In 1971, in his seminal paper entitled Unified theory of linear estimation, C.R. Rao considered the properties of best linear unbiased estimators, BLUEs, in the general linear model \( {\mathscr {M}}(\textbf{V}) = \{ \textbf{y}, \textbf{X}{\varvec{\beta }}, \textbf{V}\}\) , where \(\textbf{V}\) refers to the covariance matrix of the observable random vector \(\textbf{y}\) and \(\textbf{X}\) is the model matrix. Both \(\textbf{X}\) and the nonnegative definite covariance matrix \(\textbf{V}\) are known. Citing Rao, “In Section 5 [of his paper] we raise the question of identification of \(\textbf{V}\) given the class of BLUE’s of all estimable functions”. It is precisely Section 5 of Rao’s paper which is in our focus. In particular, we will take a good look at Rao’s Theorems 5.2 and 5.3 which answer the following question: Given the model \({\mathscr {M}}(\textbf{V}_{0}) = \{ \textbf{y}, \textbf{X}{\varvec{\beta }}, \textbf{V}_{0} \}\) , how to characterize the set of all covariance matrices \(\textbf{V}\) such that every representation of the BLUE of \(\textbf{X}{\varvec{\beta }}\) under \({\mathscr {M}}(\textbf{V}_{0})\) remains BLUE under \({\mathscr {M}}(\textbf{V})\) . Our attempt is to provide some new insight into this problem area.
