An Improved Solution to the Two Normal Means Problem via Regularization

TL;DR

This paper introduces a regularized inferential method for estimating one mean in a two-normal-means problem with a closeness constraint, improving efficiency over existing approaches.

Contribution

It develops a novel possibilistic inferential model based on regularized ML estimation for the two-normal-means problem with a proximity constraint.

Findings

The new IM is valid and more efficient than standard marginal inference.

It outperforms recent partial conditioning IM solutions.

The approach leverages a regularized estimator for improved inference.

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).