Estimating the suspected larger of two normal means

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