On relations between BLUEs under two transformed linear models

For a given general linear model ℳ={y,Xβ,Σ}, we investigate relationships between the best linear unbiased estimations (BLUEs) under its two transformed models ℳ1={Ay,AXβ,AΣA′} and ℳ2={By,BXβ,BΣB′}. We first establish some expansion formulas for calculating the ranks and inertias of the covariance matrices of BLUEs and their operations under ℳ1ℳ1 and ℳ2ℳ2. We then derive from the rank and inertia formulas necessary and sufficient conditions for equalities and inequalities of BLUEs’ covariance matrices to hold. We also give applications of the rank and inertia formulas to two sub-sample models of ℳℳ.