Spatial quantiles on the hypersphere

Abstract

We propose a concept of quantiles for probability measures on the unit hypersphere S d − 1 of R d . The innermost quantile is the Fréchet median, that is, the L 1 -analog of the Fréchet mean. The proposed quantiles μ mα,u are directional in nature: they are indexed by a scalar order α ∈ [ 0 , 1 ] and a unit vector u in the tangent space T m S d − 1 to S d − 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 S d − 1 onto T m S d − 1 . Despite this link with Euclidean spatial quantiles, studying the proposed spherical 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.