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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 Qn . Asymptotic distributions of extrinsic sample means are derived. Explicit computations of these indices of Qn 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). Introduction.The aim of this article is to develop nonparametric statistical inference procedures for measures of location of distributions on general manifolds, which are complete as metric spaces. Although the main applications are to distributions on (i) spheres S d (spaces of directions), (ii) real projective spaces RP N−1 (axial spaces) and (iii) complex projective spaces CP k−2 (planar shape spaces), a general theory for both compact and noncompact manifolds is sought. In this introduction a summary of the main results is presented, along with a brief review of the literature on the subject.A natural index of location for a probability measure Q on a metric space M with the distance ρ is the so-called Fréchet mean which minimizes F (p) = ρ 2 (p, x)Q(dx), if there is a unique minimizer. In general, the set of all minimizers is called the Fréchet mean set. In the case M is a d-dimensional connected C ∞ Riemannian manifold with a metric tensor g and geodesic distance d g , we will assume that (M, d g ) is complete and we will refer to the Fréchet mean (set) as the intrinsic mean (set). We say that the intrinsic mean exists if there is a unique minimizer, and denote it by µ I (Q). It is shown in Theorem 2.1 that (i) the intrinsic mean set is compact, (ii) for each point m in the intrinsic mean set, the Euclidean mean of the distribution on the tangent space at m of the inverse of the exponential map is zero and (iii) in the case of simply connected M of nonpositive curvature, the intrinsic mean exists if F is finite; a particular case of this result, when M is a Bookstein's shape space of labeled triangles, with a Riemannian metric of constant negative curvature is due to Le and Kume (2000). From a result of Karcher (1977) it follows that if the distribution is

This article develops nonparametric inference procedures for estimation and testing problems for means on manifolds. A central limit theorem for Fréchet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on Riemannian manifolds. Central limit theorems are also obtained for extrinsic sample means w.r.t. an arbitrary embedding of a differentiable manifold in a Euclidean space. Bootstrap methods particularly suitable for these problems are presented. Applications are given to distributions on the sphere S d (directional spaces), real projective space RP N−1 (axial spaces), complex projective space CP k−2 (planar shape spaces) w.r.t. Veronese-Whitney embeddings and a three-dimensional shape space Σ 4 3 .This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Statistics, 2005, Vol. 33, No. 3, 1225-1259. This reprint differs from the original in pagination and typographic detail. 1 2 R. BHATTACHARYA AND V. PATRANGENARU sample means and confidence regions based on them. We provide classical CLT-based confidence regions and tests based on them, as well as those based on Efron's bootstrap [17].Measures of location and dispersion for distributions on a manifold M were studied in [7,8] as Fréchet parameters associated with two types of distances on M . If j : M → R k is an embedding, the Euclidean distance restricted to j(M ) yields the extrinsic mean set and the extrinsic total variance. On the other hand, a Riemannian distance on M yields the intrinsic mean set and intrinsic total variance.Recall that the Fréchet mean of a probability measure Q on a complete metric space (M, ρ) is the minimizer of the function F (x) = ρ 2 (x, y)Q(dy), when such a minimizer exists and is unique [21]. In general the set of minimizers of F is called the Fréchet mean set. The intrinsic mean µ I (Q) is the Fréchet mean of a probability measure Q on a complete d-dimensional Riemannian manifold M endowed with the geodesic distance d g determined by the Riemannian structure g on M . It is known that if Q is sufficiently concentrated, then µ I (Q) exists [see Theorem 2.2(a)]. The extrinsic mean µ E (Q) = µ j,E (Q) of a probability measure Q on a manifold M w.r.t. an embedding j : M → R k is the Fréchet mean associated with the restriction to j(M ) of the Euclidean distance in R k . In [8] it was shown that the extrinsic mean of Q exists if the ordinary mean of j(Q) is a nonfocal point of j(M ), that is, if there is a unique point x 0 on j(M ) having the smallest distance from the mean of j(Q). In this case µ j,E (Q) = j −1 (x 0 ).It is easier to compute the intrinsic mean if the Riemannian manifold has zero curvature in a neighborhood containing supp Q [45]. In particular this is the case for distributions on linear projective shape spaces [42]. If the manifold has nonzero curvature around supp Q, it is easier to compute the extrinsic sample mean. It may be pointed out that if Q is highly concentrated as in our medical examples in [8] ...

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