The Fundamental BLUE Equation in Linear Models Revisited

In the world of linear statistical models there is a particular matrix equation, G(X :VX⊥) =(X:0), which is sufficiently important that it is sometimes called the fundamental BLUE equation. In this equation, X is a model matrix, V is the covariance matrix of y in the linear model y = Xβ+ε, and we are interested in finding the best linear estimator, BLUE, of Xβ. Any solution G for this equation has the property that Gy provides a representation for the BLUE of Xβ: this is the message of the the fundamental BLUE equation, whose main developer was the late Professor C. R. Rao in early 1970s. In this article we revisit some interesting features and consequences of this equation. We do not provide essentially new results– the aim is to offer a compact easy-to-follow review including also some recent related results by the authors.