Deriving posterior distributions for a location parameter: A decision theoretic approach. (English) Zbl 0544.62008

Gatsonis, Constantine A.

Ann. Stat. 12, 958-970 (1984).

Let x be an observation from an n-dimensional location family with density f(x-\(\theta)\), and let \(\pi\) (\(\theta)\) be the prior density of the location parameter \(\theta\). Let \(\phi_{\pi}(\theta | x)\) denote the posterior density of \(\theta\) given x, and let \(\delta(x)=\delta(\theta | x)\) be an estimate of the posterior density with the loss being \(L(\pi,\delta,x)=\| \delta(\theta | x)-\phi_{\pi}(\theta | x)\|^ 2,\) where \(\| \cdot \|\) denotes the usual \(L_ 2\) norm. The risk of \(\delta\) is given by \(R(\delta,\pi)=EL(\pi,\delta(x),x),\) where the expectation is taken with respect to the marginal distribution of x. It is easy to show that \({\bar \delta}\)(\(\theta | x)=f(x-\theta)\) is the best translation invariant estimator of the posterior density.
It is shown in this paper that \({\bar \delta}\) is admissible for \(n=1,2\) and inadmissible for \(n\geq 3\). In fact, if \(n\geq 3\), then \({\bar \delta}\) is dominated by estimates of the form \(\delta(\theta | x)=f(x+\gamma(x)-\theta)\), where \(x+\gamma(x)\) is a point estimate of \(\theta\). Explicit choices of \(\gamma\) are given for the case in which \(\theta\) denotes the mean of a normal distribution.

Reviewer: K.Alam

MSC:

62C15	Admissibility in statistical decision theory
62C10	Bayesian problems; characterization of Bayes procedures
62A01	Foundations and philosophical topics in statistics
62F15	Bayesian inference
62F10	Point estimation
62H12	Estimation in multivariate analysis

Keywords:

noninformative priors; Stein phenomenon; n-dimensional location family; best translation invariant estimator; posterior density; normal distribution