Nonparametric Measure-Transportation-Based Methods for Directional Data

This paper proposes various nonparametric tools based on measure transportation for directional data. We use optimal transports to define new notions of distribution and quantile functions on the hypersphere, with meaningful quantile contours and regions yielding closed-form formulas under the classical assumption of rotational symmetry. The empirical versions of our distribution functions enjoy the expected Glivenko-Cantelli property of traditional distribution functions. They provide fully distribution-free concepts of ranks and signs and define data-driven systems of (curvilinear) parallels and (hyper)meridians. Based on this, we also construct a universally consistent test of uniformity which, in simulations, outperforms the ``projected'' Cram\' er-von Mises, Anderson-Darling, and Rothman procedures recently proposed in the literature. We also propose fully distribution-free rank- and sign-based tests for directional MANOVA. Two real-data examples involving the analysis of sunspots and proteins structures conclude the paper.