The efficiency of the method of moments estimates for hyperparameters in the empirical Bayes binomial model

Summary

In the empirical Bayes binomial model of Morris (1983), the Bayes estimate of the binomial proportion parameter has a shrinkage pattern with prior mean p and shrinkage factor b as the hyperparameters. Formulating an empirical Bayes estimate, Morris employs the method of moments to estimate these two hyperparameters. This paper investigates the efficiency/asymptotic efficiency of these estimates relative to the Cramér-Rao lower bound for the variance of unbiased estimates. The efficiency of the estimate of p and the asymptotic efficiency of the estimate of b are mathematically formulated. The estimate of p is at least 95% efficient in all cases, and it becomes perfectly efficient as the parameter b approaches 0 or 1. The estimate of b has a very high asymptotic efficiency in most cases, for example at least 83% when the binomial sample size m ≤ 10; at least 90% when m ≤ 5, and it becomes perfectly asymptotically efficient as the parameter b approaches 1.