Tests of Concentration for Low-Dimensional and High-Dimensional Directional Data

We consider asymptotic inference for the concentration of directional data. More precisely, wepropose tests for concentration (i) in the low-dimensional case where the sample size n goes to infinity andthe dimension p remains fixed, and (ii) in the high-dimensional case where both n and p become arbitrarilylarge. To the best of our knowledge, the tests we provide are the first procedures for concentration thatare valid in the (n; p)-asymptotic framework. Throughout, we consider parametric FvML tests, that areguaranteed to meet asymptotically the nominal level constraint under FvML distributions only, as well as“pseudo-FvML” versions of such tests, that are validity-robust within the class of rotationally symmetricdistributions.We conduct a Monte-Carlo study to check our asymptotic results and to investigate the finitesamplebehavior of the proposed tests.