On testing the equality of latent roots of scatter matrices under ellipticity

In the present paper, we tackle the problem of testing $H_{0q}:{\lambda }_q>{\lambda }_{q+1}=\cdots ={\lambda }_p$, where ${\lambda }_1,\dots ,{\lambda }_p$ are the scatter matrix eigenvalues of an elliptical distribution on $R^p$. This is a classical problem in multivariate analysis which is very useful in dimension reduction. We analyse the problem using the Le Cam asymptotic theory of experiments and show that contrary to the testing problems on eigenvalues and eigenvectors of a scatter matrix tackled in Hallin et al. (2010), the non-specification of nuisance parameters has an asymptotic cost for testing $H_{0q}$. We moreover derive signed-rank tests for the problem that enjoy the property of being asymptotically distribution-free under ellipticity. The van der Waerden rank test uniformly dominates the classical pseudo-Gaussian procedure for the problem. Numerical illustrations show the nice finite-sample properties of our tests.