Abstract
In the framework of axial data, the most classical test of uniformity on the unit sphere of $\mathbb{R}^p$ is the Bingham (Ann. Statist. 2 (1974) 1201-1225) test. In this work, we study the non-null behaviour of this test in high-dimensional asymptotic scenarios where $p = p_n$ diverges to infinity with the sample size $n$. We consider a semiparametric class of alternatives that includes Watson alternatives and we derive a local asymptotic normality property. Le Cam’s third lemma reveals that the Bingham test is blind to the corresponding contiguous alternatives, though. By using martingale central limit theorems, we therefore study the behaviour of the Bingham test under more severe alternatives. Far from restricting to the aforementioned semiparametric alternatives, our results cover a broad class of rotationally symmetric alternatives, which allows us to consider non-axial alternatives, too. In every distributional framework we consider, the “detection threshold” of the Bingham test is identified and a comparison with the classical test of uniformity for non-axial data, namely the Rayleigh (Philos. Mag. 37 (1919) 321-346) test, is made possible. In the framework of axial data, we derive a lower bound on the minimax separation rate and establish that the Bingham test is minimax rate-optimal in the class of Watson distributions.
