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We propose a novel approach to uniformity testing on the $d$-dimensional unithypersphere $\mathcal{S}^{d-1}$ based on maximal projections. This approachgives a unifying view on the classical uniformity tests of Rayleigh andBingham, and it links to measures of multivariate skewness and kurtosis. Wederive the limiting distribution under the null hypothesis using limit theoremsfor Banach space valued stochastic processes and we present strategies tosimulate the limiting processes by applying results on the theory of sphericalharmonics. We examine the behavior under contiguous and fixed alternatives andshow the consistency of the testing procedure for some classes of alternatives.For the first time in uniformity testing on the sphere, we derive local Bahadurefficiency statements. We evaluate the theoretical findings and empiricalpowers of the procedures in a broad competitive Monte Carlo simulation studyand, finally, apply the new tests to a data set on midpoints of large craterson the moon.

A general maximal projection approach to uniformity testing on the hypersphere