Journal of Systems Science and Complexity

, Volume 30, Issue 5, pp 1173–1188

A new estimator of covariance matrix via partial Iwasawa coordinates

Article

Abstract

This paper is concerned with the problem of improving the estimator of covariance matrix under Stein’s loss. By the partial Iwasawa coordinates of covariance matrix, the corresponding risk can be split into three parts. One can use the information in the weighted matrix of weighted quadratic loss to improve one part of risk. However, this paper indirectly takes advantage of the information in the sample mean and reuses Iwasawa coordinates to improve the rest of risk. It is worth mentioning that the process above can be repeated. Finally, a Monte Carlo simulation study is carried out to verify the theoretical results.

Keywords

Covariance matrix James–Stein estimator partial Iwasawa coordinates Stein’s loss weighted quadratic loss

References

1. [1]
James W and Stein C, Estimation with quadratic loss, Proceedings of the Fourth Berleley Symposium on Mathematical Statistics and Probability, Vol.I. University of California Press, Berkeley, 1961, 361–379.Google Scholar
2. [2]
Efron B and Morris C, Multivariate empirical Bayes and estimation of covariance matrices, Annals of Statistics, 1976, 4: 22–32.
3. [3]
Daniels M J and Kass R E, Nonconjugate Bayesian estimation of covariance matrices and its use in hierarchical models, Journal American Statistical Association, 1999, 94: 1254–1263.
4. [4]
Yang R and Berger J O, Estimation of a covariance matrix using the reference prior, Annals of Statistics, 1994, 22: 1195–1211.
5. [5]
Kubokawaa T and Srivastava M S, Estimating the covariance matrix: A new approach, Journal of Multivariate Analysis, 2003, 86: 28–47.
6. [6]
Xu K and He D J, Further results on estimation of covariance matrix, Statistics Probability Letters, 2015, 101: 11–20.
7. [7]
Haff L R, Empirical Bayes estimation of the multivariate normal covariance matrix, Annals of Statistics, 1980, 8: 586–597.
8. [8]
Dey D K and Srinivasan C, Estimation of a covariance matrix under Stein’s loss, Annals of Statistics, 1985, 13: 1581–1591.
9. [9]
Anderson T W, An Introduction to Multivariate Statistical Analysis, John Wiley and Sons, Hoboken, New Jersey, 2003.
10. [10]
Konno Y, Estimation of a normal covariance matrix with incomplete data under Stein’s loss, Journal of Multivariate Analysis, 1995, 52: 308–324.
11. [11]
Ma T F, Jia L J, and Su Y S, A new estimator of covariance matrix, Journal of Statistical Planning and Inference, 2012, 142: 529–536.
12. [12]
Bilodeau M and Kariya T, Minimax estimators in the normal MANOVA model, Journal of Multivariate Analysis, 1989, 28: 260–270.
13. [13]
Tsukuma H and Kubokawa T, Methods for improvement in estimation of a normal mean matrix, Journal of Multivariate Analysis, 2007, 98: 1592–1610.
14. [14]
Casella G and Berger R L, Statistical Inference, 2nd Edition, Thomson Learning, 2002.
15. [15]
Muirhead R J, Aspects of Multivariate Theory, John Wiley and Sons, New York, 1982.