SPATIAL QUANTILES ON THE HYPERSPHERE

• 成果类型: Article
• 署名作者: Konen, Dimitri; Paindaveine, Davy
• 署名单位: Universite Libre de Bruxelles; Universite Libre de Bruxelles
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/23-AOS2332
• 发表日期: 2023
• 页码: 2221-2245
• 关键词: directional-data; hilbert-spaces; data depth; MANIFOLDS; regression; efficient; notion; ranks

摘要

We propose a concept of quantiles for probability measures on the unit hypersphere Sd-1 of Rd. The innermost quantile is the Frechet median, that is, the L1-analog of the Frechet mean. The proposed quantiles mu au are di-rectional in nature: they are indexed by a scalar order alpha e [0, 1] and a unit vector u in the tangent space TmSd-1 to Sd-1 at m. To ensure computability in any dimension d, our quantiles are essentially obtained by considering the Euclidean (Chaudhuri (J. Amer. Statist. Assoc. 91 (1996) 862-872)) spatial quantiles in a suitable stereographic projection of Sd-1 onto TmSd-1. De-spite this link with Euclidean spatial quantiles, studying the proposed spher-ical quantiles requires understanding the nature of the (Chaudhuri (1996)) quantiles in a version of the projective space where all points at infinity are identified. We thoroughly investigate the structural properties of our quan-tiles and we further study the asymptotic behavior of their sample versions, which requires controlling the impact of estimating m. Our spherical quantile concept also allows for companion concepts of ranks and depth on the hy-persphere. We illustrate the relevance of our construction by considering two inferential applications, related to supervised classification and to testing for rotational symmetry.