Orthogonal Block Structure and Uniformly Best Linear Unbiased Estimators

Abstract

Models with orthogonal block structure, OBS, have variance covariance matrices that are linear combinations $\sum _{j=1}^{m} \gamma _{j} Q_{j}$ of known pairwise orthogonal–orthogonal projection matrices that add up to I _n. We are interested in characterizing such models with least square estimators that are best linear unbiased estimator whatever the variance components, assuming that γ ∈ ∇_≥, with ∇_≥ the set of vectors with nonnegative components of a subspace ∇. This is an extension of the usual concept of OBS in which we require $\boldsymbol {\gamma } \in \mathbb {R}^{m}_{\geq }.$ Thus as we shall see it is usual when we apply our results to mixed models.