| Abstract: | We consider the problem of simultaneously estimating k + 1 related proportions, with a special emphasis on the estimation of Hardy-Weinberg (HW) proportions. We prove that the uniformly minimum-variance unbiased estimator (UMVUE) of two proportions which are individually admissible under squared-error loss are inadmissible in estimating the proportions jointly. Furthermore, rules that dominate the UMVUE are given. A Bayesian analysis is then presented to provide insight into this inadmissibility issue: The UMVUE is undesirable because the two estimators are Bayes rules corresponding to different priors. It is also shown that there does not exist a prior which yields the maximum-likelihood estimators simultaneously. When the risks of several estimators for the HW proportions are compared, it is seen that some Bayesian estimates yield significantly smaller risks over a large portion of the parameter space for small samples. However, the differences in risks become less significant as the sample size gets larger. |
