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 and 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 and a class of fully distribution-free and universally consistent tests for directional MANOVA which, in simulations, outperform all their existing competitors. A real-data example involving the analysis of sunspots concludes the paper.

翻译：本文提出了多种基于测度传输的非参数工具，用于处理定向数据。我们利用最优传输在超球面上定义了新的分布函数和分位数函数概念，在经典的旋转对称假设下，得到了具有明确分位数等高线和区域的结果，并给出了闭式公式。所提出的分布函数的经验版本具有传统分布函数预期的Glivenko-Cantelli性质。它们提供了完全分布自由的秩和符号概念，并定义了数据驱动的（曲线）平行线和（超）子午线系统。在此基础上，我们还构建了一个普遍一致的均匀性检验，以及一类完全分布自由且普遍一致的定向多元方差分析检验，这些检验在模拟中优于所有现有方法。最后，通过一个涉及太阳黑子分析的真实数据实例对本文进行了总结。