A Connection between Minimum Variance Unbiased Estimation and Maximum Likelihood

C. R. Rao (1973, problem 5.11) constructed an example where a minimum variance unbiased estimator exists for the parameter function (1 – α)2 but no uniformly minimum variance unbiased estimator exists for α itself. The example was further discussed by Romano and Siegel (1986, example 9.6) who exhibited the family of unbiased estimators for α; the nonexistence of a uniformly minimum variance unbiased estimator comes from the fact that for a fixed value α0 of α the unbiased estimator with smallest variance depends on α0 and hence no unbiased estimator has uniformly smallest variance. In this note we show that a natural continuation of the example leads to an estimating equation for α that produces the maximum likelihood estimator. A second example, taken from Fisher (1958), exhibits a similar property. This example is both simpler and less artificial than the first, and we have found it very useful for teaching purposes. The general result is closely related to Fisher's motivation of maximum likelihood estimation and provides a good teaching illustration of Fisher's argument.