Intrinsic distance lower bound for unbiased estimators on Riemannian manifolds

We consider statistical models parameterized over connected Riemannian manifolds. We present a lower bound on the mean-square distance of unbiased estimators about their mean values. The derived bound depends both on the curvature of the parameter manifold and a coordinate-free extension of the classical Fisher information matrix. Our study can be applied in estimation problems with smooth parametric constraints, and in statistical models indexed over coset spaces. Illustrative examples concerning inferences on the unit-sphere and the complex projective space are worked out.