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We consider the linear regression problem of estimating an unknown, deterministic parameter vector based on measurements corrupted by colored Gaussian noise. We present and analyze blind minimax estimators (BMEs), which consist of a bounded parameter set minimax estimator, whose parameter set is itself estimated from measurements. Thus, our approach does not require any prior assumption or knowledge, and the proposed estimator can be applied to any linear regression problem. We demonstrate analytically that the BMEs strictly dominate the least-squares (LS) estimator, i.e., they achieve lower mean-squared error (MSE) for any value of the parameter vector. Both Stein's estimator and its positive-part correction can be derived within the blind minimax framework. Furthermore, our approach can be readily extended to a wider class of estimation problems than Stein's estimator, which is defined only for white noise and nontransformed measurements. We show through simulations that the BMEs generally outperform previous extensions of Stein's technique.