Theoretical Research And Applications For Several Classes Of Functional Linear Models

Advancements of modern technology have enabled experimenters to collect datasets with functional features,such as curves,images or other objects over multiple time or spatial points.This type of data is called functional data and has been encountered frequently in many scientific fields such as biology,medical science,econometrics,environtology and ecology.The study of functional data has become one of the hottest research topics in modern statistics.As an important tool in functional data analysis,functional regression has been extensively applied in various application areas.Functional linear model characterizing the linear relationship between functional variables is one of the most important models in functional regressions.This dissertation mainly focuses on the estimation and variable selection for several classes of functional linear models,including functional varying-coefficient mixed effects model,function-on-function linear model,scalar-on-function linear model and partial functional linear model.More specifically,the research contents of this dissertation include the following aspects.We begin with exploring the estimation for varying-coefficient mixed effects models with longitudinal and sparse functional response data.We construct the estimators of the varying-coefficient functions in this model by using the generalized least squares method coupling a modified local linear technique.Under the regularity conditions,we establish both uniform consistency and pointwise asymptotic normality for the proposed estimator.The proposed approach provides a useful framework that simultaneously takes into account the within-subject covariance and all observation information in the estimation to improve efficiency.The finite sample performance of the proposed procedure is illustrated with simulation studies and a real data example.For the function-on-function linear model,firstly,we introduce a variable selection procedure for this model using the functional principal component analysis(FPCA)-based estimation method coupling the group smoothly clipped absolute deviation(SCAD)regularization.This approach enables us to select significant functional predictors and estimate the functional coefficients simultaneously.A data-driven procedure is provided for choosing the tuning parameters of the proposed method to achieve high efficiency.Under some mild conditions,we establish the estimation and selection consistencies for the proposed procedure.Secondly,we consider the robust estimation for the function-on-function linear model.We construct the robust penalized M-estimators of the functional coefficients in this model using M-estimation and penalized spline regression.The efficiency of the proposed variable selection procedure and the robust penalized M-estimation method are investigated with several simulation studies and real data examples.For the functional linear models with scalar responses,firstly,we consider the robust estimation for the model when there are functional predictors only.A modified Huber's function with exponential squared loss(ESL)tail function coupling the functional principal component analysis method is applied to construct the robust estimators of the functional coefficients in this model.Moreover,we also investigate the asymptotic properties of the proposed estimators.Simulation studies and a real data analysis are carried out to illustrate the finite sample performance of the proposed procedure.Secondly,we explore the robust estimation for the model when there are both functional and scalar predictors.We construct the robust estimators of parametric coefficients and functional coefficients in this model by adopting the previous loss function combining with the B-spline estimation method.Several theoretical properties including the consistency and asymptotical normality of the resulting estimators are established.Simulation studies are carried out to illustrate the efficiency of the proposed method.Finally,we apply the proposed model and estimation method to analyze the Tecator dataset and obtain a good prediction performance.