Immigrated urn models—theoretical properties and applications

Urn models have been widely studied and applied in both scientific and social science disciplines. In clinical studies, the adoption of urn models in treatment allocation schemes has proved to be beneficial to researchers, by providing more efficient clinical trials, and to patients, by increasing the likelihood of receiving the better treatment. In this paper, we propose a new and general class of immigrated urn (IMU) models that incorporates the immigration mechanism into the urn process. Theoretical properties are developed and the advantages of the IMU models are discussed. In general, the IMU models have smaller variabilities than the classical urn models, yielding more powerful statistical inferences in applications. Illustrative examples are presented to demonstrate the wide applicability of the IMU models. The proposed IMU framework, including many popular classical urn models, not only offers a unify perspective for us to comprehend the urn process, but also enables us to generate several novel urn models with desirable properties.

[1]  Paul Windrum,et al.  Leveraging technological externalities in complex technologies: Microsoft’s exploitation of standards in the browser wars , 2004 .

[2]  Julio M. Singer,et al.  Central Limit Theorems , 2011, International Encyclopedia of Statistical Science.

[3]  Anastasia Ivanova,et al.  A play-the-winner-type urn design with reduced variability , 2003 .

[4]  A. Slinko,et al.  Exploratory Analysis of Similarities Between Social Choice Rules , 2006 .

[5]  I. Higueras,et al.  Central limit theorems for generalized Pólya urn models , 2006, Journal of Applied Probability.

[6]  W. Rosenberger,et al.  The theory of response-adaptive randomization in clinical trials , 2006 .

[7]  Robert T. Smythe,et al.  Central limit theorems for urn models , 1996 .

[8]  Svante Janson,et al.  Functional limit theorems for multitype branching processes and generalized Pólya urns , 2004 .

[9]  Feifang Hu,et al.  Asymptotic theorems of sequential estimation-adjusted urn models , 2006 .

[10]  L. J. Wei,et al.  The Randomized Play-the-Winner Rule in Medical Trials , 1978 .

[11]  F. Hu,et al.  The Gaussian approximation for multi-color generalized Friedman’s urn model , 2008, 0812.3697.

[12]  Feifang Hu,et al.  Asymptotic normality of urn models for clinical trials with delayed response , 2004 .

[13]  N Stallard,et al.  Optimal Adaptive Designs for Binary Response Trials , 2001, Biometrics.

[14]  M. Aloia,et al.  Randomizing research participants: promoting balance and concealment in small samples. , 2010, Research in nursing & health.

[15]  Nancy Flournoy,et al.  A Birth and Death Urn for Ternary Outcomes: Stochastic Processes Applied to Urn Models , 2000 .

[16]  F. Hu,et al.  An Adaptive Design for Multi-Arm Clinical Trials , 2002 .

[17]  Feifang Hu,et al.  Implementing optimal allocation for sequential continuous responses with multiple treatments , 2009 .

[18]  Jay Neitz,et al.  An urn model of the development of L/M cone ratios in human and macaque retinas , 2006, Visual Neuroscience.

[19]  A. Ivanova.  Urn designs with immigration: Useful connection with continuous time stochastic processes , 2006 .

[20]  William F Rosenberger,et al.  Response-adaptive randomization for clinical trials with continuous outcomes. , 2006, Biometrics.

[21]  Feifang Hu,et al.  Optimal Adaptive Designs for Binary Response Trials With Three Treatments , 2010 .

[22]  D. A. Sprott.  Urn Models and Their Application—An Approach to Modern Discrete Probability Theory , 1978 .

[23]  William F. Rosenberger,et al.  Implementing Optimal Allocation in Sequential Binary Response Experiments , 2007 .

[24]  Sebastian J. Schreiber,et al.  Generalized URN models of evolutionary processes , 2004 .

[25]  Feifang Hu,et al.  Asymptotics in randomized urn models , 2005 .

[26]  P. Hall,et al.  Martingale Limit Theory and Its Application , 1980 .

[27]  Samuel Kotz,et al.  Advances in Urn Models during the Past Two Decades , 1997 .

[28]  S. D. Durham,et al.  A sequential design for maximizing the probability of a favourable response , 1998 .

[29]  Douglas E. Faries,et al.  A Case Study of an Adaptive Clinical Trial in the Treatment of Out-Patients with Depressive Disorder , 1994 .

[30]  William F. Rosenberger,et al.  Optimality, Variability, Power , 2003 .

[31]  Olgica Milenkovic,et al.  Probabilistic Transforms for Combinatorial Urn Models , 2004, Combinatorics, Probability and Computing.

[32]  Norman L. Johnson,et al.  Urn models and their application , 1977 .

[33]  Robert K. Niven,et al.  Generalized classical, quantum and intermediate statistics and the Pólya urn model , 2009 .

[34]  L. J. Wei,et al.  The Generalized Polya's Urn Design for Sequential Medical Trials , 1979 .

[35]  Peter Donnelly,et al.  The asymptotic behavior of an urn model arising in population genetics , 1996 .

[36]  G. Pólya,et al.  Über die Statistik verketteter Vorgänge , 1923 .

[37]  F. Hu,et al.  A GENERALIZED DROP-THE-LOSER URN FOR CLINICAL TRIALS WITH DELAYED RESPONSES , 2007 .

[38]  Feifang Hu,et al.  Asymptotic theorems for urn models with nonhomogeneous generating matrices , 1999 .

[39]  Feifang Hu,et al.  Asymptotic properties of doubly adaptive biased coin designs for multitreatment clinical trials , 2003 .

[40]  K. Athreya,et al.  Embedding of Urn Schemes into Continuous Time Markov Branching Processes and Related Limit Theorems , 1968 .

[41]  Alan W. Beggs,et al.  On the convergence of reinforcement learning , 2005, J. Econ. Theory.

[42]  William F. Rosenberger,et al.  Asymptotically best response-adaptive randomization procedures , 2006 .

[43]  W. Stout.  Almost sure convergence , 1974 .

[44]  Rahul Bhattacharya,et al.  Urn-Based Response Adaptive Procedures and Optimality , 2008 .

[45]  F. Hoppe.  Pólya-like urns and the Ewens' sampling formula , 1984 .

[46]  A. Roth,et al.  Predicting How People Play Games: Reinforcement Learning in Experimental Games with Unique, Mixed Strategy Equilibria , 1998 .