## The Annals of Statistics

### Admissibility Results for Generalized Bayes Estimators of Coordinates of a Location Vector

James O. Berger

#### Abstract

Let $X$ be an $n$-dimensional random vector with density $f(x - \theta)$. It is desired to estimate $\theta_1$, under a strictly convex loss $L(\delta - \theta_1)$. If $F$ is a generalized Bayes prior density, the admissibility of the corresponding generalized Bayes estimator, $\delta_F$, is considered. An asymptotic approximation to $\delta_F$ is found. Using this approximation, it is shown that if (i) $f$ has enough moments, (ii) $L$ and $F$ are smooth enough, and (iii) $F(\theta) \leqq K(|\theta_1| + \sum^n_{i=2} \theta_i^2)^{(3-n)/2}$, then $\delta_F$ is admissible for estimating $\theta_1$. For example, assume that $F(\theta) \equiv 1$ and that $L$ is squared error loss. Under appropriate conditions it can be shown that $\delta_F(x) = x_1$, and that $\delta_F$ is the best invariant estimator. If, in addition, $f$ has 7 absolute moments and $n \leqq 3$, it can be concluded that $\delta_F$ is admissible.

#### Article information

Source
Ann. Statist. Volume 4, Number 2 (1976), 334-356.

Dates
First available in Project Euclid: 12 April 2007

http://projecteuclid.org/euclid.aos/1176343410

Digital Object Identifier
doi:10.1214/aos/1176343410

Mathematical Reviews number (MathSciNet)
MR400486

Zentralblatt MATH identifier
0323.62005

JSTOR