Several Types Of Mixed Functional Data Modeling

With the advances of measuring instruments and the improvement of data collect-ing and storage capacity, in many fields of applied sciences (such as chemometrics, biometrics, medicine, econometrics, etc.), we can collect and deal with the observa-tion data at finer and finer resolution. For example, For example, in order to under-stand the daily temperature curve, we can write down the temperature in every minute. And then, we can get an approximate daily the temperature curve. For this kind of data, the traditional statistical models to face great challenges, such as over-fitting and dimension curse problem. To overcome this difficulty, statistics have put forward functional data analysis methods. That is, each individual observation will be seen as a curve. Then, we can do statistical analysis based on the curve data. Regression analysis is aways an important analytical methods in the statistical analysis. It has been widely studied by lots of scientists. Recently, some statisticians extended the traditional regression model to the functional data case. For instance, Ramsay and Sliverman,([81],[83]) Studied the various functional linear regression models in their monograph. Ferraty and Vieu [52] considered the nonparametric functional regression model. In addition, in order to improve the power of interpretation and prediction of regression model, some statisticians introduced some additional random variables in the functional regression model, which is called the mixed-functional data regression model.In this thesis, we study the problem of mixed functional data modeling and con-sider two kinds of nonparametric methods:polynomial spline method and penalized spline method.In chapter2, we introduced the semi-functional linear model. In this model, we studied the polynomial spline estimation. Under some regularity conditions, we got the global and uniform convergence rates. By simulation study, we considered the finite-sample properties of the proposed estimator. We also compared our model with semi-functional partial linear regression [3] and partial functional linear model [96], which showed the feasibility of our model.In chapter3, we considered the polynomial spline estimation of partial functional linear model [96]. Under some regularity conditions, we also got the asymptotic nor-mality on the parameter estimators and the global convergence rate of functional co-efficient. The finite-sample properties of the proposed estimators have been examined by a simulation. Compared with the estimation method of [96], the results showed our method is superior to [96].In chapter4, we put forward a more robust estimation method for partial func-tional linear model-penalized spline estimation. Similarly, under some regularity con-ditions, we also got the asymptotic normality on the parameter estimators and the global convergence rate of functional coefficient. We studied the finite-sample prop-erties of penalized spline estimation by a simulation. Meanwhile, we compared the three methods of estimation, the results showed the penalized spline estimation is the best of all.In chapter5, we introduced a new mixed functional data modeling-varying coef-ficient partially functional linear model. We studied the polynomial spline estimators of this model. Under the given conditions, we obtained the global and uniform con-vergence rates. By simulation experiment, we considered the finite-sample properties of the proposed estimator.