The efficiency of the method of moments estimates for hyperparameters in the empirical Bayes binomial model (Q1966371)

The paper deals with the Bayes estimate of the binomial proportion parameter \(p^*\) in the Bayes binomial model due to \textit{C. N. Morris} [Ann. Stat. 11, 515-529 (1983; Zbl 0521.62014)]. Under this model the posterior mean of \(p^*\) has a shrinkage pattern \(E(p^*_i |y_1,\dots,y_n)=(1-b)y_i/m+bp\) that shrinks the observed proportion \(y_i/m\) towards the prior mean \(p\) with shrinkage factor \(b\). The parameters \(p\) and \(b\) are estimated by the method of moments. The efficiency resp. asymptotic efficiency of this estimate relative to the Cramér-Rao lower bound is investigated for various combinations of parameter values. The estimate of \(p\) is at least \(95\%\) efficient in all cases, the estimate of \(b\) is at least \(90\%\) efficient when the sample size \(m\leq 10\), and both estimates become perfectly asymptotically efficient as \(b\to 1\).