Large sample theory of intrinsic and extrinsic sample means on manifolds. I

Abstract

Sufficient conditions are given for the uniqueness of intrinsic and extrinsic means as measures of location of probability measures Q on Riemannian manifolds. It is shown that, when uniquely defined, these are estimated consistently by the corresponding indices of the empirical Q̂ n. Asymptotic distributions of extrinsic sample means are derived. Explicit computations of these indices of Q̂ n and their asymptotic dispersions are carried out for distributions on the sphere S d (directional spaces), real projective space ℝP N-1 (axial spaces) and ℂP k-2 (planar shape spaces).