Geodesic lower bound for parametric estimation with constraints

We consider parametric statistical models indexed by embedded submanifolds /spl otimes/ of /spl Ropf//sup p/. This setup occurs in practical applications whenever the parameter of interest /spl theta/ is known to satisfy a priori deterministic constraints, encoded herein by /spl otimes/. We assume that the submanifold /spl otimes/ is connected and endowed with the Riemannian structure inherited from the ambient space /spl Ropf//sup p/. This turns /spl otimes/ 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 /spl otimes/. A numerical example involving the special group of orthogonal matrices SO(n, /spl Ropf/) is worked out.