On the large sample optimality of sequential designs for comparing two or more treatments

The present paper deals with optimal (in Kiefer’s sense) response adaptive designs for parametric inference on v ≥ 2 treatments. Sometimes (e.g., for nonlinear models) a sequential estimation procedure combined with an adaptive experiment suggests itself as the “natural” best design. One of the questions is whether, since we proceed sequentially, we should infer conditionally on the design. Another question is whether such an adaptive design is really optimal for the chosen type of inference. The main purpose of this paper is to give proofs of the asymptotic optimality for inferring both conditionally and unconditionally of a large class of such designs, incorporating response-adaptive randomization as well. The asymptotic optimality of the Maximum Likelihood design, namely that based on the step-by-step updating of the parameter estimates by maximum likelihood, is proved for responses belonging to the exponential family. Under this procedure the MLEs retain the strong consistency and asymptotical normality properties. Furthermore, such properties still hold approximately for suitable inverse sampling stopping rules.