Uniformly Minimum Variance Unbiased Estimators (UMVUE) Not Attaining Cramer-Rao Lower Bounds

The main thrust of this article is to provide counterexamples where the variance of the UMVUE does not achieve the Cramer-Rao lower bound. We provided many motivating counterexamples and showed that these UMVU estimators are, in fact, asymptotically efficient estimators. All counterexamples are new or may not be available in standard textbooks. To illustrate the entire process, we supplied many definitions related to UMVUE and described various methods and step-by-step approaches for finding UMVUE’s. In concluding remarks, we also gave a short biography of Professor C.R. Rao. It is hoped that the article will have pedagogical value in courses on statistical inference. IJSS, Vol. 24(2) Special, December, 2024, pp 1-18