Tests of concentration for low-dimensional and high-dimensional directional data

Abstract

We consider asymptotic inference for the concentration of directional data. More precisely, we propose tests for concentration (1) in the low-dimensional case where the sample size $n$ goes to infinity and the dimension $p$ remains fixed, and (2) in the high-dimensional case where both $n$ and $p$ become arbitrarily large. To the best of our knowledge, the tests we provide are the first procedures for concentration that are valid in the $(n,p)$-asymptotic framework. Throughout, we consider parametric FvML tests, that are guaranteed to meet asymptotically the nominal level constraint under FvML distributions only, as well as “pseudo-FvML” versions of such tests, that meet asymptotically the nominal level constraint within the whole class of rotationally symmetric distributions. We conduct a Monte-Carlo study to check our asymptotic results and to investigate the finite-sample behavior of the proposed tests.