Adaptive Randomly Reinforced Urn design and its asymptotic properties

In response-adaptive designs for clinical trials the probability to assign patients to treatments depends on previous patients’ allocations and responses. Some of these designs are based on urn models to construct randomized procedures for the allocation of the subjects to treatments and to incorporate information on treatment performances. In particular, we consider the class of randomly reinforced urn (RRU) models in which the proportion of subjects allocated to the superior treatment converges to one as the sample size increases to infinity. For this reason, RRU designs are not able to target any desired allocation proportion to satisfy different optimality criteria. To address this issue, we propose a new adaptive randomly reinforced urn (ARRU) design in which the allocation proportion converge to desired values defined as functions of the response distribution. In addition, we establish the convergence rate and a central limit theoremfor the allocation proportion in both the cases of equal and different responses means.