A note on a previous lemma in the theory of least squares and some further results

Let Y be a vector of random variables such that E(Y)=Xβ where β is a vector of unknown parameters and Σ be the covariance matrix of Y. A linear function L'Y is said to be best linear unbiased estimator (BLUE) of a parametric function p'β with respect to Σ if L'ΣL is a minimum subject to p'=L'X. The paper deals with necessary and sufficient conditions that, for every estimable parametric function or for a given subset, the BLUE with respect to Σ is the same as the BLUE with respect to Σ =I (identity matrix) or the same as the BLUE with respect to Σ=Σ0 (a given matrix). Let Z be a matrix of maximum rank such that X'Z=0. It is shown that when Σ=Σ0 is non-singular, or rank (X:Z) =rank (X:Σ0Z), then a NAS condition for the equality of BLUE's of all estimable functions for Σ and Σ0 is that Σ is of the form Σ=XθX'+Σ0ZΓZ'Σ0+Σ0 where θ, Γ are arbitrary. The representations of Σ in other situations where Σ0 is singular have also been obtained