Minimum Mean Square Error Adjustment, Part 1:The Empirical BLE and the repro-BLE for Direct Observations

Abstract

It has long been argued that Minimum Mean Square Error Estimation , although theoreti cally superior to the least-squares adjustment, is impractical in the absence of any prior infor mation on the unknown parameters. The Empirical BLE therefore applies another estimate from the same dataset, e.g. the BLUUE (Best Linear Uniformly Unbiased Estimate) or the ridge estimate, in order to overcome this problem. Here, we introduce the repro-BLE (Best Linear Estimate with the reproducing property) which if it exists belongs to the same class of (nonlinear) estimates, but with the provision that the vector used to form the empirical mean square error risk coincides with the eventual estimate , thus fulfilling the "reproducing property". A few elementary examples for the case of direct observations clarify this approach and may help to understand the behavior of repro-BLE in comparison to the more commonly used Empirical BLE, or to the BLUUE that is generated by a (weighted) least-squares adjustment . The more general Gauss-Markov model will be treated in a second part .