A general maximal projection approach to uniformity testing on the hypersphere

We propose a novel approach to uniformity testing on the d-dimensional unit hypersphere $S$$^{d−1}$ based on maximal projections. This approach gives a unifying view on the classical uniformity tests of Rayleigh and Bingham, and it links to measures of multivariate skewness and kurtosis. We derive the limit distribution under the null hypothesis of the newly proposed test statistics using limit theorems for Banach-space-valued stochastic processes and we present strategies to simulate the limit processes by applying results on the theory of spherical harmonics. We examine the behaviour of the test statistics under contiguous and fixed alternatives and show the consistency of the testing procedure for some classes of alternatives. For the first time in uniformity testing on the hypersphere, we derive local Bahadur efficiency statements. Finally, we evaluate the theoretical findings and empirical power of the procedures in a broad competitive Monte Carlo simulation study.