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Minimax estimation of linear combinations of restricted location parameters

Tatsuya Kubokawa

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Abstract

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.

Chapter information

Source
Dominique Fourdrinier, Éric Marchand and Andrew L. Rukhin, eds., Contemporary Developments in Bayesian Analysis and Statistical Decision Theory: A Festschrift for William E. Strawderman (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2012), 24-41

Dates
First available in Project Euclid: 14 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1331731609

Digital Object Identifier
doi:10.1214/11-IMSCOLL802

Zentralblatt MATH identifier
1356.62012

Subjects
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]

Keywords
admissibility decision theory generalized Bayes estimator minimaxity restricted parameters Stein estimation

Rights
Copyright © 2012, Institute of Mathematical Statistics

Citation

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


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