Title: Existence and uniqueness of maximum likelihood estimators of Kronecker product covariances
Authors: Mathias Drton - University of Copenhagen (Denmark)
Satoshi Kuriki - The Institute of Statistical Mathematics (Japan) [presenting]
Abstract: Consider iid samples from a vector-valued Gaussian distribution. When the covariance matrix of the Gaussian distribution has no structure, the MLE uniquely exists with probability one if and only the sample size $n$ is equal to or greater than the size of the covariance matrix. However, this is not the case where the covariance matrix has a structure and is described with a fewer number of parameters. We consider the case where the covariance matrix is the Kronecker product of two matrices ($m_1\times m_1$ and $m_2\times m_2$ matrices). We show that the existence and the uniqueness of the MLE are characterized by a rank of an $m_1\times m_2\times n$ tensor. In particular, when $n=2$, the problem can be reduced by Kronecker's canonical form of two matrices. The tensor rank is given explicitly as the solution of an integer programming. The Groebner basis computation is also useful when $m_1$ and $m_2$ are small.