General Admissibility and Inadmissibility Results for Estimation in a Control Problem

Abstract

Let $\mathbf{X} = (X_1, \cdots, X_p)^t$ be an observation from a $p$-variate normal distribution with unknown mean $\mathbf{\theta} = (\theta_1, \cdots, \theta_p)^t$ and identity covariance matrix. We consider a control problem which, in canonical form, is the problem of estimating $\mathbf{\theta}$ under the loss $L(\mathbf{\theta, \delta}) = (\mathbf{\theta}^t \mathbf{\delta} - 1)^2$, where $\mathbf{\delta(x)} = (\delta_1(\mathbf{x}), \cdots, \delta_p(\mathbf{x}))^t$ is the estimate of $\mathbf{\theta}$ for a given $\mathbf{x}$. General theorems are given for establishing admissibility or inadmissibility of estimators in this problem. As an application, it is shown that estimators of the form $\mathbf{\delta(x)} = (|\mathbf{x}|^2 + c)^{-1}\mathbf{x} + |\mathbf{x}|^{-4}w(|\mathbf{x}|)\mathbf{x}$, where $w(|\mathbf{x}|)$ tends to zero as $|\mathbf{x}| \rightarrow \infty$, are inadmissible if $c > 5 - p$, but are admissible if $c \leq 5 - p$ and $\mathbf{\delta}$ is generalized Bayes for an appropriate prior measure. Also, an approximation to generalized Bayes estimators for large $|\mathbf{x}|$ is developed.