On an asymptotic optimality property of play-the-winner and vector-at-a-time samplig

Summary

Simon andWeiss (1975) consider the formulation of the clinical trial as a selection procedure (Bechhoffer, Kiefer andSobel, 1968). The object of the trial is to choose the better treatment with probability ≥P ^*, whereP ^* is assigned, when the difference in sucess probabilities is ≥ Δ^*,Δ^* also being assigned. They consider a family of single step allocation methods for the reduction of the number of patients given the poorer treatment. Using numerical results,Simon andWeiss conclude that if the stopping rule is based on the difference in successes then either alternating allocation or play-the-winner allocation appears to be optimal (Robbins, 1956;Sobel, Weiss, 1970). We make precise the above statement and the in our main theorem prove it to be true for all Δ^* sufficiently small andP ^*→ 1.