Sequential Allocation for an Estimation Problem with Ethical Costs

Abstract

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.