Softly unbiased estimation, part 1: The Gauss-Markov model Original Research Article

In the Gauss-Markov Model, weighted least-squares adjustment generates the BLUUE (Best Linear Uniformly Unbiased Estimator). Sometimes the requirement of unbiasedness is questioned as to its usefulness. Without it the corresponding principle of minimizing the mean square estimation error among all linear estimates, including the biased ones, leads to the BLE (Best Linear Estimator). Here we present a way to gradually soften the unbiasedness constraint in order to allow a continuous transition from BLUUE to BLE, thereby relying heavily on matrix algebra.