Models with a Kronecker Product Covariance Structure: Estimation and Testing

In this article we consider a pq-dimensional random vector (x) over bar distributed normally with mean vector (theta) over bar and covariance matrix Lambda assumed to be positive definite. On the basis of N independent observations on the random vector (x) over bar, we want to estimate parameters and test the hypothesis H : Lambda = Psi circle times Sigma where Psi = (Psi(ij)): q x q, Psi(qq:) = 1, and > p x p, and Lambda = (Psi(ij) Sigma), the Kronecker product of Psi and Sigma. That is instead of 1/2pq(pq + 1) parameters, it has only,1/2p(p + 1) + 1/2q(q + 1) - 1 parameters. A test based on the likelihood ratio is given to check if this model holds. intraclass correlation And, when this model holds, we test the hypothesis that klf is a matrixwith structure. The maximum likelihood estimators (MLE) are obtained under the hypothesis as well as tinder the alternatives. Using these estimators the likelihood ratio tests (Lta) are obtained. One of the main objects of the paper is to show that the likelihood equations provide unique estimators.