Abstract
A new class of nonparametric tests, based on random projections, is proposed. They can be used for several null hypotheses of practical interest, including uniformity for spherical (directional) and compositional data, sphericity of the underlying distribution and homogeneity in two-sample problems on the sphere or the simplex. The proposed procedures have a number of advantages, mostly associated with their flexibility (for example, they also work to test "partial uniformity" in a subset of the sphere), computational simplicity and ease of application even in high-dimensional cases. This paper includes some theoretical results concerning the behaviour of these tests, as well as a simulation study and a detailed discussion of a real data problem in astronomy.
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