Estimation equations for multivariate linear models with Kronecker structured covariance matrices

The aim of the paper is to determine maximum-likelihood estimation equations. Observations follow a multivariate normal distribution, Xi∼Np,q(μ,Ψ,Σ)$\mbox{\boldmath $ {\bf X}$}_i \sim N_{p,q}(\mbox{\boldmath $ {\bf \mu }$},\mbox{\boldmath $ {\bf \Psi }$},\mbox{\boldmath $ {\bf \Sigma }$})$, where D[Xi]=Σ⊗Ψ$D[\mbox{\boldmath $ {\bf X}$}_i]= \mbox{\boldmath $ {\bf \Sigma }$}\otimes \mbox{\boldmath $ {\bf \Psi }$}$, Ψ$\mbox{\boldmath $ {\bf \Psi }$}$ and Σ$\mbox{\boldmath $ {\bf \Sigma }$}$ describe the unknown covariance structure between rows and columns of Xi$\mbox{\boldmath $ {\bf X}$}_i$, respectively. Imposing restrictions on Ψ$\mbox{\boldmath $ {\bf \Psi }$}$ and Σ$\mbox{\boldmath $ {\bf \Sigma }$}$ four types of covariance structures will be considered.