Estimating the suspected larger of two normal means

Courtney Drew () and Éric Marchand ()
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Courtney Drew: Statistics Canada
Éric Marchand: Université de Sherbrooke

Metrika: International Journal for Theoretical and Applied Statistics, 2025, vol. 88, issue 2, No 5, 245 pages

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

Keywords: Bayes estimator; Hierarchical prior; Point estimation; Interval estimation; Skew-normal; Additional information; Uncertain constraint; 62F15; 62F30; 62F10; 62C20 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s00184-024-00961-5

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