Models with a Kronecker Product Covariance Structure: Estimation and Testing

In this article we consider a pq-dimensional random vector x distributed normally with mean vector θ and the covariance matrix Λ, assumed to be positive definite. On the basis of N independent observations on the random vector x, we wish to estimate parameters and test the hypothesis H: Λ = Ψ ⊗Σ, where Ψ = (ψij) : q × q and Σ = (σij) : p × p, and Λ = (ψijΣ), the Kronecker product of Ψ and Σ. That is instead of 12pq(pq+1) parameters, it has only 12p(p + 1) + 1 2q(q + 1) − 1 parameters. When this model holds, we test the hypothesis that Ψ is an identity matrix, a diagonal matrix or of intraclass correlation structure. The maximum likelihood estimators (MLE) are obtained under the hypothesis as well as under the alternatives. Using these estimators the likelihood ratio tests (LRT) are obtained. Moreover, it is shown that the estimators are unique.