Kendall和Spearman秩相关系数矩阵的极限谱性质及其在高维独立性检验中的应用

In multivariate analysis, it is commonly assumed that the components of the observation vectors are totally independent, so there’s a huge demand to test the independence assumption before conducting any statistical inference procedures. This project aims at proposing new test statistics to test total independence among the components of high dimensional data sets, which are based on the eigenvalues of two high dimensional non-parametric matrix models called Kendall's tau and Spearman's rho. Since the eigenvalue-based statistics can be viewed as a special case of linear spectral statistics, in the first step, under the null hypothesis, we will derive the central limit theorem of linear spectral statistics for Kendall's tau and Spearman's rho when the dimension goes to infinity proportionally with the sample size. Furthermore, under some specific alternative hypothesis, to study the power function, we will derive some limiting spectral properties regarding to these two matrix models, e.g. the convergence of empirical spectral distributions, limiting behaviors of the extreme eigenvalues and central limit theorem of linear spectral statistics, which are in fact of independent interest in random matrix theory.

在多元统计推断中，一个普遍的前提假设条件是观测数据的各个分量之间具有独立性，所以有必要在做统计推断之前进行独立性假设检验。本项目基于两个非参高维随机矩阵模型（Kendall秩相关系数矩阵和Spearman秩相关系数矩阵）的特征根，提出新的检验统计量用来检验高维数据各个分量之间的独立性。基于特征根的检验统计量可以看成是线性谱统计量的一种特殊形式，因此本项目首先在原假设下研究高维框架下（维数与样本量成比例地趋于正无穷）Kendall秩相关系数矩阵和Spearman秩相关系数矩阵线性谱统计量的中心极限定理；另外，在某些特定的备则假设下，我们将会从理论上研究这两个高维随机矩阵模型的一些极限谱性质，比如其经验谱分布的收敛性、极值特征根的渐近分布以及线性谱统计量的中心极限定理，这方面的理论结果除了可以用来研究检验统计量的效用函数之外，在随机矩阵理论中也将具有其独特的理论价值。