Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T18:13:59.917Z Has data issue: false hasContentIssue false

Minimax Estimators for the Location Vectors of Spherically Symmetric Densities

Published online by Cambridge University Press:  18 October 2010

George Judge
Affiliation:
University of Illinois
Shigetaka Miyazaki
Affiliation:
University of Illinois
Thomas Yancey
Affiliation:
University of Illinois

Abstract

The estimation of K (K ≥ 3) location parameters is considered under quadratic loss when the coordinates of the best invariant estimators are spherically symmetrically distributed. Under these stochastic mechanisms traditional Stein estimators are evaluated for finite samples and shown to have a risk performance superior to some conventional rules.

Type
Brief Report
Copyright
Copyright © Cambridge University Press 1985 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1. Baranchik, A. J. Multiple regression and estimation of the mean of the multivariate normal distribution. Technical Report No. 51, Department of Statistics, Stanford University, 1964.Google Scholar
2. Berger, J. O. Minimax estimation of location vectors for a wide class of densities. The Annals of Statistics 3 (1975): 13181328.10.1214/aos/1176343287Google Scholar
3. Bickel, P. J. Parametric robustness: Small biases can be worthwhile. The Annals of Statistics 12 (1984): 864879.Google Scholar
4. Bock, M. E. Minimax estimators of the mean of a multivariate normal distribution. The Annals of Statistics 3 (1975). 209218.Google Scholar
5. Bock, M. E. Minimax estimators that shift towards a hypersphere for location vectors of spherically symmetric distributions. Department of Statistics, Purdue University, 1983.Google Scholar
6. Brandwein, A. C. Minimax estimation of the mean of spherically symmetric distributions under general quadratic loss. Journal of Multivariate Analysis 9 (1979): 579588.Google Scholar
7. Brandwein, A. C. and Strawderman, W. E.. Minimax estimation of location parameters for spherically symmetric unimodal distributions under quadratic loss. The Annals of Statistics 6 (1978): 377416.Google Scholar
8. Brandwein, A. C. and Strawderman, W. E.. Minimax estimation of location parameters for spherically symmetric distributions with concave loss. The Annals of Statistics 8 (1980): 279284.Google Scholar
9. Brownstone, David. Review of Handbook of Econometrics Vol. I, Z., Griliches. and Intriligator, Michael D., Eds. Journal of Economic Literature, XXIII, (1985): 120.Google Scholar
10. Huber, P. J. Robust Statistical Procedures. Philadelphia: SIAM, 1977.Google Scholar
11. James, W. and Stein., C. Estimation with quadratic loss. In Proceedings of the Fourth Berkeley Symposium, Vol. 1, pp. 361379. Berkeley: University of California, 1961.Google Scholar
12. Judge, G. C. and Bock, M. E.. The Statistical Implication of Pre-Test and Stein-Rule Estimators in Econometrics. Amsterdam: North-Holland, 1978.Google Scholar
13. Judge, G. C, Griffiths, W., Hill, C., Lutkepohl, H., and Lee., T. C. The Theory and Practice of Econometrics, 2nd ed. New York: Wiley, 1985.Google Scholar
14. Kelejian, H. H. and Prucha., I. R. Independent and uncorrelated disturbances in linear regression: An illustration of the difference. Working Paper, University of Maryland, 1983.Google Scholar
15. Knight, J. L. The distribution of the Stein rule estimator in a model with non-normal disturbances. Westerm Ontario University, Department of Economics, 1984.Google Scholar
16. Koenker, R. W. and Bassett., G. Regression quantiles. Econometrica 46 (1978): 3350.Google Scholar
17. Koenker, R. W. Robust methods in econometrics. Econometric Reviews 1 (1982): 213255.Google Scholar
18. Shinozaki, W. Simultaneous estimation of location parameters under squared error loss. The Annals of Statistics 12 (1984): 322335.Google Scholar
19. Stein, C. Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In Proceedings of the Third Berkeley Symposium, Vol. 1, pp. 197206. Berkeley: University of California, 1955.Google Scholar
20. Stein, C. Estimation of the mean of a multivariate normal distribution. The Annals of Statistics 9 (1981): 11351151.Google Scholar
21. Strawderman, W. E. Minimax estimation of location parameters for certain spherically symmetric distributions. Journal of Multivariate Analysis 4 (1974): 255264.10.1016/0047-259X(74)90032-3Google Scholar
22. Zellner, A. Bayesian and non-Bayesian Analysis of the regression model with multivariate student-t error terms. Journal of the American Statistical Association 71 (1976): 400405.Google Scholar