Estimating the suspected larger of two normal means

For \(X_1, X_2\) $X_1,X_2$ independently and normally distributed with means \(\theta _1\) ${\theta }_1$ and \(\theta _2\) ${\theta }_2$ , variances \(\sigma ^2_1\) ${\sigma }_1^2$ and \(\sigma ^2_2\) ${\sigma }_2^2$ , we consider Bayesian inference about \(\theta _1\) ${\theta }_1$ with the difference \(\theta _1-\theta _2\) ${\theta }_1-{\theta }_2$ being lower-bounded by an uncertain m. We obtain a class of minimax Bayes estimators of \(\theta _1\) ${\theta }_1$ , based on a posterior distribution for \((\theta _1, \theta _2)^{\top }\) ${({\theta }_1,{\theta }_2)}^{\top }$ taking values on \(\mathbb {R}^2\) ${\mathbb{R}}^2$ , which dominate the unrestricted MLE under squared error loss for \(\theta _1-\theta _2 \ge 0\) ${\theta }_1-{\theta }_2\ge 0$ . We also construct and study an ad hoc credible set for \(\theta _1\) ${\theta }_1$ with approximate credibility \(1-\alpha \) $1-\alpha$ and provide numerical evidence of its frequentist coverage probability closely matching the nominal credibility level. A spending function is incorporated which further increases the coverage.