Monte Carlo (MC) methods have become an important tool for
inferences in non-Gaussian and non-Euclidean settings. We study
the use of MC methods in those signal/image processing scenarios
where the parameter spaces are certain Riemannian manifolds
(finite-dimensional Lie groups and their quotient sets). We
investigate the estimation of {\it means} and {\it variances}, of
the manifold-valued parameters, using two popular sampling
methods: independent and importance sampling. Using Euclidean
embeddings, we specify a notion of {\it extrinsic means}, employ
Monte Carlo methods (independent and importance sampling) to
estimate these means, and utilize large-sample asymptotics to
approximate the estimator covariances. Experimental results are
presented for target pose estimation (orthogonal groups) and
signal subspace estimation (Grassmann manifolds). Asymptotic
covariances are utilized to construct confidence regions, to
compare estimators, and to determine the sample size for MC
methods.