Asymptotic Properties of an Adaptive Randomly Reinforced Urn Model

Abstract

In the design of experiments, urn models have been widely used as randomization devices to allocate subjects to treatments and incorporate ethical constraints. We propose a new adaptive randomly reinforced urn design, in a clinical trial context. The design consists of a randomly reinforced urn wherein a sequential allocation of patients to treatments is performed and the associated responses are collected. The model is based on two stochastic sequences representing random and time-dependent thresholds for the urn proportion process. These thresholds are defined as functions of the estimators of unknown parameters modeling the response distributions, so that they change any time a new response is collected. First and second-order asymptotic results under different conditions have been investigated. Specifically, we present the limit, the rate of convergence and the asymptotic distribution of the proportion of subjects assigned to the treatments.