Statistical inference of the mixed linear model with incorrect stochastic linear restrictions

We considered the general mixed linear model $ \mathscr{N} $ subject to two competing stochastic linear restrictions, $ \mathscr{M}_{0} $ and $ \mathscr{M} $, where the restrictions $ \mathscr{M} $ are the correct information whereas restrictions $ \mathscr{M}_{0} $ may be incorrect. Statistical inference conclusions of using the above two competing restrictions are not necessarily the same, so it is prominent to discuss the relationships between incorrect restrictions $ \mathscr{M}_{0} $ and the corresponding correct restrictions $ \mathscr{M} $ in the context of model $ \mathscr{N} $. In this article, we first present some properties on the best linear unbiased predictors (BLUPs) under model $ \mathscr{N} $ with restrictions $ \mathscr{M} $. We then provide necessary and sufficient conditions under which the BLUPs under $ \mathscr{N} $ with the incorrect restrictions $ \mathscr{M}_{0} $ continue to be BLUPs associated with correct restrictions.