Minimax regret estimation in linear models

We develop a new linear estimator for estimating an unknown vector x in a linear model, in the presence of bounded data uncertainties. The estimator is designed to minimize the worst-case regret across all bounded data vectors, namely the worst-case difference between the MSE attainable using a linear estimator that does not know the true parameters x, and the optimal MSE attained using a linear estimator that knows x. We demonstrate through several examples that the minimax regret estimator can significantly increase the performance over the conventional least-squares estimator, as well as several other least-squares alternatives.