Selecting Estimators When MSE Favors Bias Over UMVU

Determine, within the frequentist framework that evaluates estimators by mean square error (MSE), which estimator should be selected in settings where biased estimators have smaller MSE than the uniformly minimum variance unbiased (UMVU) estimator for all parameter values, given that an estimator that minimizes MSE uniformly across the parameter space generally does not exist.

Background

The paper discusses the frequentist approach to inference based on hypothetical repeated sampling, where estimators are assessed by bias and mean square error (MSE). In many classical problems, UMVU estimators are preferred due to their unbiasedness and minimum variance among unbiased estimators. However, phenomena such as James–Stein illustrate that biased estimators can uniformly reduce MSE compared to UMVU across all parameter values.

The authors point out a core difficulty in this framework: if MSE is the evaluation criterion and biased estimators uniformly dominate UMVU in MSE, then UMVU should not be used, but there is no general estimator that minimizes MSE uniformly over the entire parameter space. This creates an unresolved decision point within the MSE-based frequentist paradigm about which estimator to choose. The paper later advocates an information-based alternative, but the question remains open within the traditional MSE criterion.