Improving on Inadmissible Estimators in the Control Problem

Abstract

Let $X$ have a $p$-variate normal distribution with unknown mean $\theta$ and identity covariance matrix. The following transformed version of a control problem (Zaman, 1981) is considered: estimate $\theta$ by $d$ subject to incurring a loss $L(d, \theta) = (\theta^t d - 1)^2$. The comparison of decision rules in terms of expected loss is reduced to the study of differential inequalities. Results establishing the minimaxity of a large class of estimators are obtained. Special attention is given to the proposition of admissible, generalized Bayes rules which dominate the uniform prior, generalized Bayes controller when $p \geq 5$.