Power Enhancement for Testing the Equality of Shape Matrix Eigenvalues Under Ellipticity

In this work, we consider the problem of testing the null hypotheses \(\mathcal {H}_{0q}: \lambda _{q,\mathbf {V}} > \lambda _{q+1,\mathbf {V}}= \ldots = \lambda _{p,\mathbf {V}}\) where \(\lambda _{1,\mathbf {V}} \geq \ldots \geq \lambda _{p,\mathbf {V}}\) are the ordered eigenvalues of the shape matrix \(\mathbf {V}\) of an elliptical distribution. We propose a class of tests based on signed-rank statistics. Our new tests are constructed (i) to keep the nice properties of the tests introduced in Bernard and Verdebout T (J Multivar Anal, 2023) for the problem, (ii) to improve the detection ability of the same tests in Bernard and Verdebout T (J Multivar Anal, 2023) against alternatives of the form \(\mathcal {H}_{1q} : \lambda _{q,\mathbf {V}} = \lambda _{q+1,\mathbf {V}}=\ldots =\lambda _{p,\mathbf {V}}\) , and (iii) to improve the robustness to outliers and heavy tails in the data-generating process of the pseudo-Gaussian test proposed in Bernard and Verdebout (Stat Sin, 2024). We show through Monte-Carlo simulations that our new tests achieve these objectives.