Reducing the gap between linear biased classical and linear Bayesian estimation

In classical estimation usually unbiased estimators are used. This is mainly because the bias term in classical biased estimators in general depends on the parameter to be estimated. However, recently a considerable amount of research has been spent on improving unbiased estimators by introducing a bias, e.g. based on a minimax optimization strategy. In this work we follow this idea of introducing a bias, but we describe a different strategy for optimizing the estimators' performance. Although we stick to classical estimation, we show that the Bayesian linear minimum mean square error estimator can be brought into the same algebraic form as the resulting biased estimator improving the best linear unbiased estimator. This not only emphasizes the fact that this approach leads to betters estimators than the minimax approach on average over all parameters, but also can be seen as another way of reducing the gap between classical and Bayesian estimation.