Exploring Cramer-Rao Lower Bounds and Uniformly Minimum Variance Unbiased Estimators (UMVUE): Counterexamples

Cramer-Rao Lower Bound (CR-LB) is a fundamental tool for determining the minimum variance of unbiased estimators. The main goal of this chapter is to present counterexamples where the variance of the UMVUE does not reach the Cramer-Rao lower bound. We provided many motivating counterexamples and demonstrated that these UMVU estimators are, in fact, asymptotically efficient. All counterexamples are either new or not typically found in standard textbooks. To illustrate the process, we included numerous definitions related to UMVUE and explained various methods and step-by-step approaches for finding UMVUEs. This chapter will be valuable for senior undergraduates and first-year graduate students taking courses in statistical inference. The material should also interest teachers of statistical estimation theory. They could include the examples from this paper in various exams. Certainly! The article also has significant pedagogical value.