A New Randomized Pólya Urn Model

Affiliation(s)

Université de Montréal, Département de mathématiques et de statistique, Montréal, Canada.

Université de Montréal, Département de mathématiques et de statistique, Montréal, Canada.

ABSTRACT

In this paper, we propose a new class of discrete time stochastic processes generated by a two-color generalized Pólya urn, that is reinforced every time. A single urn contains*a* white balls, *b* black balls and evolves as follows: at discrete times *n*=1,2,…, we sample* M*_{n }balls and note their colors, say* R*_{n }are white and *M*_{n}- R_{n} are black. We return the drawn balls in the urn. Moreover, *N*_{n}R_{n }new white balls and *N*_{n} (*M*_{n}- R_{n})* *new black balls are added in the urn. The numbers* M*_{n }and* N*_{n }are random variables. We show that the proportions of white balls forms a bounded martingale sequence which converges almost surely. Necessary and sufficient conditions for the limit to concentrate on the set {0,1} are given.

In this paper, we propose a new class of discrete time stochastic processes generated by a two-color generalized Pólya urn, that is reinforced every time. A single urn contains

Cite this paper

D. Aoudia and F. Perron, "A New Randomized Pólya Urn Model,"*Applied Mathematics*, Vol. 3 No. 12, 2012, pp. 2118-2122. doi: 10.4236/am.2012.312A292.

D. Aoudia and F. Perron, "A New Randomized Pólya Urn Model,"

References

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[2] N. L. Johnson and S. Kotz, “Urn Models and Their Application,” John Wiley & Sons, New York, 1977.

[3] F. Eggenberger and G. Pólya, “über Die Statistik Verketetter Vorg?ge,” Journal of Applied Mathematics and Mechanics, Vol. 3, No. 4, 1923, pp. 279-289. doi:10.1002/zamm.19230030407

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[6] R. Pemantle, “A Time-Dependent Version of Pòlya’s Urn,” Journal of Theoretical Probability, Vol. 3, No. 4, 1990, pp. 627-637. doi:10.1007/BF01046101

[7] R. Gouet, “A Martingale Approach to Strong Convergence in a Generalized Pólya-Eggenberger Urn Model,” Statistics & Probability Letters, Vol. 8, No. 3, 1993, pp. 225-228. doi:10.1016/0167-7152(89)90126-0

[8] S. Kotz, H. Mahmoud and P. Robert, “On Generalized Pólya Urn Models,” Statistics & Probability Letters, Vol. 49, No. 2, 2000, pp. 163-173. doi:10.1016/S0167-7152(00)00045-6

[9] A. Paganoni and P. Secchi, “A Numerical Study for Comparing Two Response-Adaptive Designs for Continuous Treatment Effects,” Statistical Methods and Applications, Vol. 16, No. 3, 2007, pp. 321-346. doi:10.1007/s10260-006-0042-4

[10] C. May, A. Paganoni and P. Secchi, “On a Two Color Generalized Pólya Urn,” Metron, Vol. 63, 2005, pp. 115-134.

[11] M. Chen and C. Wei, “A New Urn Model,” Journal of Applied Probability, Vol. 42, No. 4, 2005, pp. 964-976. doi:10.1239/jap/1134587809

[12] P. Flajolet, H. Gabbaroó and H. Pekari, “Analytic Urns,” Annals of Probability, Vol. 33, No. 3, 2005, pp. 1200-1233. doi:10.1214/009117905000000026

[13] P. Muliere, A. Paganoni and P. Secchi, “A Randomly Reinforced Urn,” Journal of Statistical Planning and Inference, Vol. 136, No. 6, 2006, pp. 1853-1874. doi:10.1016/j.jspi.2005.08.009

[14] C. May and N. Flournoy, “Asymptotics in Response-Adaptive Designs Generated by a Two-Color, Randomly Reinforced Urn,” Annals of Statistics, Vol. 32, 2010, pp. 1058-1078.

[15] P. Hall and C. Heyde, “Martingale Limit Theory and Its Applications,” Academic Press, New York, 1980.

