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.

中文翻译：

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阵列正态模型中 MLE 的等变极小极大支配者

可以使用数组正态模型来推断多路数据数组中的相关性，该模型对应于具有可分离协方差矩阵的多元正态分布类。在阵列正态模型中进行推理的最大似然和贝叶斯方法已经出现在文献中，但没有关于此类估计量的最优性的任何结果。在本文中，我们获得了数组正态模型的结果，这些结果类似于有关多元正态模型协方差估计的一些经典结果。我们表明，在下三角乘积组下，可以通过广义贝叶斯过程获得一致的最小风险等变估计量 (UMREE)。虽然这个 UMREE 是 minimax 并且在 MLE 中占主导地位，可以通过正交等变修改对其进行改进。这些估计量的风险的数值比较表明，等变估计量的风险比 MLE 低得多。