Asymptotic optimality of experimental designs in estimating a product of means

In nonlinear estimation problems with linear models, one difficulty in obtaining optimal design is their dependence on the true value of the unknown parameters. A Bayesian approach is adopted with the assumption the means are independent a priori and have conjugate prior distributions. The problem of designing an experiment to estimate the product of the means of two normal populations is considered. The main results determine an asymptotic lower bound for the Bayes risk, and a necessary and sufficient condition for any sequential procedure to achieve the bound.