Geodesic lower bound for parametric estimation with constraints

We consider parametric statistical models indexed by embedded submanifolds ⊗ of R^p. This setup occurs in practical applications whenever the parameter of interest θ is known to satisfy a priori deterministic constraints, encoded herein by ⊗. We assume that the submanifold ⊗ is connected and endowed with the Riemannian structure inherited from the ambient space R^p. This turns ⊗ into a metric space in which the distance between points corresponds to the geodesic distance. We discuss a lower bound for the intrinsic variance (that is, measured in terms of the geodesic distance) of unbiased estimators taking values in ⊗. A numerical example involving the special group of orthogonal matrices SO(n, R) is worked out.