The Annals of Applied Probability

Randomized urn models revisited using stochastic approximation

Sophie Laruelle and Gilles Pagès

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Abstract

This paper presents the link between stochastic approximation and clinical trials based on randomized urn models investigated by Bai and Hu [Stochastic Process. Appl. 80 (1999) 87–101], Bai and Hu [Ann. Appl. Probab. 15 (2005) 914–940] and Bai, Hu and Shen [J. Multivariate Anal. 81 (2002) 1–18]. We reformulate the dynamics of both the urn composition and the assigned treatments as standard stochastic approximation (SA) algorithms with remainder. Then, we derive the a.s. convergence and the asymptotic normality [central limit theorem (CLT)] of the normalized procedure under less stringent assumptions by calling upon the ODE and SDE methods. As a second step, we investigate a more involved family of models, known as multi-arm clinical trials, where the urn updating depends on the past performances of the treatments. By increasing the dimension of the state vector, our SA approach provides this time a new asymptotic normality result.

Article information

Source
Ann. Appl. Probab., Volume 23, Number 4 (2013), 1409-1436.

Dates
First available in Project Euclid: 21 June 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1371834033

Digital Object Identifier
doi:10.1214/12-AAP875

Mathematical Reviews number (MathSciNet)
MR3098437

Zentralblatt MATH identifier
06205797

Subjects
Primary: 62L20: Stochastic approximation 62E20: Asymptotic distribution theory 62L05: Sequential design
Secondary: 62F12: Asymptotic properties of estimators 62P10: Applications to biology and medical sciences

Keywords
Stochastic approximation extended Pólya urn models nonhomogeneous generating matrix strong consistency asymptotic normality multi-arm clinical trials adaptive asset allocation

Citation

Laruelle, Sophie; Pagès, Gilles. Randomized urn models revisited using stochastic approximation. Ann. Appl. Probab. 23 (2013), no. 4, 1409--1436. doi:10.1214/12-AAP875. https://projecteuclid.org/euclid.aoap/1371834033


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