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Asymptotic Properties of Multicolor Randomly Reinforced Pólya Urns

Part of: Limit theorems Nonparametric inference

Published online by Cambridge University Press:  22 February 2016

Li-Xin Zhang ,
Feifang Hu ,
Siu Hung Cheung  and
Wai Sum Chan
Li-Xin Zhang*
Affiliation:
Zhejiang University
Feifang Hu*
Affiliation:
University of Virginia
Siu Hung Cheung*
Affiliation:
The Chinese University of Hong Kong
Wai Sum Chan*
Affiliation:
The Chinese University of Hong Kong
*
∗ Postal address: Department of Mathematics, Zhejiang University, Yuquan Campus, Hangzhou 310027, PR China. Email address: stazlx@zju.edu.cn
∗∗ Postal address: Department of Statistics, George Washington University, Washington, DC 20052, USA. Email address: feifang@gwu.edu
∗∗∗ Postal address: Department of Statistics, The Chinese University of Hong Kong, Shatin, Hong Kong, PR China. Email address: shcheung@cuhk.edu.hk
∗∗∗∗ Postal address: Department of Finance, The Chinese University of Hong Kong, Shatin, Hong Kong, PR China. Email address: chanws@cuhk.edu.hk
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Abstract

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The generalized Pólya urn has been extensively studied and is widely applied in many disciplines. An important application of urn models is in the development of randomized treatment allocation schemes in clinical studies. The randomly reinforced urn was recently proposed, but, although the model has some intuitively desirable properties, it lacks theoretical justification. In this paper we obtain important asymptotic properties for multicolor reinforced urn models. We derive results for the rate of convergence of the number of patients assigned to each treatment under a set of minimum required conditions and provide the distributions of the limits. Furthermore, we study the asymptotic behavior for the nonhomogeneous case.

Keywords

Response-adaptive designasymptotic normalityclinical trialurn modelbranching process with immigrationrate of convergence

MSC classification

Primary: 60F15: Strong theorems 62G10: Hypothesis testing
Secondary: 60F05: Central limit and other weak theorems 60F10: Large deviations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 2 , June 2014 , pp. 585 - 602
Copyright
© Applied Probability Trust 

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