Abstract: We consider the classical problem of testing , where , . . . , are the ordered latent roots of covariance matrices . We show that the usual Gaussian procedure, , for this problem essentially shows no power against alternatives of weaker signals of the form , which is problematic if it is used to perform inference on the true dimension of the signal. We show that the same test enjoys some local and asymptotic optimality properties for detecting alternatives to the equality of the p-q smallest roots of , provided that and are sufficiently separated. We obtain tests, , for the problem that retain the local and asymptotic optimality properties of when and are sufficiently separated and properly detect alternatives of the form . We illustrate the performances of our tests using simulations and on a gene expression data set, where we also discuss the problem of estimating the dimension of the signal.

Key words and phrases: Hypothesis testing, latent roots, signal dimension.