New results about the relationship between optimally weighted least squares estimate and linear minimum variance estimate

The optimally weighted least squares estimate and the linear minimum variance estimate are two of the most popular estimation methods for a linear model. In this paper, the authors make a comprehensive discussion about the relationship between the two estimates. Firstly, the authors consider the classical linear model in which the coefficient matrix of the linear model is deterministic, and the necessary and sufficient condition for equivalence of the two estimates is derived. Moreover, under certain conditions on variance matrix invertibility, the two estimates can be identical provided that they use the same a priori information of the parameter being estimated. Secondly, the authors consider the linear model with random coefficient matrix which is called the extended linear model; under certain conditions on variance matrix invertibility, it is proved that the former outperforms the latter when using the same a priori information of the parameter.