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Generalized Pólya urn Designs with Null Balance

Published online by Cambridge University Press:  14 July 2016

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.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2007 

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