Asymptotic theorems of sequential estimation-adjusted urn models

Zhang, Li Xin; Hu, Feifang; Cheung, Siu Hung

doi:10.1214/105051605000000746

Cited by 55 publications

(67 citation statements)

References 31 publications

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“…For the delayed Friedman's urn problem, Hu and Zhang (2004) prove the asymptotic normality of the fraction assigned to each choice. In Zhang et al (2006), the authors derive the asymptotic properties for the sequential estimation Generalized Polya Urn (GPU). For the delayed GPU, Crimaldi and Leisen (2008) derive asymptotic results.…”

Section: Incorporation Of Lengths Of Paths In the K-path Resultsmentioning

confidence: 99%

Trail formation in ants. A generalized Polya urn process

et al. 2010

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Faced with a choice of paths, an ant chooses a path with a higher concentration of pheromone. Subsequently, it drops pheromone on the path chosen. The reinforcement of the pheromone-following behavior favors the selection of an initially discovered path as the preferred path. This may cause a long path to emerge as the preferred path, were it discovered earlier than a shorter path. However, the shortness of the shorter path offsets some of the pheromone accumulated on the initially discovered longer path. In this paper, we model the trail formation behavior as a generalized Polya urn process. For k equal length paths, we give the distribution of pheromone at any time and highlight its sole dependence on the initial pheromone concentrations on paths. Additionally, we propose a method to incorporate the lengths of paths in the urn process and derive how the pheromone distribution alters on its inclusion. Analytically, we show that it is possible, under certain conditions, to reverse the initial bias that may be present in favor of paths that were discovered prior to the discovery of more efficient (shorter) paths. This addresses the Plasticity-Stability dilemma for ants, by laying out the conditions under which the system will remain stable or become plastic and change the path. Finally, we validate our analysis and results using simulations.

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Section: Incorporation Of Lengths Of Paths In the K-path Resultsmentioning

confidence: 99%

Trail formation in ants. A generalized Polya urn process

et al. 2010

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“…Also known in the literature as estimation-adjusted urn models, these adaptive designs involve updating rules of the urn which rely on the estimation of some parameters based on data accrued so far. Recently, Zhang, Hu, and Cheung (2006) established the consistency and asymptotic normality for both the randomization scheme and the estimator of the parameter of interest in this challenging framework, under weak assumptions. Targeting a specific user-supplied optimal unknown choice of randomization scheme is at the core of adaptive designs based on flipping a (data-adaptively) biased coin.…”

Section: Bibliographymentioning

confidence: 99%

Targeting the Optimal Design in Randomized Clinical Trials with Binary Outcomes and No Covariate: Theoretical Study

Chambaz

Laan²

2011

The International Journal of Biostatistics

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This article is devoted to the asymptotic study of adaptive group sequential designs in the case of randomized clinical trials (RCTs) with binary treatment, binary outcome and no covariate. By adaptive design, we mean in this setting a RCT design that allows the investigator to dynamically modify its course through data-driven adjustment of the randomization probability based on data accrued so far, without negatively impacting on the statistical integrity of the trial. By adaptive group sequential design, we refer to the fact that group sequential testing methods can be equally well applied on top of adaptive designs. We obtain that, theoretically, the adaptive design converges almost surely to the targeted unknown randomization scheme. In the estimation framework, we obtain that our maximum likelihood estimator of the parameter of interest is a strongly consistent estimator, and it satisfies a central limit theorem. We can estimate its asymptotic variance, which is the same as that it would feature had we known in advance the targeted randomization scheme and independently sampled from it. Consequently, inference can be carried out as if we had resorted to independent and identically distributed (iid) sampling. In the testing framework, we obtain that the multidimensional t-statistic that we would use under iid sampling still converges to the same canonical distribution under adaptive sampling. Consequently, the same group sequential testing can be carried out as if we had resorted to iid sampling.Furthermore, a comprehensive simulation study that we undertake in a companion article validates the theory.A three-sentence take-home message is "Adaptive designs do learn the targeted optimal design and inference, and testing can be carried out under adaptive sampling as they would under the targeted optimal randomization probability iid sampling. In particular, adaptive designs achieve the same efficiency as the fixed oracle design. This is confirmed by a simulation study, at least for moderate or large sample sizes, across a large collection of targeted randomization probabilities.'"

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“…For instance, in response-adaptive experiments the adding rule depends at each step on the observed responses (see, e.g. [17], [28], and [29]). Note that, if R n is random, a key quantity that governs the evolution of the process is the generating matrix…”

Section: The Random Addition Matrixmentioning

confidence: 99%

“…Other recent contributions include [2], [3], [4], [17], and [29], where the asymptotic normality of {W n } n∈N and {N n } n∈N is derived for nonhomogeneous GPU models by assuming that the total number of balls added at each step is a positive constant, namely that 2…”

Section: The Random Addition Matrixmentioning

confidence: 99%

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Generalized Pólya urn Designs with Null Balance

Antognini

Giannerini

2007

J. Appl. Probab.

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In this paper we propose a class of sequential urn designs based on generalized Pólya urn (GPU) models for balancing the allocations of two treatments in sequential clinical trials. In particular, we consider a GPU model characterized by a 2 × 2 random addition matrix with null balance (i.e. null row sums) and replacement rule depending upon the urn composition. Under this scheme, the urn process has a Markovian structure and can be regarded as a random extension of the classical Ehrenfest model. We establish almost sure convergence and asymptotic normality for the frequency of treatment allocations and show that in some peculiar cases the asymptotic variance of the design admits a natural representation based on the set of orthogonal polynomials associated with the corresponding Markov process.

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Asymptotic theorems of sequential estimation-adjusted urn models

Cited by 55 publications

References 31 publications

Trail formation in ants. A generalized Polya urn process

Trail formation in ants. A generalized Polya urn process

Targeting the Optimal Design in Randomized Clinical Trials with Binary Outcomes and No Covariate: Theoretical Study

Generalized Pólya urn Designs with Null Balance

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