Estimation in Models with Commutative Orthogonal Block Structure

A model with variance-covariance matrix V = Sigma(v)(i=1)sigma(2)(i) P-i, where P-1,...,P-v are known pairwise orthogonal orthogonal projection matrices, will have Orthogonal Block Structure with variance components sigma(2)(1),...,sigma(2)(v). Moreover, if matrices P-1,...,P-v commute with the orthogonal projection matrix T on the space spanned by the mean vector, the model will have Commutative Orthogonal Block Structure (COBS). In this paper we will use Commutative Jordan Algebras to study the algebraic properties of these models as well as optimal estimators. We show that once normality is assumed, sufficient complete statistics are obtained and estimators are Uniformly Minimum Variance Unbiased Estimators.