CENTRAL LIMIT THEOREM FOR AN ADAPTIVE RANDOMLY REINFORCED URN MODEL

• 成果类型: Article
• 署名作者: Ghiglietti, Andrea; Vidyashankar, Anand N.; Rosenberger, William F.
• 署名单位: University of Milan; George Mason University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/16-AAP1274
• 发表日期: 2017
• 页码: 2956-3003
• 关键词: sequential estimation; asymptotic theorems; branching-processes; polya urns; designs; trials; Allocations

摘要

The generalized Polya urn (GPU) models and their variants have been investigated in several disciplines. However, typical assumptions made with respect to the GPU do not include urn models with a diagonal replacement matrix, which arise in several applications, specifically in clinical trials. To facilitate mathematical analyses of models in these applications, we introduce an adaptive randomly reinforced urn model that uses accruing statistical information to adaptively skew the urn proportion toward specific targets. We study several probabilistic aspects that are important in implementing the urn model in practice. Specifically, we establish the law of large numbers and a central limit theorem for the number of sampled balls. To establish these results, we develop new techniques involving last exit times and crossing time analyses of the proportion of balls in the urn. To obtain precise estimates in these techniques, we establish results on the harmonic moments of the total number of balls in the urn. Finally, we describe our main results in the context of an application to response-adaptive randomization in clinical trials. Our simulation experiments in this context demonstrate the ease and scope of our model.