A note on necessary and sufficient conditions of existence and uniqueness for the maximum likelihood estimator of a Kronecker-product variance–covariance matrix

다국어 초록 (Multilingual Abstract)

In this note, the author discusses the problem of maximum likelihood (ML) estimation for the Kronecker product of two positive definite variance–covariance matrices, with focus on necessary and sufficient conditions of existence and uniqueness for the ML estimator. Recent and earlier references, from the author and others, are scrutinized. The discussion includes computational aspects, as an estimation algorithm is used in the absence of analytical solutions. A timely update is provided as a result.

참고문헌 (Reference)

1 Manceur, A. M., "“sEparaTe”, Version 0.2.1: Maximum likelihood estimation and likelihood ratio test functions for separable variance–covariance structures. R package, The Comprehensive R Archive Network (CRAN)"

2 Lu, N., "The likelihood ratio test for a separable covariance matrix" 73 : 449-457, 2005

3 Dutilleul, P., "The MLE algorithm for the matrix normal distribution" 64 : 105-123, 1999

4 Filipiak, K., "Score test for a separable covariance structure with the first component as compound symmetric correlation matrix" 150 : 105-124, 2016

5 Fuhrmann, D., "On the existence of positive-definite maximum-likelihood estimates of structured covariance matrices" 34 : 722-729, 1988

6 Mardia, K. V., "Multivariate environmental statistics), Series in Probability and Statistics" North-Holland 347-385, 1993

7 Srivastava, M., "Models with a Kronecker product covariance structure : estimation and testing" 17 : 357-370, 2008

8 Manceur, A. M., "Maximum likelihood estimation for the tensor normal distribution : algorithm, minimum sample size, and empirical bias and dispersion" 239 : 37-49, 2013

9 Graybill, F. A., "Matrices with applications in statistics" Wadsworth 1983

10 Soloveychik, I., "Gaussian and robust Kronecker product covariance estimation : existence and uniqueness" 149 : 92-113, 2016

11 Roś, B., "Existence and uniqueness of the maximum likelihood estimator for models with a Kronecker product covariance structure" 143 : 345-361, 2016

12 Burg, J. P., "Estimation of structured covariance matrices" 70 : 963-974, 1982

13 Dutilleul, P., "Estimation and testing for separable variance-covariance structures" 10 (10): 2018

14 de Munck, J. C., "Estimating stationary dipoles from MEG/EEG data contaminated with spatially and temporally correlated background noise" 50 : 1565-1572, 2002

15 Lee, C. H., "Comment on"Models with a Kronecker product covariance structure : estimation and testing"by M. S. Srivastava, T. von Rosen and D. von Rosen" 19 : 88-90, 2010

16 Leiva, R., "Classification of higher-order data with separable covariance and structured multiplicative or additive mean models" 43 : 989-1012, 2014

17 Dutilleul, P., "Apport en Analyse Spectrale d’un Périodogramme Modifié et Modélisation des Séries Chronologiques avec Répétitions en vue de leur Comparaison en Fréquence" Université catholique de Louvain 1990

18 Filipiak, K., "A comparison of likelihood ratio tests and Rao’s score test for three separable covariance matrix structures" 59 : 192-215, 2017