A Complete Class Theorem for the Control Problem and Further Results on Admissibility and Inadmissibility

Abstract

The following decision problem is studied. The statistician observes a random $n$-vector $y$ normally distributed with mean $\beta$ and identity covariance matrix. He takes action $\delta\in\mathbb{R}^n$ and suffers the loss $L(\beta, \delta) = (\beta'\delta - 1)^2.$ It is shown that this is equivalent to the linear control problem and closely related to the calibration problem. Among the invariant estimators, it is shown that the formal Bayes rules together with some of their limits include all admissible invariant rules. Other results on admissibility and inadmissibility of some commonly used estimators for the problem are obtained.