.