处理大维数据时，许多经典的多元统计方法存在一定程度的低效率甚至是不可应用。为适用于大维数据分析，近年来一些研究者应用随机矩阵理论改进传统的多元统计方法，建立更精确的极限理论。但关于非中心化F矩阵在大维情况下的谱分析还处于理论空白。基于非中心化F矩阵在工具变量回归中的重要性，本项目拟应用随机矩阵理论建立大维非中心化F矩阵的谱分析及其极限理论。具体研究内容叙述涵括三个方面：（1）建立大维非中心化F随机矩阵的极限谱分布和极端特征根的极限定理；（2）建立此类矩阵线性谱统计量的中心极限定理；（3）首次将大维非中心化F矩阵的谱分析应用于多元工具变量回归方法中，建立有效工具变量识别和弱工具变量检测的统一精确框架。申请人在随机矩阵理论及其在多元统计中的应用和多元回归分析方面具有丰富经验，本项目的结果不仅能够填补大维随机矩阵极限理论的空白，也能够完善多元工具变量回归的分析框架，从而指导实际数据分析。

Many traditional multivariate statistical methods will become inefficient and generate misleading results when applied to high dimensional data situations. Recently, some researchers focus on improving the classical multivariate statistical procedures based on the random matrix theory under the large dimensional framework. But, there is no existing research on the spectral analysis of large dimensional non-central Fisher matrix. As the non-central Fisher matrix is very important in multivariate instrumental variables regression, this project is aimed to study its spectral analysis and the corresponding limit theories based on the random matrix theory. Specifically, the project consists of three parts: (1) to construct the limit spectral distribution and the limit theories of the largest and smallest eigenvalues of the non-central Fisher matrix; (2) to construct the central limit theory for linear spectral statistics; (3) to develop a precise framework for the identification of valid instruments and the detection of weak instruments for large dimensional instrumental variables regression, based on the spectral analysis of large dimensional Fisher matrix. The applicant is experienced in the random matrix theory and its applications in multivariate statistical analysis. The theoretical results yielded in this project will add to the limit theory of large dimensional random matrix on the one hand, and advance the analytical framework for multivariate instrumental variables regression, thereby providing useful guidance for real data analysis.