Statistical Inference For Some Functional Models

Nowadays,with the rapid development of science and technology,the progress of computational tools(both memory and capacity increasing)allows to create,store and work with large databases.In many cases,the dataset is made up of observations from a finite dimensional distribution,measured over a period of time or recorded at different spatial locations.When the temporal or spatial grid is fine enough,the sample can be considered as an observation of a random variable on a certain functional space.Data of such type are termed functional.Analyzing this kind of data with standard multivariate methods and ignoring its functional feature may fail dramatically because of the curse of dimensionality,collinearity,or valuable information loss,etc.Considering the data observed as an element of an infinite dimensional functional space is the basic idea of functional data analysis to avoid these problems.At the same time,functional data analysis has a wide application in psychology,economics,meteorology,chemistry and Life science,and so on.In this dissertation,we focus on inference for some functional models,in-cluding varying-coefficient partially functional linear quantile regression models,functional partial linear regression model,single-index partially functional lin-ear model,partial functional linear regression model and partial functional linear quantile regression model.The main contents include estimation,statistical test-s,and so on.More specifically,the research contents of this thesis contain the following three parts.The first part studies the quantile estimation and composite quantile esti-mation about functional models.In Chapter 2,we introduce varying-coefficient partially functional linear quantile regression model,which combines varying-coefficient quantile regression model with functional linear quantile regression model.The functional principal component basis and regression splines are em-ployed to estimate the slope function and varying-coefficient functions,respective-ly,and the convergence rates of the estimators are obtained under some regularity conditions.Simulations and an illustrative example of the Tecator data are pre-sented.Since the estimation efficiency may fluctuate with the level of quantile,composite quantile regression by combining the information from multiple quan-tile regression to obtain more efficient estimator.Thus,in Chapter 3,we study composite quantile estimation in functional partial linear regression model.The functional principal component analysis and regression splines are employed to estimate the slope function and the nonparametric function respectively,and the convergence rates of the estimators are obtained under some regularity conditions.In the second part,we study two kinds of generalized forms about functional linear model.By reducing the dimensionality from multivariate predictors to a univariate index,single index model avoid the "curse of dimensionality",while still capturing important features in high-dimensionality.Thus,in Chapter 4,we propose a flexible single-index partially functional linear regression model,which combines single-index model with functional linear regression model.All the coef-ficient functions are approximated by B-spline basis functions.Under some mild conditions,the convergence rates and asymptotic normality of the estimators are obtained.Finally,simulation studies and real data analysis are conducted to inves-tigate the performance of the proposed methodologies.In chapter 5,we generalize the form of functional partial linear model in chapter 3 to partial functional lin-ear additive regression model.We obtain the convergence rates and asymptotic normality of the estimators by B-spline technique and modal modal regression method.The third part studies the hypothesis test problem about partial function-al linear regression model.In Chapter 6,we investigate the hypothesis test of the parametric effect in partial functional linear regression.We propose a test procedure based on the residual sums of squares under the null and alternative hypothesis,and establish the asymptotic properties of the resulting test.A simu-lation study shows that the proposed test procedure has good size and power with finite sample sizes.Finally,we present an illustration through fitting the Berkeley growth data with a partial functional linear regression model and testing the effect of gender on the height of kids.In Chapter 7,we further discuss the hypothesis test of the parametric effect in partial functional linear quantile regression mod-el.We develop quantile rank score test based on functional principal component analysis,and establish the asymptotic properties of the resulting test.