## The Annals of Statistics

- Ann. Statist.
- Volume 23, Number 2 (1995), 593-597.

### A Note on Admissibility When Precision is Unbounded

Charles Anderson and Nabendu Pal

#### Abstract

The estimation of a common mean vector $\theta$ given two independent normal observations $X \sim N_p(\theta, \sigma^2_x I)$ and $Y \sim N_p(\theta, \sigma^2_y I)$ is reconsidered. It being known that the estimator $\eta X + (1 - \eta)Y$ is inadmissible when $\eta \in (0, 1)$, we show that when $\eta$ is 0 or 1, then the opposite is true, that is, the estimator is admissible. The general situation is that an estimator $X^\ast$ can be improved by shrinkage when there exists a statistic $B$ which, in a certain sense, estimates a lower bound on the risk of $X^\ast$. On the other hand, an estimator is admissible under very general conditions if there is no reasonable way to detect that its risk is small.

#### Article information

**Source**

Ann. Statist., Volume 23, Number 2 (1995), 593-597.

**Dates**

First available in Project Euclid: 11 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176324537

**Digital Object Identifier**

doi:10.1214/aos/1176324537

**Mathematical Reviews number (MathSciNet)**

MR1332583

**Zentralblatt MATH identifier**

0824.62007

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62C15: Admissibility

Secondary: 62H12: Estimation

**Keywords**

Inadmissibility shrinkage estimation Stein's normal identity

#### Citation

Anderson, Charles; Pal, Nabendu. A Note on Admissibility When Precision is Unbounded. Ann. Statist. 23 (1995), no. 2, 593--597. doi:10.1214/aos/1176324537. https://projecteuclid.org/euclid.aos/1176324537