【作者】 刘宁；

【导师】 郑术蓉；

【作者基本信息】 东北师范大学 ， 统计学， 2021， 硕士

【摘要】 近几十年来,随着科学技术的快速发展,计算机的计算速度、存储空间迅速增长,进行大维数据分析成为可能。然而,一些经典的统计方法在处理大维数据时有时会出现误差或者失效。而大维随机矩阵理论能有效处理这一问题,从而大维随机矩阵的研究引起了统计学家们的广泛关注。例如,多元线性回归分析、信号处理等也经常涉及到大维随机矩阵。此外,很多假设检验问题构建的检验统计量都与样本协方差矩阵相关(样本协方差矩阵是最常见的一种随机矩阵),因而研究样本协方差矩阵的谱性质变得十分重要。样本协方差矩阵极限谱分布研究中具有代表性的一项工作是Mar(?)enko and Pastur(1967)证明了样本协方差矩阵的经验谱分布依概率收敛到Mar(?)enko-Pastur律。在这项工作基础上,本文将其应用到研究总体协方差矩阵具有多类非零特征根时的样本协方差矩阵的极限谱密度函数。本文的主要工作内容是:根据大维样本协方差矩阵的谱分析知识,给出了详细的关于本文所提模型的样本协方差矩阵的极限谱密度函数的推导过程,并提供了如何确定该极限谱分布函数支撑集的算法以及构建该极限谱密度函数图像的算法。另外,还通过编程模拟得到了一些数值结果,以展示本文方法的可操作性和有效性。更多还原

【Abstract】 In recent decades,with the rapid development of technology and science the rapid growth of computer computing speed and storage has made it possible to analyze the large dimensional data.However,many classical statistical methods are no longer applicable when we are dealing with large dimensional data,and some of them even have serious errors.Large dimensional random matrix theory has attracted wide attention of statistician because it can deal with large dimensional data effectively.There are many research questions which can be handled by large dimensional random matrix knowledge,such as multivariate regression analysis,signal processing,etc.In addition,many test statistics constructed for hypothesis test problems are related to the sample covariance matrix,which is the most common kind of random matrix,so it is very important to study the spectral properties of sample covariance matrix.Mar(?)enko and Pastur(1967)proved that the empirical spectral distribution of sample covariance matrix converges to M-P law in probability.On the basis of the work,we apply it to study the limiting spectral density function of sample covariance matrix when the population covariance matrix has multiclass of nonzero population eigenvalues.In this paper,according to the spectral analysis knowledge of large dimensional sample covariance matrix,we give a set of detailed derivation process about the proposed model of the limiting spectral density function of sample covariance matrix,and provide the algorithm about how to determine the support set of limiting spectral distribution function and make the graph of the limiting spectral density function.In addition,some numerical results are obtained by programming to show the operability and effectiveness of this method.更多还原