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The very general results of Beran and Gine on invariant tests of uniformity are applied to Sₚ, the surface of the unit hypersphere, and Hₚ, the surface of the unit hypersphere with antipodes identified, to give a class of invariant tests of uniformity for signed and unsigned directional data in (p + 1) -dimensions. The (p + 1) -dimensional analogues of the test statistics due to Rayleigh, Bingham, Ajne, and Gine are constructed as the simplest examples, and corresponding methods are derived for particular orientation statistics as examples on H₃.
Michael J. Prentice (Sun,) studied this question.