Large sample theory of intrinsic and extrinsic sample means on manifolds

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 $\hat Q_n$. Asymptotic distributions of extrinsic sample means are derived. Explicit computations of these indices of $\hat Q_n$ and their asymptotic dispersions are carried out for distributions on the sphere $S^d$ (directional spaces), real projective space $\mathbb{R}P^{N-1}$ (axial spaces) and $\mathbb{C} P^{k-2}$ (planar shape spaces).