Inference about dependencies in a multiway data array can be made using thearray normal model, which corresponds to the class of multivariate normaldistributions with separable covariance matrices. Maximum likelihood andBayesian methods for inference in the array normal model have appeared in theliterature, but there have not been any results concerning the optimalityproperties of such estimators. In this article, we obtain results for the arraynormal model that are analogous to some classical results concerning covarianceestimation for the multivariate normal model. We show that under a lowertriangular product group, a uniformly minimum risk equivariant estimator(UMREE) can be obtained via a generalized Bayes procedure. Although this UMREEis minimax and dominates the MLE, it can be improved upon via an orthogonallyequivariant modification. Numerical comparisons of the risks of theseestimators show that the equivariant estimators can have substantially lowerrisks than the MLE.
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