Joint Semiparametric Mean-Covariance Model In Longitudinal Data

Longitudinal data are highly unbalanced because the data are collected at irregular and possibly subject-specific time points. Therefore, Semiparametric regression mod-els and estimating covariance functions are very useful in longitudinal data study. The varying-coefficient partially linear model is extensively used as one type of semiparametric regression models.In chapter 1, the longitudinal data and the varying-coefficient partially linear model are introduced, then the joint mean-covariance models and local estimation technique are introduced as well. In chapter 2, the function and coefficient of the varying-coefficient partially linear model are estimated based on joint mean-covariance models applied to longitudinal data. Firstly, the local linear estimation is employed to estimate regression functions. Secondly, we adopt the modified Cholesky decomposition approach to decompose the covariance structure and get the joint mean-covariance model. Then the estimation of covariance is converted to the estimation of generalized autoregressive parameters and innovation variance. Thirdly, we estimate the coefficient and generalized autoregressive parameters by using the quasi-likelihood and propose the local linear estimation for innovation variance. For innovation variance, the local linear estimator is proved to be prior to the kernel estimator. We also use the MCV to choose the bandwidth. Finally, introduce the Newton-Raphson iterative algorithm to get the estimators. In chapter 3, the asymptotic normality of the resulting estimators referred in chapter 2 is established. In chapter 4, the simulation study of two examples is used to illustrate the proposed approach.