Shrinkage Testimator for the Common Mean of Several Univariate Normal Populations

Abstract

The challenge of combining two unbiased estimators is a common occurrence in applied statistics, with significant implications across diverse fields such as manufacturing quality control, medical research, and the social sciences. Despite the widespread relevance of estimating the common population mean 𝜇, this task is not without its challenges. A particularly intricate issue arises when the variations within populations are unknown or possibly unequal. Conventional approaches, like the two-sample t-test, fall short in addressing this problem as they assume equal variances among the two populations. When there exists prior information regarding population variances (𝜎ᵢ² , i=1,2), with the consideration that 𝜎₁² and 𝜎₂² might be equal, a hypothesis test can be conducted: H₀: 𝜎₁² = 𝜎₂² versus H₁: 𝜎₁² ≠ 𝜎₂² . The initial sample is utilized to test H₀, and if we fail to reject H₀, we gain confidence in incorporating our prior knowledge (after testing) to estimate the common mean 𝜇. However, if H₀ is rejected, indicating unequal population variances, the prior knowledge is discarded. In such cases, a second sample is obtained to compensate for the loss of prior knowledge. The estimation of the common mean 𝜇 is then carried out using either the Graybill–Deal estimator (GDE) or the maximum likelihood estimator (MLE). A noteworthy discovery is that the proposed preliminary testimators, denoted as 𝜇̂ₚₜ₁ and 𝜇̂ₚₜ₂, exhibit superior performance compared to the widely used unbiased estimators (GDE and MLE).