Equality of the BLUPs under the mixed linear model when random components and errors are correlated

We consider a general mixed linear model ℳ without any rank assumptions to the covariance matrix and without any restrictions on the correlation between the random effects vector and the random errors vector. We get the representations of best linear unbiased estimators (BLUEs)/ best linear unbiased predictors (BLUPs) of ℳ through a particular construction from the model ℳ which uses stochastic restriction. For the general mixed linear models ℳ 1 and ℳ 2 , which have different covariance matrices, we derive the necessary and sufficient conditions for that the BLUEs and/or BLUPs under ℳ 1 continue to be the BLUEs and/or BLUPs under the ℳ 2 . And we also give the necessary and sufficient conditions for the equivalence of BLUP under ℳ 1 and ℳ 2 .