Sequential sampling with costly analysis – an application of dynamic programming

We investigate a sequential-sampling problem in which there is a cost of analysis in addition to the usual cost of the sampling and the terminal decision. This leads to a group-sampling procedure, and the aim of the paper is to discuss the application of adaptive dynamic programming to determine an optimal group-sampling scheme. Unfortunately, the approximation methods which prove so useful for single samples cannot be expected to extend to group samples, so we investigate approximation procedures for determining the optimal risk function (and the corresponding sampling scheme) based on a piecewise linear approximation method. We give the full algorithm and complete proofs in the normally distributed case. We also offer two conjectures about the nature of the optimal sampling scheme.