Stein’s method for the beta distribution and the Pólya-Eggenberger urn. (English) Zbl 1304.60033
The scaled number of white balls in the Pólya-Eggenberger urn model is approximated by the beta distribution via the Stein method. Optimal order bounds are obtained for the Wasserstein distance. For the proof, the Stein operator for ball statistics is written in a form similar to the Stein operator for the beta distribution. This allows to prove estimates containing the derivative of the solution to the characterizing Stein equation for the beta distribution.
Reviewer: Jurgita Markeviciute (Vilnius)
MSC:
| 60F05 | Central limit and other weak theorems |
| 62E20 | Asymptotic distribution theory in statistics |
| 60K99 | Special processes |
Software:
DLMFReferences:
| [1] | Argiento, R., Pemantle, R., Skyrms, B. and Volkov, S. (2009). Learning to signal: analysis of a micro-level reinforcement model. Stoch. Process. Appl. 119, 373-390. · Zbl 1166.60044 · doi:10.1016/j.spa.2008.02.014 |
| [2] | Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: the Chen-Stein method. Ann. Prob. 17, 9-25. · Zbl 0675.60017 · doi:10.1214/aop/1176991491 |
| [3] | Barbour, A. D. (1990). Stein’s method for diffusion approximations. Prob. Theory Relat. Fields 84, 297-322. · Zbl 0665.60008 · doi:10.1007/BF01197887 |
| [4] | Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation . Oxford University Press, New York. · Zbl 0746.60002 |
| [5] | Brown, T. C. and Phillips, M. J. (1999). Negative binomial approximation with Stein’s method. Methodology Comput. Appl. Prob. 1, 407-421. · Zbl 0992.62015 · doi:10.1023/A:1010094221471 |
| [6] | Chatterjee, S., Fulman, J. and Röllin, A. (2011). Exponential approximation by Stein’s method and spectral graph theory. ALEA Lat. Amer. J. Prob. Math. Statist. 8, 197-223. · Zbl 1276.60010 |
| [7] | Chauvin, B., Pouyanne, N. and Sahnoun, R. (2011). Limit distributions for large Pólya urns. Ann. Appl. Prob. 21, 1-32. · Zbl 1220.60006 · doi:10.1214/10-AAP696 |
| [8] | Chen, L. H. Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein’s Method. Springer, Heidelberg. · Zbl 1213.62027 |
| [9] | Chu, W. (2007). Abel’s lemma on summation by parts and basic hypergeometric series. Adv. Appl. Math. 39, 490-514. · Zbl 1131.33008 · doi:10.1016/j.aam.2007.02.001 |
| [10] | Chung, F., Handjani, S. and Jungreis, D. (2003). Generalizations of Pólya’s urn problem. Ann. Combinatorics 7, 141-153. · Zbl 1022.60005 · doi:10.1007/s00026-003-0178-y |
| [11] | Döbler, C. (2012). A rate of convergence for the arcsine law by Stein’s method. Preprint. Available at http://arxiv.org/abs/1207.2401v1. |
| [12] | Döbler, C. (2012). Stein’s method of exchangeable pairs for absolutely continuous, univariate distributions with applications to the Polya urn model. Preprint. Available at http://arxiv.org/abs/1207.0533v2. |
| [13] | Durrett, R. (2007). Random Graph Dynamics . Cambridge University Press. · Zbl 1116.05001 |
| [14] | Eichelsbacher, P. and Reinert, G. (2008). Stein’s method for discrete Gibbs measures. Ann. Appl. Prob. 18, 1588-1618. · Zbl 1146.60011 · doi:10.1214/07-AAP0498 |
| [15] | Feller, W. (1970). An Introduction to Probability Theory and Its Applications . Vol. I. John Wiley, New York. · Zbl 0039.13201 |
| [16] | Fisher, R. A. (1930). The Genetical Theory of Natural Selection . Clarendon Press, Oxford. · JFM 56.1106.13 |
| [17] | Gibbs, A. L. and Su, F. E. (2002). On choosing and bounding probability metrics. Internat. Statist. Rev. 70, 419-435. · Zbl 1217.62014 · doi:10.2307/1403865 |
