Generalized Functional Regression Model And Missing Data Model

With the advances of data collection tools and storage technology, how to dig out valuable information from massive data has attracted global attention. As a statistical method to deal with high dimensional and complex data, functional data analysis has become a hot issue of statistics in the last decades, and also been widely used to many applied fields, such as financial engineering, bio-chemical engineering, medicine, hydrology, brain image analysis and modeling and so on. "Missing Data" is very common in practice, for example, in a sample survey, respondents are not willing to answer some questions, or in drug tracking testing, patients drop out in the middle of testing. In parameter and semi-parameter regression model, to deal with and analyze "missing data" has been studied extensively. However, of functional data, there is only a few papers focused on this problem. In addition, the generalized linear model has been widely studied because it can be used to model many different kinds of response variables. In the context of functional data, James (2002) firstly proposed generalized functional linear model, and discussed statistical inference of this model based on functional principal components analysis method. But, there is a little literature focused on polynomial spline method. Therefore, this thesis first considers polynomial spline estimation for generalized functional regression model. Second, statistical inference of functional partial linear models with response missing at random has been discussed. These research have important theoretical significance and practical application value.The contents of this thesis are as follows:(1) By introducing additional covariates in generalized functional linear models, generalized partial functional linear models are proposed, which are generalization of Zhang et al. (2007) and Shin (2009)’s. Polynomial spline estimates of parameter vector and slope function are constructed. Under some conditions, asymptotic normality of estimator of parameter vector is proved, and global convergence rate of the estimator of slope function is also established. The feasibility of the polynomial spline method is demonstrated by some numerical simulations.(2) In the context of response variables missing at random, polynomial spline estimation for functional partial linear models proposed by Lian (2011) are constructed. Furthermore, regression imputation and partial inverse probability weighted estimators of the mean of response variable are given. Under some general conditions, global convergence rate of the spline estimator of the slope function is obtained. Also, the convergence rates of the two estimators of mean are established. Compared to Ferraty et al. (2013), convergence rates of the two estimators can’t attain the (?) consistency rate, unless the slope function is limited in some a finite dimensional function space. The finite sample behaviors of polynomial spline estimators are studied by a numerical simulation.(3) In the context of response variables missing at random, functional principal component estimation for functional partial linear models are constructed. Similarly, two kinds of estimators of mean are proposed, which are based on regression imputation and partial inverse probability weighted. Based on the incomplete data, the convergence rate of estimator of the slope function is established under given some conditions, which is similar to Lian (2011)’s conclusion. Furthermore, the convergence rates of the two estimators of mean are also obtained. The results show that the (?) consistency rate of Ferraty et al. (2013) can’t be obtained under the given conditions. Finally, the feasibility of the polynomial spline method and functional principal component method is demonstrated by two me numerical simulations.