Randomized Urn Models revisited using Stochastic Approximation

Sophie Laruelle (1) , Gilles Pagès (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

This paper presents the link between stochastic approximation and clinical trials based on randomized urn models investigated in Bai and Hu (1999,2005) and Bai, Hu and Shen (2002). 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 (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.

Keywords: strong consistency; multi-arm clinical trials; asymptotic normality; Stochastic approximation; extended Pòlya urn models; nonhomogeneous generating matrix

Domains: Probability [math.PR]

Sophie Laruelle, Gilles Pagès. Randomized Urn Models revisited using Stochastic Approximation. The Annals of Applied Probability, 2013, 23 (4), pp.1409-1436. ⟨hal-00555752v6⟩