As is the case for many curved exponential families, the computation of maximum likelihood estimates in a multivariate normal model with a Kronecker covariance structure is typically carried out with an iterative algorithm, specifically, a block-coordinate ascent algorithm. In this article we highlight a setting, specified by a coprime relationship between the sample size and dimension of the Kronecker factors, where the likelihood equations have algebraic degree one and an explicit, easy-to-evaluate rational formula for the maximum likelihood estimator can be found. A partial converse of this result is provided that shows that outside of the aforementioned special setting and for large sample sizes, examples of data sets can be constructed for which the degree of the likelihood equations is larger than one.

翻译：与许多弯曲指数族的情况类似，具有Kronecker协方差结构的多元正态模型中极大似然估计的计算通常通过迭代算法实现，特别是块坐标上升算法。本文重点探讨一种特定设置——由样本量与Kronecker因子维度之间的互质关系所定义——在该设置下似然方程具有一次代数度，且可得到显式、易于计算的有理形式的极大似然估计量表达式。本文进一步提供了该结果的部分逆命题，表明在上述特殊设置之外且当样本量较大时，可以构造出使似然方程次数大于一的数据集实例。