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Deriving Posterior Distributions for a Location Parameter: A Decision Theoretic Approach

Constantine A. Gatsonis
The Annals of Statistics
Vol. 12, No. 3 (Sep., 1984), pp. 958-970
Stable URL: http://www.jstor.org/stable/2240972
Page Count: 13
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Deriving Posterior Distributions for a Location Parameter: A Decision Theoretic Approach
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Abstract

In this paper we develop a decision theoretic formulation for the problem of deriving posterior distributions for a parameter θ, when the prior information is vague. Let π(dθ) be the true but unknown prior, Qπ(dθ∣ X) the corresponding posterior and δ(dθ∣ X) an estimate of the posterior based on an observation X. The loss function is specified as a measure of distance between Qπ(·∣ X) and δ(·∣ X), and the risk is the expected value of the loss with respect to the marginal distribution of X. When θ is a location parameter, the best invariant procedure (under translations in Rn) specifies the posterior which is obtained from the uniform prior on θ. 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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