Journal
ANNALS OF APPLIED PROBABILITY
Volume 15, Issue 1B, Pages 914-940Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/105051604000000774
Keywords
asymptotic normality; extended Polya's urn models; generalized Friedman's urn model; martingale; nonhomogeneous generating matrix; response-adaptive designs; strong consistency
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This paper studies a very general urn model stimulated by designs in clinical trials, where the number of balls of different types added to the urn at trial n depends on a random outcome directed by the composition at trials 1, 2,..., n - 1. Patient treatments are allocated according to types of balls. We establish the strong consistency and asymptotic normality for both the urn composition and the patient allocation under general assumptions on random generating matrices which determine how balls are added to the urn. Also we obtain explicit forms of the asymptotic variance-covariance matrices of both the urn composition and the patient allocation. The conditions on the nonhomogencity of generating matrices are mild and widely satisfied in applications. Several applications are also discussed.
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