Convergence Rates For Estimators Of Geodesic Distances And Frechet Expectations

Consider a sample X-n, = {X-1, ... , X-n} of independent and identically distributed variables drawn with a probability distribution P-x supported on a compact set M subset of R-d. In this paper we mainly deal with the study of a natural estimator for the geodesic distance on M. Under rather general geometric assumptions on M, we prove a general convergence result. Assuming M to be a compact manifold of known dimension d' <= d, and under regularity assumptions on P-x, we give an explicit convergence rate. In the case when M has no boundary, knowledge of the dimension d' is not needed to obtain this convergence rate. The second part of the work consists in building an estimator for the Frechet expectations on M, and proving its convergence under regularity conditions, applying the previous results.