Connecting pairwise spheres by depth: DCOPS

We extend the classical notion of the spherical depth in \mathbb{R}^k, to the important setup of data on a Riemannian manifold. We show that this notion of depth satisfies a set of desirable properties. For the empirical version of this depth function both uniform consistency and the asymptotic distribution are studied. Consistency is also shown for functional data. The behaviour of the depth is illustrated through several examples for Riemannian manifold data.