kroncov {TRES}

The covariance estimation of tensor normal distribution

Description

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).

Usage

kroncov(Tn, tol = 1e-06, maxiter = 10)

Arguments

Tn

A p_1\times\cdots p_m\times n matrix, array or tensor, where n is the sample size.

tol

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.

maxiter

The maximal number of iterations. The default value is 10.

Details

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.

Value

lambda

The normalizing constant.

S

A matrix list, consisting of each normalized covariance matrix \Sigma_1,\ldots,\Sigma_m.

References

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.