Robust Estimation of a Random Parameter in a Gaussian Linear Model With Joint Eigenvalue and Elementwise Covariance Uncertainties

We consider the estimation of a Gaussian random vector
x
observed through a linear transformation
H
and corrupted by additive Gaussian noise with a known covariance matrix, where the covariance matrix of
x
is known to lie in a given region of uncertainty that is described using bounds on the eigenvalues and on the elements of the covariance matrix. Recently, two criteria for minimax estimation called difference regret (DR) and ratio regret (RR) were proposed and their closed form solutions were presented assuming that the eigenvalues of the covariance matrix of
x
are known to lie in a given region of uncertainty, and assuming that the matrices
H

^T

Cw

^-1

H
and
C

_x
are jointly diagonalizable, where
C

_w
and
C

_x
denote the covariance matrices of the additive noise and of
x
respectively. In this work we present a new criterion for the minimax estimation problem which we call the generalized difference regret (GDR), and derive a new minimax estimator which is based on the GDR criterion where the region of uncertainty is defined not only using upper and lower bounds on the eigenvalues of the parameter's covariance matrix, but also using upper and lower bounds on the individual elements of the covariance matrix itself. Furthermore, the new estimator does not require the assumption of joint diagonalizability and it can be obtained efficiently using semidefinite programming. We also show that when the joint diagonalizability assumption holds and when there are only eigenvalue uncertainties, then the new estimator is identical to the difference regret estimator. The experimental results show that we can obtain improved mean squared error (MSE) results compared to the MMSE, DR, and RR estimators.