RANDOMIZED URN MODELS REVISITED USING STOCHASTIC APPROXIMATION

• 成果类型: Article
• 署名作者: Laruelle, Sophie; Pages, Gilles
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Sorbonne Universite; Universite Paris Cite
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/12-AAP875
• 发表日期: 2013
• 页码: 1409-1436
• 关键词: multiarm clinical-trials; central limit-theorems; branching-processes; DESIGN; algorithms

摘要

This paper presents the link between stochastic approximation and clinical trials based on randomized urn models investigated by Bai and Hu [Stochastic Process. Appl. 80 (1999) 87-101], Bai and Hu [Ann. Appl. Probab. 15 (2005) 914-940] and Bai, Hu and Shen [J. Multivariate Anal. 81 (2002) 1-18]. We reformulate the dynamics of both the urn composition and the assigned treatments as standard stochastic approximation (SA) algorithms with remainder. Then, we derive the as. convergence and the asymptotic normality [central limit theorem (CLT)] of the normalized procedure under less stringent assumptions by calling upon the ODE and SDE methods. As a second step, we investigate a more involved family of models, known as multi-arm clinical trials, where the urn updating depends on the past performances of the treatments. By increasing the dimension of the state vector, our SA approach provides this time a new asymptotic normality result.