Journal
ANNALS OF APPLIED PROBABILITY
Volume 23, Issue 4, Pages 1409-1436Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/12-AAP875
Keywords
Stochastic approximation; extended Polya urn models; nonhomogeneous generating matrix; strong consistency; asymptotic normality; multi-arm clinical trials; adaptive asset allocation
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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 as. 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.
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