Title: Spatial quantiles on the hypersphere
Authors: Dimitri Konen - Université Libre de Bruxelles (Belgium) [presenting]
Davy Paindaveine - Universite libre de Bruxelles (Belgium)
Abstract: A concept of quantiles for distributions on the unit hypersphere $R^d$ is proposed. The innermost quantiles are Frechet medians, i.e. the $L^1$-analog of Frechet means. Since these medians may be non-unique, we define a quantile field around each such median $m$. The corresponding quantiles are directional in nature: they are indexed by a scalar order between 0 and 1 and a unit vector in the tangent space to the hypersphere at the median. To ensure computability in any dimension, our quantiles are essentially obtained by considering the Euclidean Chaudhuri spatial quantiles in a suitable stereographic projection of the hypersphere onto its tangent space at the median. Despite this link with their Euclidean antecedent, studying our quantiles requires understanding the nature of the Chaudhuri quantile in a version of the projective space where all points at infinity are identified. We thoroughly investigate the properties of the proposed quantiles, and study in particular the asymptotic behaviour of their sample versions, which requires controlling the impact of estimating the median. Our spherical quantile concept also allows for companion concepts of ranks and depth on the hypersphere.