The Gauss-Markov Theorem: Beyond the BLUE

Until now the Gauss-Markov theorem has been the handmaid of least squares; it has served as a proof that the least-squares method produces the Best Linear Unbiased Estimator (BLUE). This theoretical paper shows that it can be, and should be, reformulated as the solution to the problem of the minimization of a quadratic form subject to a linear constraint. The whole theory of linear statistical modeling, from basic to complicated, receives a clean and efficient development on the basis of this reformulation; estimates and predictions based thereon are BLUE from the start, rather than BLUE by subsequent proof. With an intermediate-level background in matrix algebra the reader will understand the frequent interpretations of this development in terms of an n-dimensional projective geometry. Because this paper elevates BLUE to its true role, “Beyond the BLUE” really means “To the True BLUE.”

Keywords: Gauss-Markov, BLUE, linear model, projection, distance metric