Estimation And Inference In Semiparametric Functional Regression Models

Functional data analysis has been proven to be of great value in many applications in fields including,for example,chemometrics,biomedical studies,and econometrics.Among many problems involving functional data,functional regression has received substantial attention.In this paper,we focus on semiparametic modelling for functional data,which combine advantages of parametric and nonparametric models.Specifically,three functional regression models were proposed and developed including estimation and inference of parametric and nonparametric components.The main works are listed as follows.(1)We examine two-sample functional linear regressions with a scaling transformation of the regression functions.We estimate the intercept,slope function,and scalar parameter using a functional principal component analysis.We also establish the rate of convergence of the estimator of the slope function,which is shown to be optimal in a minimax sense under certain smoothness assumptions.In addition,we investigate the semiparametric efficiency of the estimation of the scalar parameter and the hypothesis tests.Then,we extend the proposed method to include sparsely and irregularly sampled functional data and establish the consistency of the estimators of the scalar parameter and the slope function.We evaluate the numerical performance of the proposed methods through simulation studies and illustrate their utility via an analysis of an AIDS data set.(2)We study two-sample functional linear regressions with functional responses,where the regression functions are assumed to have a scaling transformation.We estimate the intercept function,slope function,and parameter components based on the least squares method and functional principal component analysis.The proposed estimator of the parameter components are shown to be root-n consistent and asymptotically normal.We also establish a rate of convergence for the estimator of the slope function.In addition,we propose asymptotically more efficient estimators for the parameter components.The numerical performance of the proposed methods is evaluated via simulation studies and an analysis of an AIDS data set.(3)We consider the estimation and inference in partially functional linear regression with multiple functional predictors.We estimate the parameters and the slope functions by using functional principal component analysis(FPCA)approach to each functional covariate,and establish the asymptotic distribution for the proposed estimators and investigate the semiparametric efficiency.We derive the rates of convergence for the estimators of the slope functions,which can achieve the optimal rate of FPCA-based estimator of functional linear regression.We also establish an optimal rate of convergence for the prediction problem.Next,we develop a linear hypothesis test for the parametric component,and construct confidence bands centered at FPCA-based estimator for the slope functions and verify its asymptotic validity.The performance of the proposed procedures is illustrated via simulation studies and an analysis of a DTI data application.