Existence and uniqueness of the Kronecker covariance MLE

In matrix-valued datasets the sampled matrices often exhibit correlations among both their rows and columns. A useful parsimonious model of such dependence is matrix normal model, in which covariances elements a random are parameterized terms Kronecker product two covariance matrices, one representing row column covariance. An appealing feature that structure allows for standard likelihood inference even when only very small number data available. For instance, some cases ratio test may be performed with sample size one. However, more generally required to ensure boundedness or existence unique maximizer depends complicated way on dimensions. This motivates study how large needed maximum estimators exist, exist uniquely probability Our main result gives precise thresholds paradigm where columns differ by at most factor two. proof uses invariance properties allow us consider canonical form, as obtained from form pencils.