Ranks, Signs, and Quantiles for Directional Databased on Measure Transportation

• Activity Number: 530 - Recent Advances in Rank-Based Inference
• Type: Invited
• Date/Time: Thursday, August 11, 2022 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Hang Liu and Marc Hallin and Thomas Verdebout*
• Companies: University of Science and Technology of China and Université libre de Bruxelles and Université Libre de Bruxelles
• Keywords: Directional data; Measure transportation; Quantiles; Ranks; Signs

Abstract

Directional statistics are concerned with observations that are not unrestricted p-dimensional vectors as in classical multivariate statistical analysis but directions, axes or rotations. Directional data can be found in various fields, including astronomy, medicine and environmetrics to cite a few. In the present paper, we propose concepts of ranks, signs, and quantiles based on measure transportation ideas for data taking values on unit (hyper)spheres. These ranks and signs allow for distribution-free inference for directional data without the usual assumption of rotational symmetry, while the corresponding quantile contours define a data-driven system of parallels. We show the consistency of empirical versions of the quantiles and demonstrate their usefulness on simulated and real data.