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Randomly Reinforced Urn Designs with Prespecified Allocations

Part of: Limit theorems Sequential methods

Published online by Cambridge University Press:  30 January 2018

Giacomo Aletti ,
Andrea Ghiglietti  and
Anna Maria Paganoni
Giacomo Aletti*
Affiliation:
Università degli Studi di Milano
Andrea Ghiglietti*
Affiliation:
Politecnico di Milano
Anna Maria Paganoni*
Affiliation:
Politecnico di Milano
*
∗ Postal address: Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, via Saldini 50, 20133 Milano, Italy. Email address: [email protected]
∗∗ Postal address: Dipartimento di Matematica“F. Brioschi”, Politecnico di Milano Piazza Leonardo da Vinci 32, 20133 Milano, Italy.
∗∗ Postal address: Dipartimento di Matematica“F. Brioschi”, Politecnico di Milano Piazza Leonardo da Vinci 32, 20133 Milano, Italy.
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Abstract

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We construct a response adaptive design, described in terms of a two-color urn model, targeting fixed asymptotic allocations. We prove asymptotic results for the process of colors generated by the urn and for the process of its compositions. An application of the proposed urn model is presented in an estimation problem context.

Keywords

Reinforced processurn schemesequential clinical trialstochastic process

MSC classification

Primary: 60F15: Strong theorems
Secondary: 62L12: Sequential estimation
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 2 , June 2013 , pp. 486 - 498
Copyright
© Applied Probability Trust 

References

Aletti, G., May, C. and Secchi, P. (2009). A central limit theorem, and related results, for two-color randomly reinforced urn. Adv. Appl. Prob. 41, 829–844.Google Scholar
Aletti, G., May, C. and Secchi, P. (2012). A functional equation whose unknown is P([0,1]) valued. J. Theoret. Prob. 25, 1207–1232.Google Scholar
Baldi Antognini, A. and Giannerini, S. (2007). Generalized Pólya urn designs with null balance. J. Appl. Prob. 44, 661–669.Google Scholar
Durham, S. D. and Yu, K. F. (1990). Randomized play-the-leader rules for sequential sampling from two populations. Prob. Eng. Inf. Sci. 4, 355–367.Google Scholar
Durham, S. D., Flournoy, N. and Li, W. (1998). A sequential design for maximizing the probability of a response. Canad. J. Statist. 26, 479–495.Google Scholar
Eisele, J. R. and Woodroofe, M. B. (1995). Central limit theorems for doubly adaptive biased coin designs. Ann. Statist. 23, 234–254.Google Scholar
Flournoy, N., May, C. and Secchi, P. (2012). Asymptotically optimal response-adaptive designs for allocating the best treatment: an overview. Internat. Statist. Rev. 80, 293–305.Google Scholar
Hill, B. M., Lane, D. and Sudderth, W. (1980). A strong law for some generalized urn processes. Ann. Prob. 8, 214–226.Google Scholar
Hu, F. and Zhang, L.-X. (2004). Asymptotic proprieties of doubly adaptive biased coin designs for multitreatment clinical trials. Ann. Statist. 32, 268–301.Google Scholar
Hu, F., Zhang, L.-X. and He, X. (2009). Efficient randomized-adaptive designs. Ann. Statist. 37, 2543–2560.Google Scholar
May, C. and Flournoy, N. (2009). Asymptotics in response-adaptive designs generated by a two-color, randomly reinforced urn. Ann. Statist. 37, 1058–1078.Google Scholar
Melfi, V. F., Page, C. and Geraldes, M. (2001). An adaptive randomized design with application to estimation. Canad. J. Statist. 29, 107–116.CrossRefGoogle Scholar
Muliere, P., Paganoni, A. M. and Secchi, P. (2006). A randomly reinforced urn. J. Statist. Planning Infer. 136, 1853–1874.Google Scholar
Rosenberger, W. F. (2002). Randomized urn models and sequential design. Sequent. Anal. 21, 1–41.Google Scholar
Zhang, Li-X., Hu, F. and Cheung, S. H. (2006). Asymptotic theorems of sequential estimation-adjusted urn models. Ann. Appl. Prob. 16, 340–369.Google Scholar