A Note on Bayes Estimates
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Throughout this paper we are concerned with the problem of estimating a real parameter when the loss function is such that the Bayes estimate exists, is unique, and satisfies a simple Equation, (1.5). If the estimate is unbiased (in the general sense of Lehmann [3]) we show under weak conditions that it must satisfy another Equation, (1.14). The main result of Section 1, Theorem 1.3, shows that, in general, these two equations are incompatible unless the Bayes risk is 0. This extends Theorem 11.24 of [1] which states that in estimation with quadratic loss, unbiased Bayes estimates have Bayes risk 0. Some counter-examples at the end of the section indicate the limits of this incompatibility result.
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There are no always-nonnegative classical unbiased estimators of σα or σ2α in the hierarchical model.
“Similarly, there are no always-nonnegative classical unbiased estimators of σ α or σ 2 α in the hierarchical model.”
From Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper) (2006) · cites this paper as Bickel and Blackwell (1967)
The introduction (p. 1) and Theorem 1.3 (p. 3) concern when an unbiased Bayes estimate must have zero Bayes risk. Searches for hierarchical models, σₐ, σ²ₐ, and always-nonnegative classical unbiased estimators found no passage addressing the claimed nonexistence result. (LLM, conf=0.77)
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“Similar issues are discussed by Bickel and Blackwell (1967) and Meng and Zaslavsky (2002).”
BackgroundPrior distributions for variance parameters in hierarchical modelsPreprint
Gelman, A. (2004) · RePEc: Research Papers in Economics · Cited by 1,558 · see citing works
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- Authors
- Bickel, P. J., & Blackwell, D.
- ORCID
- Bickel, Peter J.
- Venue
- The Annals of Mathematical Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 1967-12
- Volume
- 38
- Issue
- 6
- Pages
- 1907-1911
- ISSN
- 0003-4851 (ISSN-L)
- OpenAlex
- W2003398575
- Cited by
- 25
- References
- 2
- Citations by year
- 2020: 3 · 2015: 2 · 2014: 1 · 2013: 1
- FWCI
- 0.00
- Type
- journal article
- Language
- en
- Open access
- diamond
- doi.org
- Locations
- The Annals of Mathematical Statistics (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Keywords
- Bayesian estimation, Bayes risk, loss functions, squared error loss, counterexample
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