Randomly Reinforced Urn Designs with Prespecified Allocations

Abstract

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.

Networks of reinforced stochastic processes: Asymptotics for the empirical means

“…These convergence results are also discussed from the point of view of the statistical applications. In particular, they lead to the construction of asymptotic confidence intervals for the common limit random variable Z ∞ based on the random variables X n,j through the empirical means (4), that specifically require neither the knowledge of the initial random variables {Z 0,j : j ∈ V } nor of the exact expression of the sequence (r n ) n . For the case γ = 1, that for instance includes the case of interacting standard Pólya's urns, we also provide a statistical test, based on the random variables X n,j through the empirical means (4), to make inference on the weighted adjacency matrix W of the network.…”

“…In particular, they lead to the construction of asymptotic confidence intervals for the common limit random variable Z ∞ based on the random variables X n,j through the empirical means (4), that specifically require neither the knowledge of the initial random variables {Z 0,j : j ∈ V } nor of the exact expression of the sequence (r n ) n . For the case γ = 1, that for instance includes the case of interacting standard Pólya's urns, we also provide a statistical test, based on the random variables X n,j through the empirical means (4), to make inference on the weighted adjacency matrix W of the network. The fact that the confidence intervals and the inferential procedures presented in this work are based on X n,j , instead of Z n,j as done in [2], represents a great improvement in any area of application, since the "actions" X n,j adopted by the agents of the network are much more likely to be observed than their "personal inclinations" Z n,j of adopting these actions.…”

Asymptotic Properties of Multicolor Randomly Reinforced Pólya Urns

“…n+1,k = Y (1) n,k +X n+1,k U (1) n+1,k with Y (1) 0,k = Y 0,k , U (2) n,k = U n,k −U (1) n,k , and Y (2) n,k = Y n,k −Y (1) n,k . Then Y (2) n+1,k = Y (2) n,k +X n+1,k U (2) n+1,k with Y (2) 0,k = 0.…”

“…According to the Borel-Cantelli lemma, P(U (2) n,k = 0 infinitely often) = 0. It follows that Y (2) n,k = O (1) and Y n,k = Y (1) n,k + O(1) a.s. Hence,…”

“…For many adaptive designs in the literature, the proportion of patients allocated to each treatment converges to a limit in (0, 1). Besides rare exceptions such as the response adaptive design of Aletti et al [1] that a two-color RRPU can target fixed asymptotic allocations (see also [17]), a design driven by the RRPU usually allocates patients in an optimal manner so that the proportion of patients assigned to the best treatment converges to 1. However, it is important to know the (expected) number of patients in each treatment when the statistical test for the treatment differences and the power of the test are considered.…”