A decision‐theoretic approach to sample size determination under several priors

In this article, we consider sample size determination for experiments in which estimation and design are performed by multiple parties. This problem has relevant applications in contexts involving adversarial decision makers, such as control theory, marketing, and drug testing. Specifically, we adopt a decision‐theoretic perspective, and we assume that a decision on an unknown parameter of a statistical model involves two actors, Ee and Eo, who share the same data and loss function but not the same prior beliefs on the parameter. We also suppose that Ee has to use Eo's optimal action, and we finally assume that the experiment is planned by a third party, Pd. In this framework, we aim at determining an appropriate sample size so that the posterior expected loss incurred by Ee in taking the optimal action of Eo is sufficiently small. We develop general results for the one‐parameter exponential family under quadratic loss and analyze the interactive impact of the prior beliefs of the three different parties on the resulting sample sizes. Relationships with other sample size determination criteria are explored. Copyright © 2016 John Wiley & Sons, Ltd.