  Inference about dependencies in a multiway data array can be made using the
array normal model, which corresponds to the class of multivariate normal
distributions with separable covariance matrices. Maximum likelihood and
Bayesian methods for inference in the array normal model have appeared in the
literature, but there have not been any results concerning the optimality
properties of such estimators. In this article, we obtain results for the array
normal model that are analogous to some classical results concerning covariance
estimation for the multivariate normal model. We show that under a lower
triangular product group, a uniformly minimum risk equivariant estimator
(UMREE) can be obtained via a generalized Bayes procedure. Although this UMREE
is minimax and dominates the MLE, it can be improved upon via an orthogonally
equivariant modification. Numerical comparisons of the risks of these
estimators show that the equivariant estimators can have substantially lower
risks than the MLE.
