### Minimax Estimation of the Mean Matrix of the Matrix Variate Normal Distribution under the Divergence Loss Function

#### Abstract

The problem of estimating the mean matrix of a matrix-variate normal distribution with a covariance matrix is considered under two loss functions. We construct a class of empirical Bayes estimators which are better than the maximum likelihood estimator under the first loss function and hence show that the maximum likelihood estimator is inadmissible. We find a general class of minimax estimators. Also we give a class of estimators that improve on the maximum likelihood estimator under the second loss function and hence show that the maximum likelihood estimator is inadmissible.

#### Keywords

#### Full Text:

PDF (English)#### References

S. AMARI (1982). Differential geometry of curved exponential families-curvatures and information loss. Annals of Statistics, 10, pp. 357–387.

M. BILODEAU, T. KARIYA (1989). Minimax estimators in the normal manova model. Journal of Multivariate Analysis, 28, pp. 260–270.

C. R. BLYTH (1951). On minimax statistical decision procedures and their admissibility. Annals of Mathematical Statistics, 22, pp. 22–42.

B. CHEN, S. H. LIN (2012). A risk-aware modeling framework for speech summarization. IEEE Transactions on Audio, Speech, and Language Processing, 20, pp. 211–222.

M. CORDUAS (1985). On the divergence between linear processes. Statistica, 45, pp. 393–401.

N. CRESSIE, T. R. C. READ (1984). Multinomial goodness-of-fit tests. Journal of the Royal Statistical Society Series B, 46, pp. 440–464.

B. EFRON, C. MORRIS (1972). Empirical Bayes on vector observations: An extension of Stein’s method. Biometrika, 59, pp. 335–347.

M. GHOSH, V. MERGEL (2009). On the stein phenomenon under divergence loss and an unknown variance-covariance matrix. Journal of Multivariate Analysis, 100, pp. 2331–2336.

M. GHOSH, V. MERGEL, G. S. DATTA (2008). Estimation, prediction and the stein phenomenon under divergence loss. Journal of Multivariate Analysis, 99, pp. 1941–1961.

M. GHOSH, G. SHIEH (1991). Empirical bayes minimax estimators of matrix normal means. Journal of Multivariate Analysis, 38, pp. 306–318.

S. JI, W. ZHANG, R. LI (2013). A probabilistic latent semantic analysis model for coclustering the mouse brain atlas. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 10, pp. 1460–1468.

R. L. KASHYAP (1974). Minimax estimation with divergence loss function. Information Sciences, 7, pp. 341–364.

Y. KONNO (1990). Families of minimax estimators of matrix of normal means with unknown covariance matrix. Journal of the Japan Statistical Society, 20, pp. 191–201.

Y. KONNO (1991). On estimation of a matrix of normal means with unknown covariance matrix. Journal of Multivariate Analysis, 36, pp. 44–55.

Y. KONNO (1992). Improved estimation of matrix of normal mean and eigenvalues in the multivariate F -distribution. Ph.D. thesis, Institute of Mathematics, University of Tsukuba.

E. L. LEHMANN, G. CASELLA (1998). Theory of Point Estimation. Springer Verlag, New York, 2 ed.

J. MALKIN, X. LI, S. HARADA, J. LANDAY, J. BILMES (2011). The vocal joystick engine v1.0. Computer Speech and Language, 25, pp. 535–555.

J. PAUL, P. Y. THOMAS (2016). Sharma-mittal entropy properties on record values. Statistica, 76, pp. 273–287.

C. P. ROBERT (2001). The Bayesian Choice, second edition. Springer Verlag, New York.

G. SHIEH (1993). Empirical Bayes minimax estimators of matrix normal means for arbitrary quadratic loss and unknown covariance matrix. Statistics and Decisions, 11, pp. 317–341.

C. STEIN (1973). Estimation of the mean of a multivariate normal distribution. In J.H. AJEK (ed.), Proceedings of the Prague Symposium on Asymptotic statistics. Universita Karlova, Prague, pp. 345–381.

H. TSUKUMA (2008). Admissibility and minimaxity of Bayes estimators for a normal mean matrix. Journal of Multivariate Analysis, 99, pp. 2251–2264.

H. TSUKUMA (2010). Proper Bayes minimax estimators of the normal mean matrix with common unknown variances. Journal of Statistical Planning and Inference, 140, pp. 2596–2606.

L. YANG, B. GENG, A. HANJALIC, X. S. HUA (2012). A unified context model for web image retrieval. ACM Transactions on Multimedia Computing, Communications, and Applications, 8. Article No. 28.

Z. ZHANG (1986a). On estimation of matrix of normal mean. Journal of Multivariate Analysis, 18, pp. 70–82.

Z. ZHANG (1986b). Selecting a minimax estimator doing well, at a point. Journal of Multivariate Analysis, 19, pp. 14–23.

S. ZINODINY, S. REZAEI, S. NADARAJAH (2017). Bayes minimax estimation of the mean matrix of matrix-variate normal distribution under balanced loss function. Statistics and Probability Letters, 125, pp. 110–120.

DOI: 10.6092/issn.1973-2201/6956