Journal of the Korean Data and Information Science Society

The Korean Data and Information Science Society (한국데이터정보과학회)
권호 표지
DOI QR코드

DOI QR Code

An improvement of estimators for the multinormal mean vector with the known norm

  • Kim, Jaehyun (Department of Computer Engineering, Seokyeong University) ;
  • Baek, Hoh Yoo (Division of Mathematics and Informational Statistics, Wonkwang University)
  • Received : 2017.01.04
  • Accepted : 2017.03.20
  • Published : 2017.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

Consider the problem of estimating a $p{\times}1$ mean vector ${\theta}$ (p ${\geq}$ 3) under the quadratic loss from multi-variate normal population. We find a James-Stein type estimator which shrinks towards the projection vectors when the underlying distribution is that of a variance mixture of normals. In this case, the norm ${\parallel}{\theta}-K{\theta}{\parallel}$ is known where K is a projection vector with rank(K) = q. The class of this type estimator is quite general to include the class of the estimators proposed by Merchand and Giri (1993). We can derive the class and obtain the optimal type estimator. Also, this research can be applied to the simple and multiple regression model in the case of rank(K) ${\geq}2$.

Keywords

References

  1. Abramowitz, M. and Stegun, I. (1965). Handbook of Mathematical functions, Dover, New York.
  2. Baek, H. Y. (2000). Lindley type estimators with the known norm. Journal of the Korean Data & Infor-mation Science Society, 11, 37-45.
  3. Baek, H. Y. and Lee, J.M. (2005). Lindley type estimators when the norm is restricted to an interval. Journal of the Korean Data & Information Science Society, 16, 1027-1039.
  4. Berger, J. (1975). Minimax estimation of location vectors for a wide class of densities. Annals of Statistics, 3, 1318-1328. https://doi.org/10.1214/aos/1176343287
  5. Egerton, M. F. and Laycock, P. J. (1982). An explicit formula for the risk of James-Stein estimators. The Canadian Journal of Statistics, 10, 199-205. https://doi.org/10.2307/3556182
  6. George, E. I. (1990). Developments in decision-theoretic variance estimation : Comment. Statistical Science, 5, 107-109. https://doi.org/10.1214/ss/1177012267
  7. James, W. and Stein D. (1961). Estimation with quadratic loss. In Proceedings Fourth Berkeley Symp. Math. Statis. Probability, 1, University of California Press, Berkeley, 361-380.
  8. Kariya, T. (1989). Equivariant estimation in a model with ancillary statistics. Annals of Statistics, 17, 920-928. https://doi.org/10.1214/aos/1176347151
  9. Kim, B. H., Baek, H. Y. and Chang, I.H. (2002), Improved estimators of the natural parameters in continuous multiparameter exponential families. Communication in Statistics-Theory and Methods, 31, 11-29. https://doi.org/10.1081/STA-120002431
  10. Lehmann, E. L. and Casella, G. (1999).Theory of Point Estimation, 2nd Ed., Springer-Verlag, New York.
  11. Lindley, D. V. (1962). Discussion of paper by C. Stein. Journal of The Royal Statistical Society B, 2, 265-296.
  12. Marchand, E. and Giri, N. C. (1993). James-Stein estimation with constraints on the norm. Communication in Statistics-Theory and Methods, 22, 2903-2924. https://doi.org/10.1080/03610929308831192
  13. Park, T. R. and Baek, H. Y. (2014). An approach to improving the James-Stein estimator shrinking towards projection vectors. Journal of the Korean Data & Information Science Society, 25, 1549-1555. https://doi.org/10.7465/jkdi.2014.25.6.1549
  14. Perron, F. and Giri, N. (1989). On the best equivariant estimator of mean of a multivariate normal population. Journal of Multivariate Analysis, 32, 1-16.
  15. Strawderman, W. E (1974). Minimax estimation of location parameters for certain spherically symmetric distributions. Journal of Multivariate Analysis, 4, 255-264. https://doi.org/10.1016/0047-259X(74)90032-3