中文摘要：

近年来，对纵向数据分析中回归均值和协方差矩阵同时进行建模研究得到越来越多的关注．为满足协方差矩阵的正定性约束，文献中常考虑对其逆矩阵进行某种分解．本文使用一种Cholesky分解方法对协方差矩阵本身进行分解，得到的参数没有取值限制且有着明确的统计意义．具体地，分解后的参数可以视为滑动平均序列的系数和对数更新方差，且在整个实轴上取值无限制．考虑到模型的稳健性和推断的有效性，提出了一种对回归均值和协方差矩阵同时进行半参数建模的方法，并利用广义估计方程和B样条给出了半参数模型的估计方法，得到了参数部分估计的渐近正态性以及非参数部分估计的最优收敛速度．最后通过模拟和实例分析对所提方法进行了数值研究．

英文摘要：

Modeling the mean and covariance simultaneously has recently received considerable attention when efficiently analyzing the longitudinal data. An unconstrained and statistically interpretable reparameterization of covariance matrix itself was presented by utilizing a novel Cholesky factor. The entries in such decomposition have moving average and log innovation interpretation and can thus be modeled as functions of covariates. With this decomposition and the consideration of model flexibility, new semiparametric models for jointly modeling the mean and covariance itself were proposed, rather than its inverse as commonly studied in literature. A spline based approach using generalized estimating equations was developed to estimate the parameters in the mean and the covariance. It was shown that the estimators for the parametric parts in both the mean and covariance are consistent and asymptotically normally distributed, and the nonparametric parts could be estimated at an optimal rate of convergence. Simulation studies and real data analysis illustrate that the proposed approach could yield highly reliable estimation of the mean and covariance matrix.