This paper discusses the approximate distributions of eigenvalues of a singular Wishart matrix. We give the approximate joint density of eigenvalues by Laplace approximation for the hyper-geometric functions of matrix arguments. Furthermore, we show that the distribution of each eigenvalue can be approximated by the chi-square distribution with varying degrees of freedom when the population eigenvalues are infinitely dispersed. The derived result is applied to testing the equality of eigenvalues in two populations

翻译：本文讨论了奇异Wishart矩阵特征值的近似分布。通过矩阵参数超几何函数的拉普拉斯近似，我们给出了特征值的近似联合密度。进一步，当总体特征值无限分散时，我们证明了每个特征值的分布可近似为具有不同自由度的卡方分布。所得结果被应用于检验两组总体特征值的相等性。