Abstract: We introduce a projection-based class of uniformity tests on the hypersphere. The class employs statistics that integrate, along all possible directions, a weighted quadratic discrepancy between the empirical cumulative distribution of the projections and the projected uniform distribution. Simple expressions for several test statistics are obtained for the circle and the sphere, as well as relatively tractable forms for higher dimensions. Despite their different origins, variants of the proposed class are shown to contain and be contained in variants of the Sobolev class of uniformity tests. Our new class proves itself advantageous by allowing the derivation of new tests that neatly extend the circular-only tests by Watson, Ajne, and Rothman, and by introducing the first instance of an Anderson?Darling-like test for directional data. We obtain usable asymptotic distributions and the local asymptotic optimality against certain alternatives of the new tests. A simulation study evaluates the theoretical findings and provides evidence that the new testing proposals are competitive. An application to the study of the crater distribution on Rhea illustrates the usage of the new tests.

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