Optimal linear estimators: an empirical Bayes version with application to the binomial distribution

An empirical Bayes estimator is one which estimates the posterior mean by making use of past data. For certain conditional distributions though, no empirical Bayes estimator can be found which converges to the posterior mean as past data are accumulated. However, an optimal linear estimator for a parameter, say θ1 can often be found. This optimal linear estimator depends upon the first two prior moments, both of which can often be estimated. The resulting estimator has been simulated under the assumption that the conditional distribution is binomial and these simulations have shown its risk substantially smaller than the risk of the maximum likelihood estimator.