MSE-ratio regret estimation with bounded data uncertainties

We consider the problem of robust estimation of a deterministic bounded parameter vector x in a linear model. While in an earlier work, we proposed a minimax estimation approach in which we seek the estimator that minimizes the worst-case mean-squared error (MSE) difference regret over all bounded vectors x, here we consider an alternative approach, in which we seek the estimator that minimizes the worst-case MSE ratio regret, namely, the worst-case ratio between the MSE attainable using a linear estimator ignorant of x, and the minimum MSE attainable using a linear estimator that knows x. The rational behind this approach is that the value of the difference regret may not adequately reflect the estimator performance, since even a large regret should be considered insignificant if the value of the optimal MSE is relatively large.