  Manifold data analysis is challenging due to the lack of parametric
distributions on manifolds. To address this, we introduce a series of
Riemannian radial distributions on Riemannian symmetric spaces. By utilizing
the symmetry, we show that for many Riemannian radial distributions, the
Riemannian $L^p$ center of mass is uniquely given by the location parameter,
and the maximum likelihood estimator (MLE) of this parameter is given by an
M-estimator. Therefore, these parametric distributions provide a promising tool
for statistical modeling and algorithmic design.
  In addition, our paper develops a novel theory for parameter estimation and
minimax optimality by integrating statistics, Riemannian geometry, and Lie
theory. We demonstrate that the MLE achieves a convergence rate of root-$n$ up
to logarithmic terms, where the rate is quantified by both the hellinger
distance between distributions and geodesic distance between parameters. Then
we derive a root-$n$ minimax lower bound for the parameter estimation rate,
demonstrating the optimality of the MLE. Our minimax analysis is limited to the
case of simply connected Riemannian symmetric spaces for technical reasons, but
is still applicable to numerous applications. Finally, we extend our studies to
Riemannian radial distributions with an unknown temperature parameter, and
establish the convergence rate of the MLE. We also derive the model complexity
of von Mises-Fisher distributions on spheres and discuss the effects of
geometry in statistical estimation.
