## The Annals of Statistics

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

Constantine A. Gatsonis

#### 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.

#### Article information

Source
Ann. Statist., Volume 12, Number 3 (1984), 958-970.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176346714

Digital Object Identifier
doi:10.1214/aos/1176346714

Mathematical Reviews number (MathSciNet)
MR751285

Zentralblatt MATH identifier
0544.62008

JSTOR