The classical methods of multivariate analysis are based on the eigenvalues of one or two sample covariance matrices. In many applications of these methods, for example, to high‐dimensional data, it is natural to consider alternative hypotheses that are a low‐rank departure from the null hypothesis. For rank 1 alternatives, this note provides a representation for the joint eigenvalue density in terms of a single contour integral. This will be of use for deriving approximate distributions for likelihood ratios and “linear” statistics used in testing. Copyright © 2014 John Wiley & Sons, Ltd.

中文翻译：

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尖峰多元模型中特征值的联合密度。

多变量分析的经典方法基于一个或两个样本协方差矩阵的特征值。在这些方法的许多应用中，例如，对于高维数据，很自然地会考虑与原假设有低秩背离的替代假设。对于排名 1 的替代方案，本注释根据单个轮廓积分提供了联合特征值密度的表示。这将用于推导出似然比和测试中使用的“线性”统计的近似分布。版权所有 © 2014 John Wiley & Sons, Ltd.