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

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).