Sequential allocation for an estimation problem with ethical costs (Q749132)

This paper deals with the problem of designing an experiment to estimate the difference \(\theta\) of the means of two normal populations with unit variance. A quasi-Bayesian, decision-theoretic approach is adopted where \(\theta\) is estimated by its maximum likelihood estimator, but the design is evaluated in Bayesian, decision-theoretic terms. The risk function of a sequential design consists of the cost of drawing a sample from either population, which may depend on unknown \(\theta\), and the loss function of the form tK[\(\sqrt{t}(\theta -{\hat \theta})]\), where t is the sample size and K is a nonnegative, nonconstant, symmetric function of polynomial growth. An asymptotic expression for integrated risks is derived. An ad hoc three-stage procedure, which takes observations in three batches, is given. The regret of the three-stage procedure is shown to be bounded as \(t\to \infty\). A series of theorems indicate that the three-stage procedure is asymptotically second-order efficient for squared error loss. Some comments on conditions used in the analysis and suggestions for further investigations are also presented.