Robust linear estimation with covariance uncertainties

In this paper, the problem of estimating a random vector x, with covariance uncertainties, that is observed through a known linear transformation H and corrupted by additive noise is considered. The linear estimator that minimizes the worst-case mean-squared error (MSE) across all possible covariance matrices is first developed. Although the minimax approach has enjoyed widespread use in the design of robust methods, its performance is often unsatisfactory as shown in the paper. A competitive minimax approach is developed in which the linear estimator that minimizes the worst-case regret, namely, the worst-case difference between the MSE attainable using a linear estimator is seek, ignorant of the signal covariance, and the optimal MSE attained using a linear estimator that knows the signal covariance. Through an example, the minimax regret approach can improve the performance over the minimax MSE approach is demonstrated.