Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-25T06:14:36.263Z Has data issue: false hasContentIssue false
  • English
  • Français

Stein's Method for the Beta Distribution and the Pólya-Eggenberger Urn

Part of: Distribution theory Special processes Limit theorems

Published online by Cambridge University Press:  30 January 2018

Larry Goldstein  and
Gesine Reinert
Larry Goldstein*
Affiliation:
University of Southern California
Gesine Reinert*
Affiliation:
University of Oxford
*
∗ Postal address: Department of Mathematics, University of Southern California, KAP 108, Los Angeles, CA 90089-2532, USA. Email address: larry@math.usc.edu
∗∗ Postal address: Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK. Email address: reinert@stats.ox.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Using a characterizing equation for the beta distribution, Stein's method is applied to obtain bounds of the optimal order for the Wasserstein distance between the distribution of the scaled number of white balls drawn from a Pólya-Eggenberger urn and its limiting beta distribution. The bound is computed by making a direct comparison between characterizing operators of the target and the beta distribution, the former derived by extending Stein's density approach to discrete distributions. In addition, refinements are given to Döbler's (2012) result for the arcsine approximation for the fraction of time a simple random walk of even length spends positive, and so also to the distributions of its last return time to 0 and its first visit to its terminal point, by supplying explicit constants to the present Wasserstein bound and also demonstrating that its rate is of the optimal order.

Keywords

Stein's methodurn modelbeta distributionarcsine distribution

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 62E20: Asymptotic distribution theory 60K99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 1187 - 1205
Copyright
© Applied Probability Trust 

References

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.Google Scholar
Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: the Chen-Stein method. Ann. Prob. 17, 9–25.Google Scholar
Barbour, A. D. (1990). Stein's method for diffusion approximations. Prob. Theory Relat. Fields 84, 297–322.Google Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford University Press, New York.CrossRefGoogle Scholar
Brown, T. C. and Phillips, M. J. (1999). Negative binomial approximation with Stein's method. Methodology Comput. Appl. Prob. 1, 407–421.Google Scholar
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.Google Scholar
Chauvin, B., Pouyanne, N. and Sahnoun, R. (2011). Limit distributions for large Pólya urns. Ann. Appl. Prob. 21, 1–32.Google Scholar
Chen, L. H. Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein's Method. Springer, Heidelberg.CrossRefGoogle Scholar
Chu, W. (2007). Abel's lemma on summation by parts and basic hypergeometric series. Adv. Appl. Math. 39, 490–514.CrossRefGoogle Scholar
Chung, F., Handjani, S. and Jungreis, D. (2003). Generalizations of Pólya's urn problem. Ann. Combinatorics 7, 141–153.CrossRefGoogle Scholar
Döbler, C. (2012). A rate of convergence for the arcsine law by Stein's method. Preprint. Available at http://arxiv.org/abs/1207.2401v1.Google Scholar
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.Google Scholar
Durrett, R. (2007). Random Graph Dynamics. Cambridge University Press.Google Scholar
Eichelsbacher, P. and Reinert, G. (2008). Stein's method for discrete Gibbs measures. Ann. Appl. Prob. 18, 1588–1618.Google Scholar
Feller, W. (1970). An Introduction to Probability Theory and Its Applications. Vol. I. John Wiley, New York.Google Scholar
Fisher, R. A. (1930). The Genetical Theory of Natural Selection. Clarendon Press, Oxford.Google Scholar
Gibbs, A. L. and Su, F. E. (2002). On choosing and bounding probability metrics. Internat. Statist. Rev. 70, 419–435.Google Scholar
Holmes, S. (2004). Stein's method for birth and death chains. In Stein's Method: Expository Lectures and Applications, eds Diaconis, P. and Holmes, S., Institute of Mathematical Statistics, Beachwood, OH, pp. 45–67.Google Scholar
Ley, C. and Swan, Y. (2013). Local Pinsker inequalities via Stein's discrete density approach. IEEE Trans. Inf. Theory 59, 5584–5591.Google Scholar
Loh, W. L. (1992). Stein's method and multinomial approximation. Ann. Appl. Prob. 2, 536–554.CrossRefGoogle Scholar
Luk, H.-M. (1994). Stein's method for the gamma distribution and related statistical applications. Doctoral Thesis, University of Southern California.Google Scholar
Mahmoud, H. M. (2003). Pólya urn models and connections to random trees: a review. J. Iranian Statist. Soc. 2, 53–114.Google Scholar
Mahmoud, H. M. (2008). Pólya Urn Models. Chapman & Hall/CRC Press, Boca Raton, FL.Google Scholar
Nourdin, I. and Peccati, G. (2011). Stein's method on Wiener chaos. Prob. Theory Relat. Fields 145, 75–118.Google Scholar
Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (eds) (2010). NIST Handbook of Mathematical Functions. Cambridge University Press.Google Scholar
Peköz, E. (1996). Stein's method for geometric approximation. J. Appl. Prob. 33, 707–713.Google Scholar
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.Google Scholar
Peköz, E. A. and Röllin, A. (2011). New rates for exponential approximation and the theorems of Rényi and Yaglom. Ann. Prob. 39, 587–608.Google Scholar
Peköz, E. A., Röllin, A. and Ross, N. (2013). Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Prob. 23, 1188–1218.Google Scholar
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.Google Scholar
Pemantle, R. (2007). A survey of random processes with reinforcement. Prob. Surveys 4, 1–79.Google Scholar
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.Google Scholar
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.Google Scholar
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 Diaconis, P. and Holmes, S., Institute of Mathematical Statistics, Beachwood, OH, pp. 1–26.Google Scholar
Wilcox, R. R. (1981). A review of the beta-binomial model and its extensions. J. Educational Behavioral Statist. 6, 3–32.CrossRefGoogle Scholar
Wright, S. (1945). The differential equation of the distribution of gene frequencies. Proc. Nat. Acad. Sci. USA 31, 382–389.Google Scholar
Wright, S. (1949). Adaptation and selection. In Genetics, Paleontology and Evolution, eds Jepson, G., Simpson, G. and Mayr, E., Princeton University Press, pp. 365–389.Google Scholar