基于非正态假设下的高维协方差矩阵检验问题研究

在大数据时代，人们面对的数据样本量巨大，维数超高，对数据的分析和处理方法变得越来越重要。本项目主要研究高维数据的统计推断新方法。本项目将对基于非正态假设下的三个重要的高维协方差矩阵假设检验问题展开深入研究。第一，拟检验两总体协方差矩阵相等的问题，通过比较两个协方差矩阵的逐条子对角线结构，并结合多重检验的方法来推断两个协方差矩阵是否相等。第二，拟采用新的非参数统计方法研究高维多总体协方差矩阵相等的检验问题，尤其是提出的方法能适用于更广泛的非正态分布的数据。第三，对高维分块对角协方差矩阵的检验问题，我们考虑利用协方差矩阵子对角线的信息推断协方差矩阵的分块结构，并基于此提出非参数检验的新方法。在不作正态性假设的条件下，当维数随着样本量发散时，建立相应的理论结果非常具有挑战性，且本项目对这三个问题的研究以提高检验功效为目标，使其具有更好的应用前景。本项目将在理论、方法和应用上实现创新。

The analysis of high-dimensional data has become more and more important facing the era of big data. We focus on the new statistical inference methods for high-dimensional data in this project. Three important hypothesis testing problems for high-dimensional covariance matrices structures under non-normality will be studied. First of all, for testing the equality of two covariance matrices, we consider comparing each of their super-diagonals individually and then using the multiple testing procedure to infer the equality of the whole matrices. Then, we will study the nonparametric method to test the equality of several covariance matrices under non-normality. Furthermore, we will test whether the covariance matrix is block diagonal. For this problem, we consider inferring the block diagonal structure of the covariance matrix using the information on its super-diagonals, and then propose a nonparametric test procedure. It is challenging to establish the theoretical properties of the three tests under non-normality as the dimension diverges with the sample size. We intend to increase the testing powers so that the three test can be better applied to practice. This project will bring innovation theoretically, methodologically and practically.