Statistical Inference Of Functional Partial Linear Varying Coefficient Model

In recent years,with the rapid development of information technology and statistical software,we are able to collect and store data which time point is very dense and the data shows a certain functional characteristics in the statistical analysis.This kind of data will change with space or time changes.Intuitively,we call this type of data as functional data.Functional data is widely used in different fields,such as criminology,biostatistics,econometrics,archeology,environmental metrology,medicine,and neurobiology and so on.In many cases,the mean model is used to estimate Functional linear model parameter.However,the mean model has limitations.It only describes the overall average characteristics of the whole population and can't be very sensitive to capture the specific changing of each variable.Quantile regression make up for the deficiency of the mean model since the concept of quantile regression was proposed.It can not only depict the central characteristics of variables in the distribution,but also depict characteristics of variables in the marginal distribution.When it comes to solving practical problems,the choice of quantile point is often embarrassing.If we consider more than one quantile point,we can well avoid this problem.As a result,the composite quantile regression came into being.In addition,the data obtained is more and more detailed with high dimensions.If we directly use them to model,it will make the prediction's result poor.Therefore,the selection of variables in the statistical modeling process is important.The appropriate variables can't only make the model simpler,make estimation accurate,but also explicit robust models.In this paper,quantile regression and composite quantile regression are applied to the functional partial linear variable coefficient model respectively,and the sparse Group LASSO method is used to achieve variable selection to improve the ability of the interpretation of the model and the estimation's precision.And under certain assumptions,the large sample properties of the estimator are proved.Finally,the estimator is verified by the data's simulation and the empirical analysis.The main contents are followed:Firstly,we briefly introduce the overview of the research background and significance of this article.According to the existing literature at home and abroad,the paper systematically introduces the linear model of functional part,quantile regression,composite quantile regression and variable selection.Then,methods of quantile regression and composite quantile regression were used to study the parameter estimation of functional linear partial variable coefficient model.In the third and fourth chapters of the paper,we give the objective function to estimate the functional linear part of the coefficient-based linear variable coefficient model based on quantile regression and composite quantile regression,and give the estimation method of the paper.Under the assumption,the large sample property of the estimated parameters is obtained,and the theoretical proofs are given.Based on the results of the numerical simulation,the correctness of the model and the feasibility of the estimation method are verified.In the fourth chapter,the sparse Group LASSO method is used for composite quantile regression,the parameter estimation and the variable selection are performed to improve the estimation accuracy of the model.For the theoretical results of the third and fourth chapter,the paper designs a variety of combination of numerical simulation for verification and comparison.According to the numerical simulation results,the rationality of the model is examed and the estimation method is feasible.Finally,the “Tecator” data was analyzed,which was used the functional partial linear variable coefficient model presented in this paper to explore the relationship between fat content and water content,protein content and absorption spectrum,and to predict the fat content.