Some Studies On Functional Linear Regression

Functional data,as an important data type,have been widely used in sociology,economics,biomedicine,epidemiology as well as other fields of natural science.Regression models are often used to deal with the relationship between covariates and response variables.Recently,various functional regression models have been a research focus of widespread interest due to its flexibility and ability to investigate the potential relationship between response variables and predictor variables in real problems.This dissertation studies the local estimation of functional-response linear models.The contents of this dissertation are as follows.(1)Based on data analysis and studies on related variables of interest,it is seen that the infant height data from Shanghai Children's Medical Center are characteristic of longitudinal data.Chapter 2 introduces the varying coefficient model often used in longitudinal data analysis and a new estimation method of functional data analysis.The new method,combining functional principal component analysis with local least square estimation method,estimates the varying coefficient and considers the correlation within the subject.The chapter also establishes the weak convergence of the estimated varying coefficient.Lastly,the effectiveness of the proposed procedure is demonstrated via a simulation study and the aforementioned application to infant height data.(2)To reveal the influence of a set of covariates of interest on multivariate growth curves and measure the correlation between different curves,Chapter 3 presents the multivariate varying-coefficient functional response model used in data analysis.The model is processed by a two-step estimation method,combining a local linear estimation under unified weight with combined functional principal component analysis.This method,which can be applied to both sparse and dense data,is able to accurately estimate the varying coefficient function and analyze the interdependence of multivariate functional responses.In numerical simulation analysis and real data analysis,the proposed estimation method performs better and more stably.(3)Different from the analysis of infant sleep time based on single sample in the previous two chapters,Chapter 4 is concerned with the Meta-analysis of functional data.Firstly,mean-covariance model is employed to fit the infant sleep time over age in multiple countries.Then,based on inverse variance weighting and the least square estimation method,the dissertation proposes an effective estimation method,considering the correlation within the subject and standard error.Next,it establishes the large sample properties of the estimation.Compared with conventional estimation methods,the new method is demonstrated to be more effective and performs better in simulation analysis.Lastly,the approach is applied to analyze the infant sleep time from 32 countries,which produce meaningful and interesting results.(4)Chapter 5 focuses on the convergence rate of functional principal component through local linear smoothing method under uniform weight.This theoretical study enables functional data,whether sparse or dense,to do functional principal component analysis under different weights.Specifically,the first part is a brief introduction of estimation on mean-covariance model under uniform weight.Then,for the functional principal components under uniform weight,namely eigenvalues and eigenfunctions,the dissertation provides their corresponding estimates and presents corresponding strong uniform convergence rate.Finally,a simulation study is conducted to verify the convergence rate.The estimation methodologies and conclusions in the thesis enrich the studies of functional regression models,which also help to analyse the complicated and volatile problems in many application fields,such as economics and biometrics.