Abstract
For location families with densitiesf 0(x−θ), we study the problem of estimating θ for location invariant lossL(θ,d)=ρ(d−θ), and under a lower-bound constraint of the form θ≥a. We show, that for quite general (f 0, ρ), the Bayes estimator δ U with respect to a uniform prior on (a, ∞) is a minimax estimator which dominates the benchmark minimum risk equivariant (MRE) estimator. In extending some previous dominance results due to Katz and Farrell, we make use of Kubokawa'sIERD (Integral Expression of Risk Difference) method, and actually obtain classes of dominating estimators which include, and are characterized in terms of δ U . Implications are also given and, finally, the above dominance phenomenon is studied and extended to an interval constraint of the form θ∈[a, b].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berger, J. O. (1985).Statistical Decision Theory and Bayesian Analysis, 2nd ed., Springer-Verlag, New York.
Blumenthal, S. and Cohen, A. (1968). Estimation of two ordered translation parameters,Annals of Mathematical Statistics,39, 517–530.
Farrell, R. H. (1964). Estimators of a location parameter in the absolutely continuous case,Annals of Mathematical Statistics,35, 949–998.
Gatsonis, C., MacGibbon, B. and Strawderman, W. E. (1987). On the estimation of a restricted normal mean,Statistics & Probability Letters,6, 21–30.
Katz, M. (1961). Admissible and minimax estimates of parameters in truncated spaces,Annals of Mathematical Statistics,32 136–142.
Hartigan, J. (2004). Uniform priors on convex sets improve risk,Statistics & Probability Letters,67, 285–288.
Kubokawa, T. (1994a). A unified approach to improving equivariant estimators,Annals of Statistics,22, 290–299.
Kubokawa, T. (1994b). Double shrinkage estimation of ratio of scale parameters,Annals of the Institute of Statistical Mathematics,46, 95–119.
Kubokawa, T. (1998). The Stein phenomenon in simultaneous estimation: A review,Applied Statistical Science III (eds. S. E. Ahmed, M. Ahsanullah and B. K. Sinha), 143–173, NOVA Science Publishers, New York.
Kubokawa, T. (1999). Shrinkage and modification techniques in estimation of variance and the related problems: A review,Communications in Statistics: Theory and Methods,28, 613–650.
Kubokawa, T. and Saleh, A. K. MD. E. (1994). Estimation of location and scale parameters under order restrictions,Journal of Statistical Research,28, 41–51.
Kumar, S. and Sharma, D. (1988). Simultaneous estimation of ordered parameters,Communications in Statistics: Theory and Methods,17, 4315–4336.
Lehmann, E. L. and Casella, G. (1998).Theory of Point Estimation, 2nd ed., Springer-Verlag, New York.
Marchand, É. and Perron, F. (2001). Improving on the MLE of a bounded normal mean.Annals of Statistics,29, 1078–1093.
Parsian, A. and Sanjari Farsipour, N. (1997). Estimation of parameters of exponential distribution in the truncated space using asymmetric loss function,Statistical Papers,38, 423–443.
Rukhin, A. L. (1990). Comments on ‘Developments in decision-theoretic variance estimation’, by Maatta, J. M. and Casella, G.,Statistical Science,5, 113–116.
Wald, A. (1950).Statistical Decision Functions, Wiley, New York.
Author information
Authors and Affiliations
Additional information
Research supported by NSERC of Canada.
About this article
Cite this article
Marchand, É., Strawderman, W.E. Improving on the minimum risk equivariant estimator of a location parameter which is constrained to an interval or a half-interval. Ann Inst Stat Math 57, 129–143 (2005). https://doi.org/10.1007/BF02506883
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02506883