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Mathematics and Statistics


Severien Nkurunziza



Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.


In this thesis, we consider the estimation problem of the mean matrix of a multivariate normal distribution in high-dimensional data. Building upon the groundwork laid by Chételat and Wells (2012), we extend their method to the cases where the parameter is the mean matrix of a matrix normal distribution. In particular, we propose a novel class of James-Stein’s estimators for the mean matrix of a multivariate normal distribution with an unknown row covariance matrix and independent columns. Given a realistic assumption, we establish that our proposed estimator outperforms the classical maximum likelihood estimator (MLE) in the context of high-dimensional data. Furthermore, we investigate the conditions for which this assumption remains valid. Additionally, we identify and rectify a notable error in the proofs of a crucial result presented in Chételat and Wells (2012). Notably, the novelty of the obtained results lies in the fact that the estimator for the row covariance matrix is singular almost surely and its rank is a random variable. Finally, we present simulation results that confirm the validity of our theoretical findings.

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