Connecting pairwise geodesic spheres by depth: DCOPS

We extend the classical notion of spherical depth for data in Rk to the case of data on a Riemannian manifold. We show that this notion has several desirable properties. The uniform consistency and limiting distribution of an empirical analogue of this depth function are studied. Consistency is also shown for functional data. Various illustrations involving Riemannian manifold data are included.