An Improved Solution to the Two Normal Means Problem via Regularization

2025/08/18 by Yang Liu, Liu, Yang, Jonathan P. Williams +1
Decision Sciences · Mathematics · #FOS: Computer and information sciences #Iterative Methods for Nonlinear Equations #Methodology (stat.ME) #Multi-Criteria Decision Making

paper · pdf · doi:10.48550/arxiv.2508.13012

openalex publication_date 2025/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The many-normal-means problem is a classic example that motivates the development of many important inferential procedures in the history of statistics. In this short note, we consider a further special case of the problem, which involves only two normally distributed data points with a constraint that the pair of means are not too far apart from one another. Starting with a regularized ML estimator, we construct a novel possibilistic IM for marginal inference on one of the two means. Not only does the new IM remain valid, it is also more efficient than the standard marginal inference ignoring the a priori information about the closeness of means, as well as the partial conditioning IM solution recently proposed in Yang et al. (2023).

Citations

• Possibilistic inferential models: a review
• Asymptotic efficiency of inferential models and a possibilistic Bernstein--von Mises theorem
• Valid and efficient imprecise-probabilistic inference with partial priors, III. Marginalization
• Partial Conditioning for Inference of Many-Normal-Means with Hölder Constraints
• Valid and efficient imprecise-probabilistic inference with partial priors, II. General framework
• Another Look at the Problem of Many-Normal-Means
• Valid and efficient imprecise-probabilistic inference with partial priors, I. First results
• Inferential models and possibility measures

Related

• Partial Conditioning for Inference of Many-Normal-Means with Hölder Constraints
• Another Look at the Problem of Many-Normal-Means
• Solving the Empirical Bayes Normal Means Problem with Correlated Noise
• Minimax Structured Normal Means Inference
• Valid and efficient imprecise-probabilistic inference with partial priors, III. Marginalization