Testing uniformity on high-dimensional spheres against contiguous rotationally symmetric alternatives

We consider the problem of testing for uniformity on high-dimensional unit spheres. We are primarily interested in non-null issues. To this end, we consider rotationally symmetric alternatives and identify alternatives that are contiguous to the null of uniformity. This reveals a Locally and Asymptotically Normality (LAN) structure, which, for the first time, allows to use Le Cam's third lemma in the high-dimensional setup. Under very mild assumptions, we derive the asymptotic non-null distribution of the high-dimensional Rayleigh test and show that this test actually exhibits slower consistency rates. All ($n$,$p$)-asymptotic results we derive are "universal", in the sense that the dimension $p$ is allowed to go to infinity in an arbitrary way as a function of the sample size $n$. Part of our results also cover the low-dimensional case, which allows to explain heuristically the high-dimensional non-null behavior of the Rayleigh test. A Monte Carlo study confirms our asymptotic results.