Optimal fixed size confidence procedures for a restricted parameter space (Q1057010)

Suppose Z is a single observation from \(N(\theta,1),\theta \in \Omega =[- d, d]\). Let A be the action space of the statistician and define a loss function L on \(A\times \Omega\) by \(L(a,\theta)=0\) if \(| a-\theta | \leq e\) and 1 if \(| a-\theta | >e\) where \(e>0\) is given. Minimax admissible Bayes estimators \(\delta^*(Z)\) for \(\theta\) with respect to L are determined. The results are extended to include the case where the sampling distribution has a density function which is unimodal and symmetric about the location parameter. The connection between the minimax rule \(\delta^*\) and an optimal fixed size confidence procedure is obtained by noting that \(C^*(Z) = [\delta^*(Z)-e, \delta^*(Z)+e]\) can be interpreted as a confidence procedure of size 2e which has the highest confidence coefficient equal to \(\inf_{\theta} P_{\theta} \{\theta\in C^*(Z)\}\).