The Annals of Statistics
- Ann. Statist.
- Volume 23, Number 2 (1995), 593-597.
A Note on Admissibility When Precision is Unbounded
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.
Ann. Statist., Volume 23, Number 2 (1995), 593-597.
First available in Project Euclid: 11 April 2007
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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