Joint density of eigenvalues in spiked multivariate models

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.