Title: Distance-based Depths for Directional Data,
Abstract: Directional data are constrained to lie on the unit sphere of~$\mathbb{R}^q$
for some~$q\geq 2$. To address the lack of a natural ordering for such data,
depth functions have been defined on spheres. However, the depths available
either lack flexibility or are so computationally expensive that they can only
be used for very small dimensions~$q$. In this work, we improve on this by
introducing a class of distance-based depths for directional data. Irrespective
of the distance adopted, these depths can easily be computed in high dimensions
too. We derive the main structural properties of the proposed depths and study
how they depend on the distance used. We discuss the asymptotic and robustness
properties of the corresponding deepest points. We show the practical relevance
of the proposed depths in two applications, related to (i) spherical location
estimation and (ii) supervised classification. For both problems, we show
through simulation studies that distance-based depths have strong advantages
over their competitors.