Journal of Integrative Natural Science (통합자연과학논문집)

The Basic Science Institute Chosun University (조선대학교 기초과학연구원)
권호 표지
DOI QR코드

DOI QR Code

Estimators Shrinking towards Projection Vector for Multivariate Normal Mean Vector under the Norm with a Known Interval

  • Baek, Hoh Yoo (Division of Mathematics and Informational Statistics, Wonkwang University)
  • Received : 2018.09.11
  • Accepted : 2018.09.18
  • Published : 2018.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Consider the problem of estimating a $p{\times}1$ mean vector ${\theta}(p-r{\geq}3)$, r = rank(K) with a projection matrix K under the quadratic loss, based on a sample $Y_1$, $Y_2$, ${\cdots}$, $Y_n$. In this paper a James-Stein type estimator with shrinkage form is given when it's variance distribution is specified and when the norm ${\parallel}{\theta}-K{\theta}{\parallel}$ is constrain, where K is an idempotent and symmetric matrix and rank(K) = r. It is characterized a minimal complete class of James-Stein type estimators in this case. And the subclass of James-Stein type estimators that dominate the sample mean is derived.

Keywords

Acknowledgement

Supported by : Wonkwang University

References

  1. W. James and D. Stein, "Estimation with quadratic loss", In Proceedings Fourth Berkeley Symp. Math. Statis. Probability, Vol. 1, University of California Press, Berkeley, pp. 361-380, 1961.
  2. D. V. Lindley, "Discussion of paper by C. Stein", Journal of The Royal Statistical Society", B, Vol. 2, pp. 265-296, 1962.
  3. W. E. Strawderman, "Minimax estimation of location parameters for certain spherically symmetric distributions, Journal of Multivariate Analysis, Vol. 4, pp. 255-264, 1974. https://doi.org/10.1016/0047-259X(74)90032-3
  4. S. Amari, "Differential geometry of curved exponential families, curvature and information loss", Annals of Statistics, Vol. 10, pp. 357-385, 1982. https://doi.org/10.1214/aos/1176345779
  5. T. Kariya, "Equivariant estimation in a model with ancillary statistics", Annals of Statistics", Vol. 17, pp. 920-928, 1989. https://doi.org/10.1214/aos/1176347151
  6. F. Perron and N. Giri, "On the best equivariant estimator of mean of a multivariate normal population", Journal of Multivariate Analysis, Vol. 32, pp. 1-16, 1989.
  7. E. Marchand and N. C. Giri, "James-Stein estimation with constraints on the norm", Communication in Statistics-Theory and Methods", Vol. 22(10), pp. 2903-2924, 1993. https://doi.org/10.1080/03610929308831192
  8. H. Y. Baek, "Lindley type estimators with the known norm", Journal of the Korean Data and Information Science Society, Vol. 11, pp. 37-45, 2000.
  9. J. Berger, "Minimax estimation of location vectors for a wide class of densities", Annals of Statistics, Vol. 3, pp. 1318-1328, 1975. https://doi.org/10.1214/aos/1176343287
  10. S. C. Chow and S. C. Wang, "A note an adaptive generalized ridge regression estimator", Statistics and Probability Letters, Vol. 10, pp. 17-21, 1990. https://doi.org/10.1016/0167-7152(90)90106-H
  11. M. F. Egerton and P. J. Laycock, "An explicit formula for the risk of James-Stein estimators", The Canadian Journal of Statistics, Vol. 10, pp. 199-205, 1982. https://doi.org/10.2307/3556182
  12. G. Bravo and G. MacGibbon, "Improved shrinkage estimators for the mean of a scale mixture of normals with unknown variance", The Canadian Journal of Statistics, Vol. 16, pp. 237-245, 1988. https://doi.org/10.2307/3314730