×

Improving on the MLE of a bounded normal mean. (English) Zbl 1041.62016

Summary: We consider the problem of estimating the mean of a \(p\)-variate normal distribution with identity covariance matrix when the mean lies in a ball of radius \(m\). It follows from general theory that dominating estimators of the maximum likelihood estimator always exist when the loss is squared error. We provide and describe explicit classes of improvements for all problems \((m, p)\). We show that, for small enough \(m\), a wide class of estimators, including all Bayes estimators with respect to orthogonally invariant priors, dominates the maximum likelihood estimator. When m is not so small, we establish general sufficient conditions for dominance over the maximum likelihood estimator. These include, when \(m \leq \sqrt{p}\), the Bayes estimator with respect to a uniform prior on the boundary of the parameter space.
We also study the resulting Bayes estimators for orthogonally invariant priors and obtain conditions of dominance involving the choice of the prior. Finally, these Bayesian dominance results are further discussed and illustrated with examples, which include (1) the Bayes estimator for a uniform prior on the whole parameter space and (2) a new Bayes estimator derived from an exponential family of priors.

MSC:

62F10 Point estimation
62F15 Bayesian inference
62F30 Parametric inference under constraints
62H12 Estimation in multivariate analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Berry, C. (1990). Minimax estimation of a bounded normal mean vector. J.Multivariate Anal. 35 130-139. · Zbl 0705.62052
[2] Casella, G. and Strawderman, W. E. (1981). Estimating a bounded normal mean. Ann.Statist. 9 870-878. · Zbl 0474.62010
[3] Charras, A. and van Eeden, C. (1991). Bayes and admissibility properties of estimators in truncated parameter spaces. Canad.J.Statist.19 121-134. JSTOR: · Zbl 0735.62008
[4] DasGupta, A. (1985). Bayes minimax estimation in multiparameter families when the parameter space is restricted to a bounded convex set. Sankhy\?a Ser.A 47 326-332. · Zbl 0588.62015
[5] Gatsonis, C., MacGibbon, B. and Strawderman, W. E. (1987). On the estimation of a restricted normal mean. Statist.Probab.Lett.6 21-30. · Zbl 0647.62015
[6] Kempthorne, P. J. (1988). Dominating inadmissible procedures using compromise decision theory. In Statistical Decision Theory and Related Topics IV (S. Gupta and J. O. Berger, eds.) 1 381-396. Springer, New York. · Zbl 0647.62019
[7] Marchand, É. and MacGibbon, B. (2000). Minimax estimation of a constrained binomial proportion. Statist.Decisions 18 129-167. · Zbl 0953.62010
[8] Marchand, É. and Perron, F. (1999). Improving on the mle of a bounded normal mean. Technical Report 2628, Centre de recherches mathématiques, Univ. Montréal. · Zbl 1041.62016
[9] Moors, J. J. A. (1981). Inadmissibility of linearly invariant estimators in the truncated parameter spaces. J.Amer.Statist.Assoc.76 910-915. JSTOR: · Zbl 0484.62012
[10] Moors, J. J. A. (1985). Estimation in truncated parameter spaces. Ph.D. thesis, Tilburg Univ.
[11] Robert, C. (1990). Modified Bessel functions and their applications in probability and statistics. Statist.Probab.Lett.9 155-161. · Zbl 0686.62021
[12] Sacks, J. (1963). Generalized Bayes solutions in estimation problems. Ann.Math.Statist.34 751-768. · Zbl 0129.32402
[13] Vidakovic, B. and DasGupta, A. (1996). Efficiency of linear rules for estimating a bounded normal mean. Sankhy\?a Ser.A 58 81-100. · Zbl 0953.62514
[14] Watson, G. S. (1983). Statistics on Spheres. Wiley, New York. · Zbl 0646.62045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.