  In many applications, we encounter data on Riemannian manifolds such as torus
and rotation groups. Standard statistical procedures for multivariate data are
not applicable to such data. In this study, we develop goodness-of-fit testing
and interpretable model criticism methods for general distributions on
Riemannian manifolds, including those with an intractable normalization
constant. The proposed methods are based on extensions of kernel Stein
discrepancy, which are derived from Stein operators on Riemannian manifolds. We
discuss the connections between the proposed tests with existing ones and
provide a theoretical analysis of their asymptotic Bahadur efficiency.
Simulation results and real data applications show the validity of the proposed
methods.
