Nonparametric measure-transportation-based methods for directional data

Author:

Hallin M1ORCID,Liu H2,Verdebout T1

Affiliation:

1. ECARES and Département de Mathématique, Université libre de Bruxelles , Brussels B-1050 , Belgium

2. School of Scitech Business, School of Management, University of Science and Technology of China , Hefei 230026, Anhui Province , People’s Republic of China

Abstract

Abstract This article 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 article.

Funder

European Cooperation in Science and Technology

Czech Science Foundation

USTC

Publisher

Oxford University Press (OUP)

Reference79 articles.

1. Sine-skewed circular distributions;Abe;Statistical Papers,2011

2. A simple test for uniformity of a circular distribution;Ajne;Biometrika,1968

3. N-distance tests of uniformity on the hypersphere;Bakshaev;Nonlinear Analysis: Modelling and Control,2010

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