Intrinsic variance lower bound (IVLB): an extension of the Cramer-Rao bound to Riemannian manifolds

We consider parametric statistical models in which the parameter space θ is a connected Riemannian manifold. This mathematical structure on the parameter space subsumes, as special cases, submanifolds of Euclidean spaces appearing in parametric estimation scenarios with a priori smooth deterministic constraints, and quotient spaces (such as Grassmann manifolds) which arise in certain parametric estimation scenarios with ambiguities. The Riemannian structure on the parameter space θ turns it into a metric space and the associated Riemannian distance is used here to quantify estimation errors. We present the intrinsic variance lower bound (IVLB) which places a lower limit on the accuracy, measured in terms of the mean-square Riemannian distance, of unbiased estimators taking values in θ. The IVLB depends both on the curvature of the parameter space and a coordinate-free extension of the well-known Fisher information matrix (FIN). We show that for flat Euclidean spaces, the IVLB collapses to the Cramer-Rao bound (CRB). In this sense, we may interpret the IVLB as a generalization of the CRB for curved parameter spaces. Computer simulations illustrating the application of the IVLB are included.