×

Asymptotic theorems for urn models with nonhomogeneous generating matrices. (English) Zbl 0954.62014

A generalized Friedman’s urn model consisting of particles of \(K\) distinct types is considered in the case that the generating matrices can differ at each consecutive stage. Asymptotic properties of the urn composition after \(n\) steps are considered as \(n \to \infty \) under some assumptions on eigenvalues and eigenvectors of the generating matrices. Especially, consistency and asymptotic normality of the vector characterizing the composition of the urn are established. Examples of practical applications such as adaptive allocation rules in clinical trials and others are given.

MSC:

62E20 Asymptotic distribution theory in statistics
60G42 Martingales with discrete parameter
62L05 Sequential statistical design
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Altman, D. G.; Royston, J. P., The hidden effect of time, Statist. Med., 7, 629-637 (1988)
[2] Athreya, K. B.; Karlin, S., Embedding of urn schemes into continuous time branching processes and related limit theorems, Ann. Math. Statist., 39, 1801-1817 (1968) · Zbl 0185.46103
[3] Athreya, K.B., Ney, P.E., 1972. Branching Processes. Springer, Berlin.; Athreya, K.B., Ney, P.E., 1972. Branching Processes. Springer, Berlin. · Zbl 0259.60002
[4] Bartlett, R. H.; Roloff, D. W.; Cornell, R. G.; Andrews, A. F.; Dillon, P. W.; Zwischenberger, J. B., Extracorporeal circulation in neonatal respiratory failure: a prospective randomized study, Pediatrics, 76, 479-487 (1985)
[5] Coad, D. S., Sequential tests for an unstable response variable, Biometrika, 78, 113-121 (1991) · Zbl 0744.62110
[6] Coad, D. S., A comparative study of some data-dependent allocation rules for Bernoulli data, J. Statist. Comput. Simulation, 40, 219-231 (1992)
[7] Flournoy, N., Rosenberger, W.F. (Eds.), 1995. Adaptive Designs. Institute of Mathematical Statistics, Hayward.; Flournoy, N., Rosenberger, W.F. (Eds.), 1995. Adaptive Designs. Institute of Mathematical Statistics, Hayward.
[8] Hall, P., Heyde, C.C., 1980. Martingale Limit Theory and its Application. Academic Press, London.; Hall, P., Heyde, C.C., 1980. Martingale Limit Theory and its Application. Academic Press, London. · Zbl 0462.60045
[9] Rosenberger, W. F., New directions in adaptive designs, Statist. Sci., 11, 137-149 (1996)
[10] Rosenberger, W. F.; Flournoy, N.; Durham, S. D., Asymptotic normality of maximum likelihood estimators from multiparameter response driven designs, J. Statist. Plann. Inference, 60, 69-76 (1997) · Zbl 0900.62454
[11] Rosenberger, W. F.; Grill, S. E., A sequential design for psychophysical experiments: an application to estimating timing of sensory events, Statist. Med., 16, 2245-2260 (1997)
[12] Rosenberger, W. F.; Sriram, T. N., Estimation for an adaptive allocation design, J. Statist. Plann. Inference, 59, 309-319 (1997) · Zbl 0900.62036
[13] Serfling, R.J., 1980. Approximation Theorems of Mathematical Statistics. Wiley, New York.; Serfling, R.J., 1980. Approximation Theorems of Mathematical Statistics. Wiley, New York. · Zbl 0538.62002
[14] Smythe, R. T., Central limit theorems for urn models, Stochastic Process. Appl., 65, 115-137 (1996) · Zbl 0889.60013
[15] Smythe, R.T., Rosenberger, W.F., 1995. Play-the winner designs, generalized Pólya’s urns, and Markov branching processes. In: Flournoy, N., Rosenberger, W.F. (Eds.), Adaptive Designs. Institute of Mathematical Statistics, Hayward, CA, pp. 13-22.; Smythe, R.T., Rosenberger, W.F., 1995. Play-the winner designs, generalized Pólya’s urns, and Markov branching processes. In: Flournoy, N., Rosenberger, W.F. (Eds.), Adaptive Designs. Institute of Mathematical Statistics, Hayward, CA, pp. 13-22. · Zbl 0949.62578
[16] Tamura, R. N.; Faries, D. E.; Andersen, J. S.; Heiligenstein, J. H., A case study of an adaptive clinical trial in the treatment of out-patients with depressive disorder, J. Amer. Statist. Assoc., 89, 768-776 (1994)
[17] Wei, L. J., The generalized Pólya’s urn design for sequential medical trials, Ann. Statist., 7, 291-296 (1979) · Zbl 0399.62083
[18] Wei, L. J.; Durham, S., The randomized play-the-winner rule in medical trials, J. Amer. Statist. Assoc., 73, 840-843 (1978) · Zbl 0391.62076
[19] Wei, L. J.; Smythe, R. T.; Lin, D. Y.; Park, T. S., Statistical inference with data-dependent treatment allocation rules, J. Amer. Statist. Assoc., 85, 156-162 (1990)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.