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October 2017 Central limit theorem for an adaptive randomly reinforced urn model
Andrea Ghiglietti, Anand N. Vidyashankar, William F. Rosenberger
Ann. Appl. Probab. 27(5): 2956-3003 (October 2017). DOI: 10.1214/16-AAP1274


The generalized Pólya 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.


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Andrea Ghiglietti. Anand N. Vidyashankar. William F. Rosenberger. "Central limit theorem for an adaptive randomly reinforced urn model." Ann. Appl. Probab. 27 (5) 2956 - 3003, October 2017.


Received: 1 February 2015; Revised: 1 October 2016; Published: October 2017
First available in Project Euclid: 3 November 2017

zbMATH: 1379.60025
MathSciNet: MR3719951
Digital Object Identifier: 10.1214/16-AAP1274

Primary: 60E20 , 60F05 , 60F15 , 60G99
Secondary: 68Q87 , 97K50

Keywords: Clinical trials , crossing times , generalized Pólya urn , harmonic moments , last exit times , target allocation

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.27 • No. 5 • October 2017
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