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We treat the problem of evaluating the performance of linear estimators for estimating a deterministic parameter vector x in a linear regression model, with the mean-squared error (MSE) as the performance measure. Since the MSE depends on the unknown vector x, a direct comparison between estimators is a difficult problem. Here, we consider a framework for examining the MSE of different linear estimation approaches based on the concepts of admissible and dominating estimators. We develop a general procedure for determining whether or not a linear estimator is MSE admissible, and for constructing an estimator strictly dominating a given inadmissible method so that its MSE is smaller for all x. In particular, we show that both problems can be addressed in a unified manner for arbitrary constraint sets on x by considering a certain convex optimization problem. We then demonstrate the details of our method for the case in which x is constrained to an ellipsoidal set and for unrestricted choices of x. As a by-product of our results, we derive a closed-form solution for the minimax MSE estimator on an ellipsoid, which is valid for arbitrary model parameters, as long as the signal-to-noise-ratio exceeds a certain threshold.