Some equalities for estimations of variance components in a general linear model and its restricted and transformed models

For the unknown positive parameter σ 2 in a general linear model ℳ = { y , X β , σ 2 Σ } , the two commonly used estimations are the simple estimator (SE) and the minimum norm quadratic unbiased estimator (MINQUE). In this paper, we derive necessary and sufficient conditions for the equivalence of the SEs and MINQUEs of the variance component σ 2 in the original model ℳ , the restricted model ℳ r = { y , X β ∣ A β = b , σ 2 Σ } , the transformed model ℳ t = { A y , A X β , σ 2 A Σ A ′ } , and the misspecified model ℳ m = { y , X 0 β , σ 2 Σ 0 } .