Properties of BLUEs and BLUPs in Full vs. Small Linear Models with New Observations

Abstract

In this article we consider the partitioned linear model $\mathcal {M}_{12} = \{ \mathbf {y}, \, {\mathbf {X}}_{1}\boldsymbol {\beta }_{1} + {\mathbf {X}}_{2}\boldsymbol {\beta }_{2}, \, \mathbf {V} \}$, where μ = X ₁β ₁ + X ₂β ₂, and the corresponding small model $\mathcal {M}_{1} = \{ \mathbf {y}, \, {\mathbf {X}}_{1} \boldsymbol {\beta }_{1}, \, \mathbf {V} \}$, where μ ₁ = X ₁β ₁. These models are supplemented with the new unobservable random vector y _∗, coming from y _∗ = Kβ ₁ + ε _∗, where the covariance matrix of y _∗ is known as well as the cross-covariance matrix between y _∗ and y. We focus on comparing the BLUEs of μ ₁ and μ, and BLUPs of y _∗ and ε _∗ under $\mathcal {M}_{12}$ and $\mathcal {M}_{1}$.