A Sharp Necessary and Sufficient Condition for Inadmissibility of Estimators in a Control Problem

Abstract

Suppose $\mathbf{x} = (x_1, \cdots, x_m)^t$ is an $m$-variate normal random variable with mean vector $\mathbf{\theta} = (\theta_1, \cdots, \theta_m)^t$ and identity dispersion matrix. We consider the control problem which, in canonical form, is the problem of estimating $\mathbf{\theta}$ with respect to the loss $L(\theta, \delta) = (1 - \theta^t\delta)^2,$ where $\delta(x) = (\delta_1(x), \cdots, \delta_m(x))^t$. A necessary and sufficient condition for the admissibility of spherically symmetric generalized Bayes $\delta(x)$ is given in terms of a Dirichlet problem. This condition is also equivalent to recurrence of a diffusion process and insolubility of certain elliptic boundary value problems.