Institute of Mathematical Statistics Collections
Minimax estimation of linear combinations of restricted location parameters
The estimation of a linear combination of several restricted location parameters is addressed from a decision-theoretic point of view. A bench-mark estimator of the linear combination is an unbiased estimator, which is minimax, but inadmissible relative to the mean squared error. An interesting issue is what is a prior distribution which results in the generalized Bayes and minimax estimator. Although it seems plausible that the generalized Bayes estimator against the uniform prior over the restricted space should be minimax, it is shown to be not minimax when the number of the location parameters, k, is more than or equal to three, while it is minimax for k=1. In the case of k=2, a necessary and sufficient condition for the minimaxity is given, namely, the minimaxity depends on signs of coefficients of the linear combination. When the underlying distributions are normal, we can obtain a prior distribution which results in the generalized Bayes estimator satisfying minimaxity and admissibility. Finally, it is demonstrated that the estimation of ratio of normal variances converges to the estimation of difference of the normal positive means, which gives a motivation of the issue studied here.
First available in Project Euclid: 14 March 2012
Permanent link to this document
Digital Object Identifier
Zentralblatt MATH identifier
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Copyright © 2012, Institute of Mathematical Statistics
Kubokawa, Tatsuya. Minimax estimation of linear combinations of restricted location parameters. Contemporary Developments in Bayesian Analysis and Statistical Decision Theory: A Festschrift for William E. Strawderman, 24--41, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2012. doi:10.1214/11-IMSCOLL802. https://projecteuclid.org/euclid.imsc/1331731609
- Brown, L. D. and Hwang, J. T. (1982). A unified admissibility proof. In Statistical Decision Theory and Related Topics III (S. S. Gupta, J. Berger, eds.), 205–230. Academic Press, New York.
- Farrell, R. H. (1964). Estimators of a location parameter in the absolutely continuous case. Ann. Math. Statist., 35, 949–998.
- Girshick, M. A. and Savage, L. J. (1951). Bayes and minimax estimates for quadratic loss functions. In Proc. Second Berkeley Symp. Math. Statist. Probab., 1, 53–74. University of California Press, Berkeley.
- Hartigan, J. (2004). Uniform priors on convex sets improve risk. Statist. Prob. Letters, 67, 285–288.
- Iliopoulos, G. and Kourouklis, S. (1999). Improving on the best affine equivariant estimator of the ratio of the generalized variances. J. Multivariate Analysis, 68, 176–192.
- Karlin, S. and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. J. Multivariate Anal., 10, 467–498.
- Katz, M. (1961). Admissible and minimax estimates of parameters in truncated spaces. Ann. Math. Statist., 32, 136–142.
- Kubokawa, T. (1994a). A unified approach to improving equivariant estimators. Ann. Statist., 22, 290–299.
- Kubokawa, T. (1994b). Double shrinkage estimation of ratio of scale parameters. Ann. Inst. Statist. Math., 46, 95–119.
- Kubokawa, T. (1999). Shrinkage and modification techniques in estimation of variance and the related problems: A review. Commun. Statist.–Theory and Methods, 28, 613–650.
- Kubokawa, T. (2004). Minimaxity in estimation of restricted parameters. J. Japan Statist. Soc., 34, 229–253.
- Kubokawa, T. and Saleh, A. K. Md. E. (1998). Estimation of location and scale parameters under order restrictions. J. Statistical Research, 28, 41–51.
- Kubokawa, T. and Srivastava, M. S. (1996). Double shrinkage estimators of ratio of variances. Proceedings of the Sixth Lukacs Symposium, 139–154.
- Marchand, E. and Strawderman, W. E. (2004). Estimation in restricted parameter spaces: A review. Festschrift for Herman Rubin, IMS Lecture Notes-Monograph Series, 45, 21–44.
- Marchand, E. and Strawderman, W. E. (2005). Improving on the minimum risk equivariant estimator of a location parameter which is constrained to an interval or a half-interval. Ann. Inst. Statist. Math., 57, 129–143.
- Rukhin, A. (1992). Asymptotic risk behavior of mean vector and variance estimators and the problem of positive normal mean. Ann. Inst. Statist. Math., 44, 299–311.
- Rukhin, A. (1995). Admissibility: Survey of a concept in progress. International Statistical Review, 63, 95–115.
- Stein, C. (1964). Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean. Ann. Inst. Statist. Math., 16, 155–160.
- Tsukuma, H. and Kubokawa, T. (2008). Stein’s phenomenon in estimation of means restricted to a polyhedral convex cone. J. Multivariate Analysis, 99, 141–164.