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一样，具有Kronecker协方差结构的多元正态模型中的最大似然估计通常通过迭代算法（具体为块坐标上升算法）进行计算。本文重点阐述一种特定情境——由样本量与Kronecker因子的维度满足互质关系所界定——在此情境下，似然方程的代数次数为一，且可求得最大似然估计的显式、易评估的有理公式。本文还给出了该结果的局部逆命题，表明在所述特殊情境之外且样本量较大的情况下，可以构造出似然方程代数次数大于一的实例数据集。