The problem of designing an experiment to estimate the difference between the means of two normal populations with unit variances is considered, when the cost of drawing a sample from either population may depend on unknown parameters. A quasi-Bayesian approach is adopted in which the mean difference is estimated by its maximum likelihood estimator, but the design (allocation rule) is evaluated in Bayesian, decision-theoretic terms. A three-stage procedure is proposed and its risk evaluated, up to terms which are small compared to the cost of a single observation. This procedure is shown to be optimal to second order for squared error loss.
"Sequential Allocation for an Estimation Problem with Ethical Costs." Ann. Statist. 18 (3) 1358 - 1377, September, 1990. https://doi.org/10.1214/aos/1176347754