Nonparametric Measure-transportation-based Methods for Directional Data

This paper proposes various nonparametric tools for directional data based on measure transportation. We use optimal transports todefine new notions of distribution and quantile functions on the hypersphere, with meaningful quantile contours and regions and closedformformulas 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 yield fully distribution-free concepts of ranks and signs and define data-driven systems of (curvilinear) parallels and (hyper) meridians. Based on this, we also propose a test of uniformity and establish its universal consistency; simulations indicate that this test outperforms the “projected” Cram´er-von Mises, Anderson-Darling, and Rothman procedures recently proposed in the literature. Two real-data examples involving the analysis of Venus craters and proteins structures conclude the paper.