On the speed of convergence of some urn designs for the balanced allocation of two treatments

Abstract

The present paper deals with sequential urn designs for the balanced allocation of two treatments.Wei (1977, 1978b) was among the first authors to suggest an algorithm based on the probabilistic properties of the generalized Friedman urn and recently Chen (2000) has suggested the Ehrenfest design, namely a sequential procedure based on the Ehrenfest urn process. Some extensions of these algorithms are discussed: in particular we focus on a generalization of Chen's procedure, called the Ehrenfest-type urn designs, recently introduced by Baldi Antognini (2004). By analyzing some convergence properties of the Ehrenfest process, we show that for an Ehrenfest-type urn it is possible to evaluate the variance of the design at each step and analyze the convergence to balance. Furthermore, the generalized Friedman urn and the Ehrenfest-type procedures are compared in terms of speed of convergence. The Ehrenfest design proposed by Chen converges to balance faster than the other urn procedures.