Improvement of Least-Squares Under Arbitrary Weighted MSE

The seminal work of Stein in the 1950´s ignited a large body of research devoted to improving the total mean-squared error (MSE) of the least-squares (LS) estimator. A drawback of these methods is that they improve the total MSE at the expense of increasing the MSE of some of the individual signal components. Here we consider a framework for developing linear estimators that outperform LS over bounded norm signals, under all weighted MSE measures. We first derive an easily verifiable condition on a linear method that ensures LS domination for every weighted MSE. We then suggest a minimax estimator that minimizes the worst-case MSE over all weighting matrices and bounded norm signals subject to the universal weighted MSE domination constraint.