ABSTRACT
In a visionary paper, Fre´chet (1948) introduced the notion of random element
in a metric space, and noted that those new concepts and results “are setting the
bases of a new Chapter, that we believe will insure a great future of Probability
theory” as they have important implications in both Statistics and Experimen-
tal Sciences. In that paper, Fre´chet defined the mean of a probability measure
on a separable metric space (M,ρ), and hinted at a proof of the consistency of the sample mean for a random sample from such a distribution. Consistency,
this important feature required of an estimator, was given a complicated first
proof by Ziezold (1977) [345]. Essentially Ziezold showed that the Fre´chet
sample mean set is a consistent estimator of the Fre´chet mean set if (M,ρ) is compact, and according to Huckemann (2011)[164], Ziezold’s proof goes
through without any difficulty in the case that (M,ρ) is complete as well. Bhattacharya and Patrangenaru (2003)[42] gave a new proof of the consistency of
the Fre´chet sample mean on a complete separable metric space, which includes
the case of a random object (r.o.) on a complete space with a manifold stratifi-
cation. They also introduced the Fre´chet total variance of a r.o. in a 2002 paper
[41], and showed that this parameter is consistently estimated by the Fre´chet
total sample variance if (M,ρ) is complete. Given a probability measure Q on a manifold M, the Fre´chet moments of Q were called by Patrangenaru (1998)[266] extrinsic moments, if ρ is induced by the chord distance via an embedding of M into an Euclidean space, respectively intrinsic moments, if
ρ is the Riemannian distance induced by a Riemannian structure g onM. A generic consistency result for extrinsic sample means on submanifolds, due
to Hendricks (1990a) [151], was extended by Patrangenaru (1998) [266] and
Bhattacharya and Patrangenaru (2003) [42] to extrinsic sample means on j-
nonfocal distributions on a manifoldM, with an embedding j
4.2 Fre´chet Means and Cartan Means
In this section we consider aM-valued random object (r.o.) X , i.e. a measurable map on a probability space (Ω ,A,Pr) into (M,BM), where BM denotes the Borel sigma-algebra generated by open subsets ofM. To each r.o. X , we associate a probability measure Q = PX on BM given by Q(B) = Pr(X−1(B)), therefore from now, when we are referring to probability measures or distribu-
tions on M, we mean probability measures defined on BM. The aim of this section is to develop nonparametric statistical inference procedures for mea-
sures of location and spread of distributions on arbitrary complete manifolds.
