In many applications, data and/or parameters are supported on non-Euclidean
manifolds. It is important to take into account the geometric structure of
manifolds in statistical analysis to avoid misleading results. Although there
has been a considerable focus on simple and specific manifolds, there is a lack
of general and easy-to-implement statistical methods for density estimation and
modeling on manifolds. In this article, we consider a very broad class of
manifolds: non-compact Riemannian symmetric spaces. For this class, we provide
a very general mathematical result for easily calculating volume changes of the
exponential and logarithm map between the tangent space and the manifold. This
allows one to define statistical models on the tangent space, push these models
forward onto the manifold, and easily calculate induced distributions by
Jacobians. To illustrate the statistical utility of this theoretical result, we
provide a general method to construct distributions on symmetric spaces. In
particular, we define the log-Gaussian distribution as an analogue of the
multivariate Gaussian distribution in Euclidean space. With these new kernels
on symmetric spaces, we also consider the problem of density estimation. Our
proposed approach can use any existing density estimation approach designed for
Euclidean spaces and push it forward to the manifold with an easy-to-calculate
adjustment. We provide theorems showing that the induced density estimators on
the manifold inherit the statistical optimality properties of the parent
Euclidean density estimator; this holds for both frequentist and Bayesian
nonparametric methods. We illustrate the theory and practical utility of the
proposed approach on the space of positive definite matrices