论文标题
随机矩阵理论的统计应用：两个人群的比较i
Statistical applications of random matrix theory: comparison of two populations I
论文作者
论文摘要
本文研究了一个统计程序，用于测试两个独立估计的协方差矩阵的平等，当可能依赖的数据向量的数量较大并且与向量的大小成正比时，即变量的数量。受到随机矩阵理论中使用的尖峰模型的启发，我们集中在矩阵的最大特征值上，以确定显着性。为了避免错误的拒绝，我们必须防止残留的尖峰，并且需要对无原假设下最大特征值的行为进行足够精确的描述。在本文中，我们通过基于订单$ 1 $的扰动来处理替代方案，即一个大型特征值。我们的统计数据允许用户测试两个人群的平等。未来的工作将把结果扩展到订单$ K $的扰动，并证明了对更通用矩阵的程序的保守性。
This paper investigates a statistical procedure for testing the equality of two independent estimated covariance matrices when the number of potentially dependent data vectors is large and proportional to the size of the vectors, that is, the number of variables. Inspired by the spike models used in random matrix theory, we concentrate on the largest eigenvalues of the matrices in order to determine significance. To avoid false rejections we must guard against residual spikes and need a sufficiently precise description of the behaviour of the largest eigenvalues under the null hypothesis. In this paper, we lay a foundation by treating alternatives based on perturbations of order $1$, that is, a single large eigenvalue. Our statistic allows the user to test the equality of two populations. Future work will extend the result to perturbations of order $k$ and demonstrate conservativeness of the procedure for more general matrices.
