In this paper, the exact distribution of the largest eigenvalue of a singular random matrix for multivariate analysis of variance (MANOVA) is discussed. The key to developing the distribution theory of eigenvalues of a singular random matrix is to use heterogeneous hypergeometric functions with two matrix arguments. In this study, we define the singular beta F-matrix and extend the distributions of a nonsingular beta F -matrix to the singular case. We also give the joint density of eigenvalues and the exact distribution of the largest eigenvalue in terms of heterogeneous hypergeometric functions.
翻译:本文讨论了用于多变量差异分析的单一随机矩阵(MANOVA)的最大电子元值的准确分布情况。开发单一随机矩阵(MANOVA)电子元值分布理论的关键是使用具有两个矩阵参数的多元超几何函数。在本研究中,我们定义了单倍F矩阵,并将非单数贝贝F矩阵的分布扩展至单数。我们还给出了非单数贝贝F矩阵的共密度和最大超几何函数的最大电子元值的确切分布。