Relevant statistics for models with commutative orthogonal block structure and unbiased estimator for variance components

Abstract

A model has orthogonal block structure, OBS, when it’s covariance matrix is a linear combination of pairwise orthogonal projection matrices P 1, … , Pl . If one of this models has mean vector μ n = ΔT v and the P 1, … , P l commute with the orthogonal projection matrix on the range space of ΔT, the model will have commutative orthogonal block structure, COBS. We obtain relevant statistics for these models as well as BLUE for estimable vectors and unbiased estimators for variance components. When we assume the normality, we show that these statistics will be sufficient and completes. These statistics will be used to get confidence regions and tests of hypotheses for variance components and estimable vectors. We will also show how to induce probabilities measures in parametric spaces and how to adjust confidence ellipsoids for estimable vectors. Lastly we characterize mixed models with COBS.