This function provides the MLE of the covariance matrix of tensor normal distribution, where the covariance has a separable Kronecker structure, i.e. \(\Sigma=\Sigma_{m}\otimes \ldots \otimes\Sigma_{1}\). The algorithm is a generalization of the MLE algorithm in Manceur, A. M., & Dutilleul, P. (2013).
kroncov(Tn, tol = 1e-06, maxiter = 10)A \(p_1\times\cdots p_m\times n\) matrix, array or tensor, where \(n\) is the sample size.
The convergence tolerance with default value 1e-6. The iteration terminates when \(||\Sigma_i^{(t+1)} - \Sigma_i^{(t)}||_F <\) tol for some covariance matrix \(\Sigma_i\).
The maximal number of iterations. The default value is 10.
The normalizing constant.
A matrix list, consisting of each normalized covariance matrix \(\Sigma_1,\ldots,\Sigma_m\).
The individual component covariance matrices \(\Sigma_i, i=1,\ldots, m\) are not identifiable. To overcome the identifiability issue, each matrix \(\Sigma_i\) is normalized at the end of the iteration such that \(||\Sigma_i||_F = 1\). And an overall normalizing constant \(\lambda\) is extracted so that the overall covariance matrix \(\Sigma\) is defined as $$\Sigma = \lambda \Sigma_m \otimes \cdots \otimes \Sigma_1.$$
If Tn is a \(p \times n\) design matrix for a multivariate random variable, then lambda = 1 and S is a length-one list containing the sample covariance matrix.
Manceur, A.M. and Dutilleul, P., 2013. Maximum likelihood estimation for the tensor normal distribution: Algorithm, minimum sample size, and empirical bias and dispersion. Journal of Computational and Applied Mathematics, 239, pp.37-49.