| [18] | Holmes, S. (2004). Stein’s method for birth and death chains. In Stein’s Method: Expository Lectures and Applications , eds P. Diaconis and S. Holmes, Institute of Mathematical Statistics, Beachwood, OH, pp. 45-67. |
| [19] | Ley, C. and Swan, Y. (2013). Local Pinsker inequalities via Stein’s discrete density approach. IEEE Trans. Inf. Theory 59, 5584-5591. · Zbl 1364.94243 · doi:10.1109/TIT.2013.2265392 |
| [20] | Luk, H.-M. (1994). Stein’s method for the gamma distribution and related statistical applications. Doctoral Thesis, University of Southern California. |
| [21] | Mahmoud, H. M. (2003). Pólya urn models and connections to random trees: a review. J. Iranian Statist. Soc. 2, 53-114. · Zbl 1499.60020 |
| [22] | Mahmoud, H. M. (2008). Pólya Urn Models . Chapman & Hall/CRC Press, Boca Raton, FL. · Zbl 1149.60005 |
| [23] | Nourdin, I. and Peccati, G. (2011). Stein’s method on Wiener chaos. Prob. Theory Relat. Fields 145, 75-118. · Zbl 1175.60053 · doi:10.1007/s00440-008-0162-x |
| [24] | Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (eds) (2010). NIST Handbook of Mathematical Functions. Cambridge University Press. · Zbl 1198.00002 |
| [25] | Peköz, E. (1996). Stein’s method for geometric approximation. J. Appl. Prob. 33, 707-713. · Zbl 0865.60014 · doi:10.2307/3215352 |
| [26] | Peköz, E. and Röllin, A. (2011). Exponential approximation for the nearly critical Galton-Watson process and occupation times of Markov chains. Electron. J. Prob. 16, 1381-1393. · Zbl 1245.60083 |
| [27] | Peköz, E. A. and Röllin, A. (2011). New rates for exponential approximation and the theorems of Rényi and Yaglom. Ann. Prob. 39, 587-608. · Zbl 1213.60049 · doi:10.1214/10-AOP559 |
| [28] | Peköz, E. A., Röllin, A. and Ross, N. (2013). Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Prob. 23, 1188-1218. · Zbl 1271.60019 · doi:10.1214/12-AAP868 |
| [29] | Peköz, E. A., Röllin, A. and Ross, N. (2013). Generalized gamma approximation with rates for urns, walks and trees. Preprint. Available at http://arxiv.org/abs/1309.4183. · Zbl 1271.60019 |
| [30] | Pemantle, R. (2007). A survey of random processes with reinforcement. Prob. Surveys 4, 1-79. · Zbl 1189.60138 · doi:10.1214/07-PS094 |
| [31] | Reinert, G. (2005). Three general approaches to Stein’s method. In An Introduction to Stein’s Method (Lecture Notes Ser. Inst. Math. Sci. Nat. Univ. Singapore 4 ), Singapore University Press, pp. 183-221. · doi:10.1142/9789812567680_0004 |
| [32] | Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. 6th Berkeley Symp. Math. Statist. Prob. , Vol II, University of California Press, Berkeley, CA, pp. 586-602. · Zbl 0278.60026 |
| [33] | Stein, C., Diaconis, P., Holmes, S. and Reinert, G. (2004). Use of exchangeable pairs in the analysis of simulations. In Stein’s Method: Expository Lectures and Applications , eds P. Diaconis and S. Holmes, Institute of Mathematical Statistics, Beachwood, OH, pp. 1-26. · doi:10.1214/lnms/1196283797 |
| [34] | Wilcox, R. R. (1981). A review of the beta-binomial model and its extensions. J. Educational Behavioral Statist. 6, 3-32. |
| [35] | Wright, S. (1945). The differential equation of the distribution of gene frequencies. Proc. Nat. Acad. Sci. USA 31, 382-389. · Zbl 0063.08327 · doi:10.1073/pnas.31.12.382 |
| [36] | Wright, S. (1949). Adaptation and selection. In Genetics, Paleontology and Evolution , eds G. Jepson, G. Simpson and E. Mayr, Princeton University Press, pp. 365-389. |
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