Bounds for Estimation of Covariance Matrices From Heterogeneous Samples

This correspondence derives lower bounds on the mean-square error (MSE) for the estimation of a covariance matrix \mbi
Mp
, using samples \mbi
Zk
,
k
=1,...,
K
, whose covariance matrices \mbi
Mk
are randomly distributed around \mbi
Mp
. This framework can be encountered e.g., in a radar system operating in a nonhomogeneous environment, when it is desired to estimate the covariance matrix of a range cell under test, using training samples from adjacent cells, and the noise is nonhomogeneous between the cells. We consider two different assumptions for \mbi
Mp
. First, we assume that \mbi
Mp
is a deterministic and unknown matrix, and we derive the Cramer-Rao bound for its estimation. In a second step, we assume that \mbi
Mp
is a random matrix, with some prior distribution, and we derive the Bayesian bound under this hypothesis.