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Generalized Pólya urn Designs with Null Balance

Part of: Sequential methods Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Alessandro Baldi Antognini  and
Simone Giannerini
Alessandro Baldi Antognini*
Affiliation:
University of Bologna
Simone Giannerini*
Affiliation:
University of Bologna
*
∗Postal address: Dipartimento di Scienze Statistiche, Via delle Belle Arti 41, Bologna 40126, Italy.
∗Postal address: Dipartimento di Scienze Statistiche, Via delle Belle Arti 41, Bologna 40126, Italy.
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Abstract

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In this paper we propose a class of sequential urn designs based on generalized Pólya urn (GPU) models for balancing the allocations of two treatments in sequential clinical trials. In particular, we consider a GPU model characterized by a 2 x 2 random addition matrix with null balance (i.e. null row sums) and replacement rule depending upon the urn composition. Under this scheme, the urn process has a Markovian structure and can be regarded as a random extension of the classical Ehrenfest model. We establish almost sure convergence and asymptotic normality for the frequency of treatment allocations and show that in some peculiar cases the asymptotic variance of the design admits a natural representation based on the set of orthogonal polynomials associated with the corresponding Markov process.

Keywords

Markov chainsequential designgeneralized Pólya urn modelEhrenfest process

MSC classification

Secondary: 62L05: Sequential design 60F05: Central limit and other weak theorems 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 3 , September 2007 , pp. 661 - 669
Copyright
Copyright © Applied Probability Trust 2007 

References

[1] Athreya, K. B. and Karlin, S. (1968). Embedding of urn schemes into continuous time Markov branching processes and related limit theorems. Ann. Math. Statist. 39, 1801–1817.Google Scholar
[2] Bai, Z. and Hu, F. (1999). Asymptotic theorems for urn models with nonhomogeneous generating matrices. Stoch. Process. Appl. 80, 87–101.Google Scholar
[3] Bai, Z. and Hu, F. (2005). Asymptotics in randomized urn models. Ann. Appl. Prob. 15, 914–940.CrossRefGoogle Scholar
[4] Bai, Z., Hu, F. and Zhang, L.-X. (2002). Gaussian approximation theorems for urn models and their applications. Ann. Appl. Prob. 12, 1149–1173.CrossRefGoogle Scholar
[5] Balaji, S., Mahmoud, H. and Watanabe, O. (2006). Distributions in the Ehrenfest process. Statist. Prob. Lett. 76, 666–674.Google Scholar
[6] Baldi Antognini, A. (2005). On the speed of convergence of some urn designs for the balanced allocation of two treatments. Metrika 62, 309–322.CrossRefGoogle Scholar
[7] Brémaud, P. (1999). Markov chains. Gibbs fields, Monte Carlo simulation, and Queues (Texts Appl. Math. 31). Springer, New York.Google Scholar
[8] Chan, K. and Geyer, C. (1994). Discussion: Markov chains for exploring posterior distributions. Ann. Statist. 22, 1747–1758.CrossRefGoogle Scholar
[9] Chen, Y.-P. (2000). Which design is better? Ehrenfest urn versus biased coin. Adv. Appl. Prob. 32, 738–749.Google Scholar
[10] Chen, Y.-P. (2006). A central limit property under a modified Ehrenfest urn design. J. Appl. Prob. 43, 409–420.Google Scholar
[11] Dette, H. (1994). On a generalization of the Ehrenfest urn model. J. Appl. Prob. 31, 930–939.Google Scholar
[12] Flajolet, P., Gabarró, J. and Pekari, H. (2005). Analytic urns. Ann. Prob. 33, 1200–1233.Google Scholar
[13] Garibaldi, U. and Penco, M. (2000). Ehrenfest's urn model generalized: an exact approach for market participation models. Statistica Applicata 12, 249–272.Google Scholar
[14] Gouet, R. (1993). Martingale functional central limit theorems for a generalized Pólya urn. Ann. Prob. 21, 1624–1639.Google Scholar
[15] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.Google Scholar
[16] Higueras, I., Moler, J., Plo, F. and San Miguel, M. (2003). Urn models and differential algebraic equations. J. Appl. Prob. 40, 401–412.Google Scholar
[17] Higueras, I., Moler, J., Plo, F. and San Miguel, M. (2006). Central limit theorems for generalized Pólya urn models. J. Appl. Prob. 43, 938–951.CrossRefGoogle Scholar
[18] Hu, F. and Rosenberger, W. (2003). Optimality, variability, power: evaluating response-adaptive randomization procedures for treatment comparisons. J. Amer. Statist. Assoc. 98, 671–678.Google Scholar
[19] Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stoch. Process. Appl. 110, 177–245.Google Scholar
[20] Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Application. John Wiley, New York.Google Scholar
[21] Krafft, O. and Schaefer, M. (1993). Mean passage times for tridiagonal transition matrices and a two-parameter Ehrenfest urn model. J. Appl. Prob. 30, 964–970.Google Scholar
[22] Muliere, P., Paganoni, A. and Secchi, P. (2006). A randomly reinforced urn. J. Statist. Planning Infer. 136, 1853–1874.Google Scholar
[23] Rosenberger, W. (2002). Randomized urn models and sequential designs. Sequential Anal. 21, 1–28.Google Scholar
[24] Schouten, H. (1995). Adaptive biased urn randomization in small strata when blinding is impossible. Biometrics 51, 1529–1535.Google Scholar
[25] Smythe, R. T. (1996). Central limit theorems for urn models. Stoch. Process. Appl. 65, 115–137.Google Scholar
[26] Wei, L. (1977). A class of designs for sequential clinical trials. J. Amer. Statist. Assoc. 72, 382–386.Google Scholar
[27] Wei, L. (1978). An application of an urn model to the design of sequential controlled clinical trials. J. Amer. Statist. Assoc. 73, 559–563.Google Scholar
[28] Wei, L. (1979). The generalized Pólya's urn design for sequential medical trials. Ann. Statist. 7, 291–296.Google Scholar
[29] Zhang, L.-X., Hu, F. and Cheung, S. H. (2006). Asymptotic theorems of sequential estimation-adjusted urn models. Ann. Appl. Prob. 16, 340–369.Google Scholar