Joining Iso-Structured Models with Commutative Orthogonal Block Structure

Abstract

In this work, we focus on a special class of mixed models, named models with commutative orthogonal block structure (COBS), whose covariance matrix is a linear combination of known pairwise orthogonal projection matrices that add to the identity matrix, and for which the orthogonal projection matrix on the space spanned by the mean vector commutes with the covariance matrix. The COBS have least squares estimators giving the best linear unbiased estimators for estimable vectors. Our approach to COBS relies on their algebraic structure, based on commutative Jordan algebras of symmetric matrices, which proves to be advantageous as it leads to important results in the estimation. Specifically, we are interested in iso-structured COBS, applying to them the operation of models joining. We show that joining iso-structured COBS gives COBS and that the estimators for the joint model may be obtained from those for the individual models.