The Annals of Statistics

Immigrated urn models—theoretical properties and applications

Li-Xin Zhang, Feifang Hu, Siu Hung Cheung, and Wai Sum Chan

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Statist., Volume 39, Number 1 (2011), 643-671.

Dates
First available in Project Euclid: 15 February 2011

Permanent link to this document
https://projecteuclid.org/euclid.aos/1297779859

Digital Object Identifier
doi:10.1214/10-AOS851

Mathematical Reviews number (MathSciNet)
MR2797859

Zentralblatt MATH identifier
1226.60012

Subjects
Primary: 60F15: Strong theorems 62G10: Hypothesis testing
Secondary: 60F05: Central limit and other weak theorems 60F10: Large deviations

Keywords
Adaptive designs asymptotic normality clinical trial urn model branching process with immigration birth and death urn drop-the-loser rule

Citation

Zhang, Li-Xin; Hu, Feifang; Cheung, Siu Hung; Chan, Wai Sum. Immigrated urn models—theoretical properties and applications. Ann. Statist. 39 (2011), no. 1, 643--671. doi:10.1214/10-AOS851. https://projecteuclid.org/euclid.aos/1297779859


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References

  • 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.
  • Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
  • 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. and Hu, F. (2005). Asymptotics in randomized urn models. Ann. Appl. Probab. 15 914–940.
  • Bai, Z. D., Hu, F. and Rosenberger, W. F. (2002). Asymptotic properties of adaptive designs for clinical trials with delayed response. Ann. Statist. 30 122–139.
  • Bai, Z. D., Hu, F. and Shen, L. (2002). An adaptive design for multi-arm clinical trials. J. Multivariate Anal. 81 1–18.
  • Bhattacharya, R. (2008). Urn-based response adaptive procedures and optimality. Drug Information Journal 42 441–448.
  • Beggs, A. W. (2005). On the convergence of reinforcement learning. J. Econom. Theory 122 1–36.
  • Benaïm, M., Schreiber, S. J. and Tarrès, P. (2004). Generalized urn models of evolutionary processes. Ann. Appl. Probab. 14 1455–1478.
  • Donnelly, P. and Kurtz, T. G. (1996). The asymptotic behavior of an urn model arising in population genetics. Stochastic Process. Appl. 64 1–16.
  • Durham, S. D., Flournoy, N. and Li, W. (1998). Sequential designs for maximizing the probability of a favorable response. Canad. J. Statist. 26 479–495.
  • Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and its Applications. Academic Press, New York.
  • Higueras, I., Moler, J., Flo, F. and San Miguel, M. (2006). Central limit theorems for generalized Pólya urn models. J. Appl. Probab. 43 938–951.
  • Hoppe, F. M. (1984). Pólya-like urns and the Ewens’ sampling formula. J. Math. Biol. 20 91–94.
  • 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.
  • Hu, F. and Rosenberger, W. F. (2006). The Theory of Response-Adaptive Randomization in Clinical Trials. Wiley, Hoboken, NJ.
  • Hu, F., Rosenberger, W. F. and Zhang, L.-X. (2006). Asymptotically best response-adaptive randomization procedures. J. Statist. Plann. Inference 136 1911–1922.
  • Hu, F. and Zhang, L. X. (2004a). Asymptotic properties of doubly adaptive biased coin designs for multi-treatment clinical trials. Ann. Statist. 32 268–301.
  • Hu, F. and Zhang, L.-X. (2004b). Asymptotic normality of urn models for clinical trials with delayed response. Bernoulli 10 447–463.
  • Ivanova, A. (2003). A play-the-winner type urn model with reduced variability. Metrika 58 1–13.
  • Ivanova, A. (2006). Urn designs with immigration: Useful connection with continuous time stochastic processes. J. Statist. Plann. Inference 136 1836–1844.
  • Ivanova, A. and Flournoy, N. (2001). A birth and death urn for ternary outcomes: Stochastic processes applied to urn models. In Probability and Statistical Models with Applications (C. A. Charalambides, M. V. Koutras and N. Balakrishnan, eds.) 583–600. Chapman and Hall/CRC Press, Boca Raton, FL.
  • Ivanova, A., Rosenberger, W. F., Durham, S. D. and Flournoy, N. (2000). A birth and death urn for randomized clinical trials: Asymptotic methods. Sankhyā Ser. B 62 104–118.
  • Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl. 110 177–245.
  • Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Applications. Wiley, New York.
  • Kotz, S. and Balakrishnan, N. (1997). Advances in urn models during the past two decades. In Advances in Combinatorial Methods and Applications to Probability and Statistics (N. Balakrishnan, ed.) 203–257. Birkhäuser, Boston.
  • Knoblauch, K., Neitz, M. and Neitz, J. (2006). An urn model of the development of L/M cone ratios in human and macaque retinas. Visual Neuroscience 23 387–394.
  • Matthews, E. E., Cook, P. F., Terada, M. and Aloia, M. S. (2010). Randomizing research participants: Promoting balance and concealment in small samples. Research in Nursing and Health 33 243–253.
  • Milenkovic, O. and Compton, K. J. (2004). Probabilistic transforms for combinatorial urn models. Combin. Probab. Comput. 13 645–675.
  • Niven, R. K. and Grendar, M (2009). Generalized classical, quantum and intermediate statistics and the Pólya urn model. Phys. Lett. A 373 621–626.
  • Rosenberger, W. F., Stallard, N., Ivanova, A., Harper, C. N. and Ricks, M. L. (2001). Optimal adaptive designs for binary response trials. Biometrics 57 909–913.
  • Smythe, R. T. (1996). Central limit theorems for urn models. Stochastic Process. Appl. 65 115–137.
  • Stout, W. F. (1974). Almost Sure Convergence. Academic Press, New York.
  • Tamura, R. N., Faries, D. E., Andersen, J. S. and Heiligenstein, J. H. (1994). A case study of an adaptive clinical trial in the treatment of out-patients with depressive disorder. J. Amer. Statist. Assoc. 89 768–776.
  • Tymofyeyev, Y., Rosenberger, W. F. and Hu, F. (2007). Implementing optimal allocation in sequential binary response experiments. J. Amer. Statist. Assoc. 102 224–234.
  • 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. D. (1978). The randomized play-the-winner rule in medical trials. J. Amer. Statist. Assoc. 73 840–843.
  • Zhang, L. J. and Rosenberger, W. F. (2006). Response-adaptive randomization for clinical trials with continuous outcomes. Biometrics 62 562–569.
  • Zhang, L.-X. (2004). Strong approximations of martingale vectors and their applications in Markov-chain adaptive designs. Acta Math. Appl. Sin. Engl. Ser. 20 337–352.
  • Zhang, L.-X. and Hu, F. (2009). The Gaussian approximation for multi-color generalized Friedman’s urn model. Sci. China Ser. A 52 1305–1326.
  • Zhang, L.-X., Chan, W. S., Cheung, S. H. and Hu, F. (2007). A generalized urn model for clinical trials with delayed responses. Statist. Sinica 17 387–409.
  • Zhang, L.-X., Hu, F. and Cheung, S. H. (2006). Asymptotic theorems of sequential estimation-adjusted urn models. Ann. Appl. Probab. 16 340–369.
  • Zhu, H. and Hu, F. (2009). Implementing optimal allocation in sequential continuous response experiments. J. Statist. Plann. Inference 139 2420–2430.