In statistical decision theory, an important question is to characterize the admissible rules. In this paper, we establish complete class theorems for estimating the noncentrality parameter of noncentral chi-square and noncentral $F$ distributions under squared error loss. Under a minor assumption, any admissible estimator must be a generalized Bayes rule. Using this result, we prove that the positive part of the UMVUE is inadmissible.
"A Complete Class Theorem for Estimating a Noncentrality Parameter." Ann. Statist. 15 (2) 800 - 804, June, 1987. https://doi.org/10.1214/aos/1176350375