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Generalized Pólya urn Designs with Null Balance

Part of: Sequential methods Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

Alessandro Baldi Antognini  and
Simone Giannerini
Alessandro Baldi Antognini*
Affiliation:
University of Bologna
Simone Giannerini*
Affiliation:
University of Bologna
*
∗Postal address: Dipartimento di Scienze Statistiche, Via delle Belle Arti 41, Bologna 40126, Italy.
∗Postal address: Dipartimento di Scienze Statistiche, Via delle Belle Arti 41, Bologna 40126, Italy.
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Abstract

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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 x 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.

Keywords

Markov chainsequential designgeneralized Pólya urn modelEhrenfest process

MSC classification

Secondary: 62L05: Sequential design 60F05: Central limit and other weak theorems 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 3 , September 2007 , pp. 661 - 669
Copyright
Copyright © Applied Probability Trust 2007 

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