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

We considered the general mixed linear model $\mathcal{N}$ subject to two competing stochastic linear restrictions, ${\mathcal{M}}_0$ and $\mathcal{M}$, where the restrictions $\mathcal{M}$ are the correct information whereas restrictions ${\mathcal{M}}_0$ 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 ${\mathcal{M}}_0$ and the corresponding correct restrictions $\mathcal{M}$ in the context of model $\mathcal{N}$. In this article, we first present some properties on the best linear unbiased predictors (BLUPs) under model $\mathcal{N}$ with restrictions $\mathcal{M}$. We then provide necessary and sufficient conditions under which the BLUPs under $\mathcal{N}$ with the incorrect restrictions ${\mathcal{M}}_0$ continue to be BLUPs associated with correct restrictions.