大维样本协方差矩阵的几个统计推断问题研究

High-dimensional data have been increasingly encountered in many fields of scientific research. However, traditional methods of multivariate analysis built in fixed dimensional settings are not applicable any more. It is thus in urgency to develop new methods of large-scale statistical inference. To this end, we turn to Random matrix theory, which can be served as a strong theoretical support for such statistical problems. In this project, based on this theory, we will investigate spectral properties of sample covariance matrices in high dimensions. The results will then be apply to deal with some problems in finance and economics. 1) We consider the problem of testing the identity of integrated covariance (ICV) matrices of high dimensional diffusion processes based on high frequency observations. It is shown that we can get any limit spectral distribution (LSD) from the realized covariance when ICV is a deterministic matrix, say identity matrix, in such situations. Therefore, test statistics based on the LSD are no longer reliable. In order to fix this problem, we propose a robust procedure for the test through a time variation adjusted realized covariance matrix. 2) we consider some problems of statistical inference in spike population models, such as the estimation of ICV's eigenvalues.

随着社会的发展、科技的进步，很多领域都遇到了高维数据问题，如何对高维数据进行统计推断是本项目关注的课题。现代随机矩阵理论可追溯到上个世纪50年代，它的出现给高维推断统计提供了理论依据，且经过半个多世纪的发展已趋于成熟。本项目以在统计推断中有核心地位的协方差阵为研究对象，以现代随机矩阵理论为理论依据，拟着重探讨金融领域中的高维协方差阵的统计推断问题：1）基于金融数据的高维扩散过程，拟考虑可积协方差矩阵是否为单位矩阵的检验问题。经研究发现即使确定协方差矩阵为单位阵的情况下，由数据得到的相关矩阵的极限谱分布仍然可以是任意的，因此基于极限谱分布的检验统计量不再是可靠的。对于这个问题，拟提出用time variation adjusted realized covariance matrix构造统计量进行检验；2）对于加权spike 总体模型，我们拟考虑相关矩阵特征值的相合估计等问题。

随着社会的进步科技的发展，我们进入了人工智能、大数据的时代。时代要求我们在各领域提高处理大维数据的能力。我们从金融领域入手，抽象出几个关于大维随机矩阵的统计推断问题：加权协方差阵的单位阵（球型阵）检验问题，Spiked矩阵的统计推断问题，高维数据关联性分析等。针对这些问题，本课题解决了加权协方差矩阵的检验问题，推导出Spiked矩阵的线性谱统计量的中心极限定理，给出高维随机向量间独立性的检验方法，讨论了多元CUSUM图等，并将相关结果应用于无线电信号等领域，取得了较好的研究结果。