On the equality of the BLUPs under two linear mixed models

Abstract

In this paper we consider two mixed linear models, $${\fancyscript{M}_1}$$ and $${\fancyscript{M}_2}$$ , say, which have different covariance matrices. We review some useful concepts and results on the best linear unbiased estimators (BLUEs) and on best linear unbiased predictors (BLUPs). We give new necessary and sufficient conditions, without making any rank assumptions, that every representation of the BLUP of the random effect under the model $${\fancyscript{M}_1}$$ continues to be BLUP under the model $${\fancyscript{M}_2}$$ . These considerations are generalized to two linear models with new unobserved future observations.