Equality of BLUEs or BLUPs under two linear models using stochastic restrictions

In this paper, we consider mixed linear models, possibly with singular covariance matrices, by supplementing a particular fixed effects model with appropriate stochastic restrictions. We show that all representations of the best linear unbiased estimator (BLUE) and best linear unbiased predictor (BLUP) can be obtained through the augmented model including stochastic restrictions. Using this approach, we consider two mixed linear models, M1 and M2, say, which have different covariance matrices. We give necessary and sufficient conditions that the BLUP and/or BLUE under the the model M1 continue to be BLUP and/or BLUE also under the model M2.