Asymptotics in response-adaptive designs generated by a two-color, randomly reinforced urn

Abstract

This paper illustrates asymptotic properties for a response-adaptive design generated by a two-color, randomly reinforced urn model. The design considered is optimal in the sense that it assigns patients to the best treatment, with probability converging to one. An approach to show the joint asymptotic normality of the estimators of the mean responses to the treatments is provided in spite of the fact that allocation proportions converge to zero and one. Results on the rate of convergence of the number of patients assigned to each treatment are also obtained. Finally, we study the asymptotic behavior of a suitable test statistic.

A central limit theorem, and related results, for a two-color randomly reinforced urn

“…is strictly positive for almost every ω ∈ , as guaranteed by Theorem 3.2 of [9]; hence, the limit distribution of K n is absolutely continuous. Indeed, May and Flournoy [9] proved that equality of the means of the reinforcement distributions implies that P(Z ∞ = 0) = P(Z ∞ = 1) = 0.…”

“…Indeed, May and Flournoy [9] proved that equality of the means of the reinforcement distributions implies that P(Z ∞ = 0) = P(Z ∞ = 1) = 0. In the particular case when µ = ν, the distribution of Z ∞ has no point masses at all; this has been shown in [10].…”

“…[6]). The definition of Z ∞ , Theorem 3.2 of [9], and Theorem 1 imply the existence of ∈ A such that P( ) = 1, and, for all ω ∈ ,…”

“…and, thus, Lemma 5 implies that P(sup n √ n(sup k>n Q k ) = ∞) = 0, which in turn implies, through (9), that P(sup n √ n(sup k>n Z k ) = ∞) = 0. Hence,…”

“…This is an informal description of the randomly reinforced urn (RRU) introduced in [13] and studied in [1], [2], [4], [7]- [10] under various assumptions concerning the reinforcement distributions µ and ν. The urn has an interesting potential for applications since it describes a general model for reinforcement learning (see [2] and [7]); in clinical trials, it implements an optimal response adaptive design (see [5], [9], [11], and [14]). …”