Sequential Allocation and Optional Stopping in Bayesian Simultaneous Estimation

Abstract Suppose random variables X 1, X 2, …, with distribution depending on parameter θ1, are observable from population 1, independent of random variables Y 1, Y 2, …, with distribution depending on parameter θ2, and observed from population 2. The simultaneous estimation of parameters θ1 and θ2 is of interest. An experiment is conducted where at each stage either an X or a Y is observed, and the experiment may be stopped at any stage (allocation and decision depending on information at that stage). In such an experiment, a sequential allocation procedure or policy determines whether to observe X or Y at each stage, and a stopping rule determines when to stop, and hence the pair (policy, stopping time) constitutes the design of the experiment. The purpose of this work is the study of pairs of policies and stopping times for simultaneous estimation of θ1 and θ2 using a Bayesian approach with additive estimation loss and unit sampling costs. In this context, optimal and asymptotically optimal pairs are d...