Lower bounds for the variance of uniformly minimum variance unbiased estimators

Abstract

The object of this paper was to study lower bounds ·for the variance of uniformly minimum variance unbiased estimators. The lower bounds of Cramer and Rao, Bhattacharyya, Hammersley, Chapman and Robbins, and Kiefer were derived and discussed. Each was compared with the other, showing their relative merits and shortcomings. Of the lower bounds considered all are greater than or equal to the Cramer-Rao lower bound. The Kiefer lower bound is as good as any of the others, or better. We were able to show that the Cramer-Rao lower bound is exactly the first Bhattacharyya lower bound. The Hammersley and the Chapman and Robbins lower bounds are identical when they both have the same parameter space, i.e., when Ω = (a,b). The use of the various lower bounds is illustrated in examples throughout the paper.