Optimal Few-Stage Allocation Procedures

This paper gives very general algorithms for the design of optimal experiments involving two Bernoulli populations in which sampling is carried out in stages. It is assumed that the outcomes of the previous stage are available before the allocations for the next stage are decided. At each stage, one must decide how many observations to take and how many to sample from each of the alternative populations. Of particular interest are 2-and 3-stage experiments. To illustrate that the algorithms can be used for experiments of useful sample sizes, they are applied to estimation and optimization problems. Results indicate that, for problems of moderate size, published asymptotic analyses do not always represent the true behavior of the optimal stage sizes, and eeciency may be lost if the analytical results are used instead of the true optimal allocation. Our results also suggest that one might approach large problems by extrapolating optimal solutions for moderate sample sizes; and, that approaches of this sort could give design guidelines that are far more explicit (and hopefully closer to optimal) than those obtained through asymptotic analyses alone. The examples also show the ease with which the computational approach can solve problems that present analytical diiculties. This allows one to use models that more accurately represent important characteristics of the problem. It is also shown that in various cases the base algorithms can be modiied to incorporate simpliications. In such settings, signiicant speedups and space reductions can be obtained, permitting the exact solution of even larger problems.