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.
Publié le : 1984-09-14
Classification:
Location parameter,
noninformative priors,
best invariant procedures,
admissibility,
Stein phenomenon,
normal mean,
62C10,
62C15,
62A99,
62F15,
62H12
@article{1176346714,
author = {Gatsonis, Constantine A.},
title = {Deriving Posterior Distributions for a Location Parameter: A Decision Theoretic Approach},
journal = {Ann. Statist.},
volume = {12},
number = {1},
year = {1984},
pages = { 958-970},
language = {en},
url = {http://dml.mathdoc.fr/item/1176346714}
}
Gatsonis, Constantine A. Deriving Posterior Distributions for a Location Parameter: A Decision Theoretic Approach. Ann. Statist., Tome 12 (1984) no. 1, pp. 958-970. http://gdmltest.u-ga.fr/item/1176346714/