Existence and uniqueness of the maximum likelihood estimator for models with a Kronecker product covariance structure

TL;DR

This paper investigates the existence and uniqueness of the maximum likelihood estimator for multivariate Gaussian models with covariance matrices structured as Kronecker products, highlighting model-dependent differences.

Contribution

It provides theoretical results on the conditions under which the MLE exists and is unique for Kronecker product covariance models.

Findings

Existence of the MLE varies across models

Uniqueness of the MLE depends on model specifics

No explicit formula for the MLE in these models

Abstract

This paper deals with multivariate Gaussian models for which the covariance matrix is a Kronecker product of two matrices. We consider maximum likelihood estimation of the model parameters, in particular of the covariance matrix. There is no explicit expression for the maximum likelihood estimator of a Kronecker product covariance matrix. The main question in this paper is whether the maximum likelihood estimator of the covariance matrix exists and if it is unique. The answers are different for different models that we consider.