Simultaneous estimation of the hardy‐weinberg proportions

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.