多响应纵向数据下相关系数模型的估计方法研究

多响应纵向数据常见于医学和生物学等科学领域的前沿，其某一个体的相关系数矩阵可用于直观地刻画该个体的多元响应变量在同一时间点处和不同时间点间的复杂相关结构，从而日益引起科学研究的兴趣。本项目主要研究多响应纵向数据下相关系数模型的估计方法，其研究内容包含：分别对多响应纵向数据的相关系数矩阵与其某一时间点处新息的相关系数矩阵进行有序与无序的分解，并建立精简的相关系数模型，然后为其联合均值-协方差模型建立广义估计方程，发展估计算法，并给出该估计量的渐近性质。随后在离群值存在的情形下，利用众数回归方法为主要基于某一时间点处新息相关系数矩阵的无序分解建立的联合模型发展稳健且有效的估计，提出迭代算法，并得到其估计量的大样本性质。本项目的研究结果不但可以用于准确解释多响应纵向数据的复杂相关结构，同时还能提高均值参数的估计效率，并有望进一步推广到空间数据分析乃至时空数据分析等相关数据类型下。

Multivariate longitudinal data occurs frequently in the front edge of biology, medical science, and other fields, whose correlation matrix for some subject can be adopted to intuitively describe complex correlation structures of multivariate longitudinal variables at the same time point and between distinct time points for that subject, and thus it is of increasing interest to scientific research. This project mainly investigates estimation methodology for correlation models with multivariate longitudinal data, which includes the following researches. This project respectively carries out order dependent decomposition and order independent decomposition for the correlation matrix of multivariate longitudinal data and that of the innovation at some time point, establishes parsimonious correlation models, develops generalized estimating equations for the joint mean-covariance models, puts forward the estimation algorithm, and gives the asymptotic properties for the estimators. Then in the presence of outliers, this project develops the robust and efficient estimator for the joint model mainly based on the order dependent decomposition for the innovation correlation matrix at some time point by adopting the modal regression method, proposes the iterative algorithm, and obtains the large sample properties for the estimators. These investigations can be used to accurately interpret complex correlation structures of multivariate longitudinal data, improve the estimation efficiency of the mean parameter, and expected to be further extended to spatial data analysis, spatiotemporal data analysis and other correlated data types.