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September, 1984 Deriving Posterior Distributions for a Location Parameter: A Decision Theoretic Approach
Constantine A. Gatsonis
Ann. Statist. 12(3): 958-970 (September, 1984). DOI: 10.1214/aos/1176346714

Abstract

In this paper we develop a decision theoretic formulation for the problem of deriving posterior distributions for a parameter $\theta$, when the prior information is vague. Let $\pi(d\theta)$ be the true but unknown prior, $Q_\pi(d\theta\mid X)$ the corresponding posterior and $\delta(d\theta\mid X)$ an estimate of the posterior based on an observation $X$. The loss function is specified as a measure of distance between $Q_\pi(\cdot\mid X)$ and $\delta(\cdot\mid X)$, and the risk is the expected value of the loss with respect to the marginal distribution of $X$. When $\theta$ is a location parameter, the best invariant procedure (under translations in $R^n$) specifies the posterior which is obtained from the uniform prior on $\theta$. We show that this procedure is admissible in dimension 1 or 2 but it is inadmissible in all higher dimensions. The results reported here concern a broad class of location families, which includes the normal.

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Constantine A. Gatsonis. "Deriving Posterior Distributions for a Location Parameter: A Decision Theoretic Approach." Ann. Statist. 12 (3) 958 - 970, September, 1984. https://doi.org/10.1214/aos/1176346714

Information

Published: September, 1984
First available in Project Euclid: 12 April 2007

zbMATH: 0544.62008
MathSciNet: MR751285
Digital Object Identifier: 10.1214/aos/1176346714

Subjects:
Primary: 62C10
Secondary: 62A99, 62C15, 62F15, 62H12

Rights: Copyright © 1984 Institute of Mathematical Statistics

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Vol.12 • No. 3 • September, 1984
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