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We consider parametric statistical models indexed by embedded submanifolds ⊗ of Rp. 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 Rp. 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.
Date of Conference: 11-14 July 2004