Intrinsic Hölder classes of density functions on Riemannian manifolds and lower bounds to convergence rates

Abstract

We consider Hölder classes of density functions on Riemannian manifolds in intrinsic perspectives. We develop a theorem that links such Hölder classes on Riemannian manifolds to those on Euclidean spaces. Using the theorem, we derive lower bounds to Lp-convergence rates (1≤p≤∞) for the estimation of the underlying density on a Riemannian manifold.