Hilbert space geometry of quadratic covariance bounds

Abstract

In this paper, we study the geometry of quadratic covariance bounds on the estimation error covariance, in a properly defined Hilbert space of random variables. We show that a lower bound on the error covariance may be represented by the Grammian of the error score after projection onto the subspace spanned by the measurement scores. The Grammian is defined with respect to inner products in a Hilbert space of second order random variables. This geometric result holds for a large class of quadratic covariance bounds including the Barankin, Cramer-Rao, and Bhattacharyya bounds, where each bound is characterized by its corresponding measurement scores. When parameters consist of essential parameters and nuisance parameters, the Cramer-Rao covariance bound is the inverse of the Grammian of essential scores after projection onto the subspace orthogonal to the subspace spanned by the nuisance scores. In two examples, we show that for complex multivariate normal measurements with parameterized mean or covariance, there exist well-known Euclidean space geometries for the general Hilbert space geometry derived in this paper.