Sequential Allocation and Optional Stopping in Bayesian Simultaneous Estimation

Abstract

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 derived and then, under further conditions on the form of the estimation loss, asymptotically optimal stopping times are described when the allocation policy is fixed in advance. The theory is applied in some detail to the Poisson distribution.