Randomly Reinforced Urn Designs whose Allocation Proportions Converge to Arbitrary Prespecified Values

There are many randomization procedures in clinical trials in which the proportion of patients allocated to treatments converges to a fixed value. Many of these procedures, like those targeting the optimal Neyman allocation, are not adaptive designs and the limiting proportion of allocations is independent of the treatment behavior. In this work we construct a response adaptive design, described in terms of a two-colour urn model, targeting fixed asymptotic allocations that are a function of treatment performances. We prove some asymptotic results for the process of colours generated by the urn and for the process of its composition. Applications to sequential clinical trials and connections with response-adaptive design of experiments are considered. Additionally, we report simulation studies concerning the power function of a hypothesis testing procedure that naturally arises from this statistical framework.