Bayesian Simultaneous Estimation for Means in $k$ Sample Problems

TL;DR

This paper develops Bayesian and minimax shrinkage estimators for simultaneously estimating multiple population means, especially when the means are suspected to be nearly equal, improving upon traditional preliminary test methods.

Contribution

It introduces Bayesian and minimax shrinkage estimators that improve mean estimation by shrinking individual means toward a pooled mean, offering an alternative to existing preliminary test estimators.

Findings

Shrinkage estimators outperform preliminary test estimators in simulations.

Bayesian and minimax estimators effectively leverage near-equality of means.

Proposed methods show improved estimation accuracy in practice.

Abstract

This paper is concerned with the simultaneous estimation of $k$ population means when one suspects that the $k$ means are nearly equal. As an alternative to the preliminary test estimator based on the test statistics for testing hypothesis of equal means, we derive Bayesian and minimax estimators which shrink individual sample means toward a pooled mean estimator given under the hypothesis. It is shown that both the preliminary test estimator and the Bayesian minimax shrinkage estimators are further improved by shrinking the pooled mean estimator. The performance of the proposed shrinkage estimators is investigated by simulation.