The Annals of Applied Probability

Asymptotics in randomized urn models

Zhi-Dong Bai and Feifang Hu

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This paper studies a very general urn model stimulated by designs in clinical trials, where the number of balls of different types added to the urn at trial n depends on a random outcome directed by the composition at trials 1,2,…,n−1. Patient treatments are allocated according to types of balls. We establish the strong consistency and asymptotic normality for both the urn composition and the patient allocation under general assumptions on random generating matrices which determine how balls are added to the urn. Also we obtain explicit forms of the asymptotic variance–covariance matrices of both the urn composition and the patient allocation. The conditions on the nonhomogeneity of generating matrices are mild and widely satisfied in applications. Several applications are also discussed.

Article information

Ann. Appl. Probab., Volume 15, Number 1B (2005), 914-940.

First available in Project Euclid: 1 February 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory 62L05: Sequential design
Secondary: 62F12: Asymptotic properties of estimators

Asymptotic normality extended Pólya’s urn models generalized Friedman’s urn model martingale nonhomogeneous generating matrix response-adaptive designs strong consistency


Bai, Zhi-Dong; Hu, Feifang. Asymptotics in randomized urn models. Ann. Appl. Probab. 15 (2005), no. 1B, 914--940. doi:10.1214/105051604000000774.

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  • Altman, D. G. and Royston, J. P. (1988). The hidden effect of time. Statist. Med. 7 629–637.
  • Andersen, J., Faries, D. and Tamura, R. N. (1994). Randomized play-the-winner design for multi-arm clinical trials. Comm. Statist. Theory Methods 23 309–323.
  • Athreya, K. B. and Karlin, S. (1967). Limit theorems for the split times of branching processes. Journal of Mathematics and Mechanics 17 257–277.
  • Athreya, K. B. and Karlin, S. (1968). Embedding of urn schemes into continuous time branching processes and related limit theorems. Ann. Math. Statist. 39 1801–1817.
  • Bai, Z. D. and Hu, F. (1999). Asymptotic theorem for urn models with nonhomogeneous generating matrices. Stochastic Process. Appl. 80 87–101.
  • Bai, Z. D., Hu, F. and Shen, L. (2002). An adaptive design for multi-arm clinical trials. J. Multivariate Anal. 81 1–18.
  • Coad, D. S. (1991). Sequential tests for an unstable response variable. Biometrika 78 113–121.
  • Flournoy, N. and Rosenberger, W. F., eds. (1995). Adaptive Designs. IMS, Hayward, CA.
  • Freedman, D. (1965). Bernard Friedman's urn. Ann. Math. Statist. 36 956–970.
  • Gouet, R. (1993). Martingale functional central limit theorems for a generalized Pólya urn. Ann. Probab. 21 1624–1639.
  • Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, London.
  • Holst, L. (1979). A unified approach to limit theorems for urn models. J. Appl. Probab. 16 154–162.
  • Hu, F. and Rosenberger, W. F. (2000). Analysis of time trends in adaptive designs with application to a neurophysiology experiment. Statist. Med. 19 2067–2075.
  • Hu, F. and Rosenberger, W. F. (2003). Optimality, variability, power: Evaluating response-adaptive randomization procedures for treatment comparisons. J. Amer. Statist. Assoc. 98 671–678.
  • Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Applications. Wiley, New York.
  • Mahmoud, H. M. and Smythe, R. T. (1991). On the distribution of leaves in rooted subtree of recursive trees. Ann. Appl. Probab. 1 406–418.
  • Matthews, P. C. and Rosenberger, W. F. (1997). Variance in randomized play-the-winner clinical trials. Statist. Probab. Lett. 35 193–207.
  • Rosenberger, W. F. (1996). New directions in adaptive designs. Statist. Sci. 11 137–149.
  • Rosenberger, W. F. (2002). Randomized urn models and sequential design (with discussion). Sequential Anal. 21 1–21.
  • Smythe, R. T. (1996). Central limit theorems for urn models. Stochastic Process. Appl. 65 115–137.
  • Wei, L. J. (1979). The generalized Pólya's urn design for sequential medical trials. Ann. Statist. 7 291–296.
  • Wei, L. J. and Durham, S. (1978). The randomized play-the-winner rule in medical trials. J. Amer. Statist. Assoc. 73 840–843.
  • Zelen, M. (1969). Play the winner rule and the controlled clinical trial. J. Amer. Statist. Assoc. 64 131–146.