[16] J. F. C. Kingman, “Uses of Exchangeability,” Annals of Probability, Vol. 6, No. 2, 1978, pp. 183-197. doi:10.1214/aop/1176995566

[17] D. Aldous, “Exchangeability and Related Topics,” école d’été de Probabilités de Saint-Flour, XIII-1983, Lecture Notes in Math 1117, Springer, Berlin, 1985.

[18] F. Hu and W. F. Rosenberger, “The Theory of Response- Adaptive Randomization in Clinical Trials,” John Wiley and Sons, Hoboken, 2006.

[1] H. Mahmoud, “Polya Urn Models,” Chapman & Hall/ CRC Texts in Statistical Science, 2008.

[2] N. L. Johnson and S. Kotz, “Urn Models and Their Application,” John Wiley & Sons, New York, 1977.

[3] F. Eggenberger and G. Pólya, “über Die Statistik Verketetter Vorg?ge,” Journal of Applied Mathematics and Mechanics, Vol. 3, No. 4, 1923, pp. 279-289. doi:10.1002/zamm.19230030407

[4] B. Friedman, “A Simple Urn Model,” Communications on Pure and Applied Mathematics, Vol. 2, No. 1, 1949, pp. 59-70. doi:10.1002/cpa.3160020103

[5] B. Hill, D. Lane and W. Sudderth, “A Strong Law for Some Generalized Urn Process,” Annals of Probablity, Vol. 8, No. 2, 1980, pp. 214-226. doi:10.1214/aop/1176994772

[6] R. Pemantle, “A Time-Dependent Version of Pòlya’s Urn,” Journal of Theoretical Probability, Vol. 3, No. 4, 1990, pp. 627-637. doi:10.1007/BF01046101

[7] R. Gouet, “A Martingale Approach to Strong Convergence in a Generalized Pólya-Eggenberger Urn Model,” Statistics & Probability Letters, Vol. 8, No. 3, 1993, pp. 225-228. doi:10.1016/0167-7152(89)90126-0

[8] S. Kotz, H. Mahmoud and P. Robert, “On Generalized Pólya Urn Models,” Statistics & Probability Letters, Vol. 49, No. 2, 2000, pp. 163-173. doi:10.1016/S0167-7152(00)00045-6

[9] A. Paganoni and P. Secchi, “A Numerical Study for Comparing Two Response-Adaptive Designs for Continuous Treatment Effects,” Statistical Methods and Applications, Vol. 16, No. 3, 2007, pp. 321-346. doi:10.1007/s10260-006-0042-4

[10] C. May, A. Paganoni and P. Secchi, “On a Two Color Generalized Pólya Urn,” Metron, Vol. 63, 2005, pp. 115-134.

[11] M. Chen and C. Wei, “A New Urn Model,” Journal of Applied Probability, Vol. 42, No. 4, 2005, pp. 964-976. doi:10.1239/jap/1134587809

[12] P. Flajolet, H. Gabbaroó and H. Pekari, “Analytic Urns,” Annals of Probability, Vol. 33, No. 3, 2005, pp. 1200-1233. doi:10.1214/009117905000000026

[13] P. Muliere, A. Paganoni and P. Secchi, “A Randomly Reinforced Urn,” Journal of Statistical Planning and Inference, Vol. 136, No. 6, 2006, pp. 1853-1874. doi:10.1016/j.jspi.2005.08.009

[14] C. May and N. Flournoy, “Asymptotics in Response-Adaptive Designs Generated by a Two-Color, Randomly Reinforced Urn,” Annals of Statistics, Vol. 32, 2010, pp. 1058-1078.

[15] P. Hall and C. Heyde, “Martingale Limit Theory and Its Applications,” Academic Press, New York, 1980.

[16] J. F. C. Kingman, “Uses of Exchangeability,” Annals of Probability, Vol. 6, No. 2, 1978, pp. 183-197. doi:10.1214/aop/1176995566

[17] D. Aldous, “Exchangeability and Related Topics,” école d’été de Probabilités de Saint-Flour, XIII-1983, Lecture Notes in Math 1117, Springer, Berlin, 1985.

[18] F. Hu and W. F. Rosenberger, “The Theory of Response- Adaptive Randomization in Clinical Trials,” John Wiley and Sons, Hoboken, 2006.