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The Gaussian approximation for generalized Friedman's urn model with heterogeneous and unbalanced updating

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Science China Mathematics 2012年第11期55卷 2379-2404页

作者： ZHANG LiXin Department of Mathematics Zhejiang University Hangzhou 310027 China

The Friedman's urn model is a popular urn model which is widely used in many *** particular,it is extensively used in treatment allocation schemes in clinical *** asymptotic properties have been studied by many *** literature,it is usually assumed that the expected number of balls added at each stage is a constant in despite of what type of balls are selected,that is,the updating of the urn is assumed to be *** it is not,the asymptotic property of the Friedman's urn model is stated in the book of Hu and Rosenberger(2006) as one of open problems in the area of adaptive *** this paper,we show that both the urn composition process and the allocation proportion process can be approximated by a multi-dimensional Gaussian process almost surely for a general multi-color Friedman type urn model with heterogeneous and unbalanced *** Gaussian process is a solution of a stochastic differential *** an application,we obtain the asymptotic properties including the asymptotic normality and the exact law of the iterated logarithm.

关键词： Gaussian approximation the law of iterated logarithm asymptotic normality urn model

The Gaussian approximation for multi-color generalized Friedman’s urn model

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Science China Mathematics 2009年第6期52卷 1305-1326页

作者： ZHANG LiXin HU FeiFang Department of Mathematics Zhejiang UniversityHangzhou 310027China Department of Statistics University of VirginiaHalsey HallCharlottesvilleVirginia 22904-4135USA

The generalized Friedman’s urn model is a popular urn model which is widely used in many *** particular,it is extensively used in treatment allocation schemes in clinical *** this paper,we show that both the urn composition process and the allocation proportion process can be approximated by a multi-dimensional Gaussian process almost surely for a multi-color generalized Friedman’s urn model with both homogeneous and non-homogeneous generating *** Gaussian process is a solution of a stochastic differential *** Gaussian approximation is important for the understanding of the behavior of the urn process and is also useful for statistical *** an application,we obtain the asymptotic properties including the asymptotic normality and the law of the iterated logarithm for a multi-color generalized Friedman's urn model as well as the randomized-play-the-winner rule as a special case.

关键词： strong invariance Gaussian approximation the law of iterated logarithm asymptotic normality urn model randomized play-the-winner rule 60F15 62E20 62L05 60F05 62F12

LIMIT THEOREMS AND OPTIMAL DESIGN WITH ADAPTIVE urn modelS

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Journal of Systems Science & Complexity 2005年第3期18卷 347-360页

作者： CHENGuijing ZHUChunhua WANGYao-hung DepartmentofMathematics AnhuiUniversityHefei230039China DepartmentofFinanceandStatistics UniversityofScienceandTechnologyofChinaHefei230026China DepartmentofStatistics TunghaiUniversityTaichungTaiwan40704

In this paper we study urn model, using some available estimates of successes probabilities, and adding particle parameter, we establish adaptive models. We obtain some strong convergence theorems, rates of convergence, asymptotic normality of components in the urn, and estimates. With these asymptotical results, we show that the adaptive designs given in this paper are asymptotically optimal designs.

关键词： urn model sequential design strong convergence asymptotical normality optimal design