Pretest Estimation for the Common Mean of Several Normal Distributions: In Meta-Analysis Context

Abstract

The estimation of unknown quantities from multiple independent yet non-homogeneous samples has garnered increasing attention in various fields over the past decade. This interest is evidenced by the wide range of applications discussed in recent literature. In this study, we propose a preliminary test estimator for the common mean (𝜇) with unknown and unequal variances. When there exists prior information regarding the population mean with consideration that 𝜇 might be equal to the reference value for the population mean, a hypothesis test can be conducted: H₀ : 𝜇 = 𝜇₀ versus H₁ : 𝜇 ≠ 𝜇₀. The initial sample is used to test H₀, and if H₀ is not rejected, we become more confident in using our prior information (after the test) to estimate 𝜇. However, if H₀ is rejected, the prior information is discarded. Our simulations indicate that the proposed preliminary test estimator significantly decreases the mean squared error (MSE) values compared to unbiased estimators such as the Garybill-Deal (GD) estimator, particularly when 𝜇 closely aligns with the hypothesized mean (𝜇₀). Furthermore, our analysis indicates that the proposed test estimator outperforms the existing method, particularly in cases with minimal sample sizes. We advocate for its adoption to improve the accuracy of common mean estimation. Our findings suggest that through careful application to real meta-analyses, the proposed test estimator shows promising potential.