Electronic Journal of Statistics

Shrinkage estimation with a matrix loss function

Reman Abu-Shanab, John T. Kent, and William E. Strawderman

Full-text: Open access


Consider estimating an $n\times p$ matrix of means $\Theta$, say, from an $n\times p$ matrix of observations $X$, where the elements of $X$ are assumed to be independently normally distributed with $E(x_{ij})=\theta_{ij}$ and constant variance, and where the performance of an estimator is judged using a $p\times p$ matrix quadratic error loss function. A matrix version of the James-Stein estimator is proposed, depending on a tuning constant $a$. It is shown to dominate the usual maximum likelihood estimator for some choices of $a$ when $n\geq 3$. This result also extends to other shrinkage estimators and settings.

Article information

Electron. J. Statist., Volume 6 (2012), 2347-2355.

First available in Project Euclid: 14 December 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C99: None of the above, but in this section
Secondary: 62H12: Estimation

James-Stein estimator matrix quadratic loss function risk Stein’s Lemma


Abu-Shanab, Reman; Kent, John T.; Strawderman, William E. Shrinkage estimation with a matrix loss function. Electron. J. Statist. 6 (2012), 2347--2355. doi:10.1214/12-EJS748. https://projecteuclid.org/euclid.ejs/1355495988

Export citation


  • Baranchik, A. J. (1970). A family of minimax estimators of the mean of a multivariate normal distribution., Ann. Math. Statist. 41 642–645.
  • Brandwein, A. C. and Strawderman, W. E. (1991). Generalizations of James-Stein estimators under spherical symmetry., Ann. Statist. 19 1639–1650.
  • Cellier, D. and Fourdrinier, D. (1995). Shrinkage estimators under spherical symmetry for the general linear model., J. Multivariate Anal. 52 338–351.
  • Efron, B. and Morris, C. (1972). Empirical Bayes on vector observations: an extension of Stein’s method., Biometrika 59 335–347.
  • Efron, B. and Morris, C. (1976). Multivariate empirical Bayes and estimation of covariance matrices., Ann. Statist. 4 22–32.
  • Fourdrinier, D., Strawderman, W. E. and Wells, M. T. (2003). Robust shrinkage estimation for elliptically symmetric distributions with unknown covariance matrix., J. Multivariate Anal. 85 24–39.
  • Fourdrinier, D., Strawderman, W. E. and Wells, M. T. (2006). Estimation of a location parameter with restrictions of “vague information” for spherically symmetric distributions., Ann. Inst. Statist. Math. 58 73–92.
  • Ghosh, M. and Shieh, G. (1991). Empirical Bayes minimax estimators of matrix normal means., J. Multivariate Anal. 38 306–318.
  • Haff, L. R. (1977). Minimax estimators for a multinormal precision matrix., J. Multivariate Anal. 7 374–385.
  • James, W. and Stein, C. (1961). Estimation with quadratic loss. In, Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I 361–379. Univ. California Press, Berkeley, Calif.
  • Shinozaki, N. (1984). Simultaneous estimation of location parameters under quadratic loss., Ann. Statist. 12 322–335.
  • Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. I 197–206. University of California Press, Berkeley and Los Angeles.
  • Stein, C. M. (1981). Estimation of the mean of a multivariate normal distribution., Ann. Statist. 9 1135–1151. http://links.jstor.org/sici?sici=0090-5364(198111)9:6<1135:EOTMOA>2.0.CO;2-5&origin=MSN
  • Tsukuma, H. (2009). Generalized Bayes minimax estimation of the normal mean matrix with unknown covariance matrix., J. Multivariate Anal. 100 2296–2304.
  • Tsukuma, H. and Kubokawa, T. (2007). Methods for improvement in estimation of a normal mean matrix., J. Multivariate Anal. 98 1592–1610.
  • Zheng, Z. (1986). On estimation of matrix of normal mean., J. Multivariate Anal. 18 70–82.