A Kind Of Mean And Covariance Semi-parametric Regression Model

Longitudinal data often arise in medical, biological, social and economical stud-ies. They are often characterized by the dependence of repeated observations over time within the same subject. Observation within the same subject tended to be correlated. On one hand, estimating covariance matrix is an important issue in the longitudinal study. It can improve the efficiency of estimated regression coefficient. On the other hand, the efficient estimation of covariance matrix also need a correct specification of mean regression model in the longitudinal data. In this article, we treat the covariance matrix as being as important as the mean and consider the mean and covariance simultaneously. For mean term, we propose several semiparamet-ric model and for covariance term, parametric, semiparametric and nonparametric models are proposed. The main research works are as follows:In chapter one, we give a brief description of the semiparametric model. The backgrounds and present development of the estimation of covariance matrix are introduced. Finally, the main results of this thesis are introduced.In chapter two, we mainly consider the joint mean-covariance model in a par-tially linear model. Firstly, we propose a partially linear model for mean term. The joint mean-covariance model is then constructed based on the modified Cholesky de-composition approach. Secondly, we estimate the regression function by using the local linear technique and propose quasi-likelihood estimating equations for both the mean and the covariance structures. Thirdly, the asymptotic normality of the resulting estimators is established. Finally, simulation study and real data analysis are used to illustrate the proposed approach.In chapter three, we mainly study the joint semiparametric mean-covariance model in longitudinal study. Firstly, a semiparametric varying-coefficient partially linear model is proposed. To heed the positive-definiteness constraint, we adopt modified Cholesky decomposition approach to decompose the covariance structure. Then the covariance structure is fitted by a semiparametric model by imposing parametric within-subject correlation while allowing nonparametric variation func-tion. Secondly, we estimate regression function by using the local linear technique and propose generalized estimating equations for the mean and correlation param-eter. Kernel estimator is developed for the estimation of nonparametric variation function. Thirdly, asymptotic normality of the resulting estimators is established. Finally, simulation study and real data analysis are used to illustrate the proposed approach.In chapter four, we mainly study the joint mean-covariance model in generalized partially linear varying coefficient models for longitudinal data. Firstly, we propose a general semiparametric models for the mean and the covariance simultaneously using the modified Cholesky decomposition. Secondly, a regression spline-based approach within the framework of generalized estimating equations is proposed to estimate the parameters in the mean and the covariance. Thirdly, asymptotic nor-mality of the resulting estimators is established. Finally, simulation study and real data analysis are used to illustrate the proposed approach.In chapter five, we research the joint estimation of mean-covariance model for longitudinal data. Firstly, we propose a semiparametric regression model. After using the modified Cholesky decomposition to decompose the covariance matrix, we construct joint mean-covariance model by modeling the smooth functions using splines functions. Secondly, we estimate the associated parameters by using the quasi-likelihood approach. Finally, simulation study and real data analysis are used to illustrate the proposed approach and demonstrate the flexibility of the proposed model.In summary, estimation of joint mean and covariance model in several semi-parametric model are systematically studied in this thesis. For the mean term, we introduce several semiparametric model which includes partially linear model, semi-parametric varying coefficient partially linear model and generalized partially linear varying coefficient model. For the covariance term, we propose generalized linear model, generalized nonparametric model and generalized partially linear varying coefficient model for innovation function, and use the kernel method and B-spline to approximate the nonparametric term or varying-coefficients. We extend and de-velop the work of Pourahmadi[70][71], Pan and Mackenzie[69], Ye and Pan[91] and Leng et al.[52]. The effectivity of the proposed estimators is justified by the rich simulation studies and some real data analysis. These results are not only useful in theory but also significant in practice.