Electronic Journal of Probability

Stein's method of exchangeable pairs for the Beta distribution and generalizations

Christian Döbler

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Abstract

We propose a new version of Stein's method of exchangeable pairs, which, given a suitable exchangeable pair $(W,W')$ of real-valued random variables, suggests the approximation of the law of $W$ by a suitable absolutely continuous distribution. This distribution is characterized by a first order linear differential Stein operator, whose coefficients $\gamma$ and $\eta$ are motivated by two regression properties satisfied by the pair $(W,W')$. Furthermore, the general theory of Stein's method for such an absolutely continuous distribution is developed and a general characterization result as well as general bounds on the solution to the Stein equation are given. This abstract approach is a certain extension of the theory developed in previous works, which only consider the framework of the density approach, i.e. $\eta\equiv1$. As an illustration of our technique we prove a general plug-in result, which bounds a certain distance of the distribution of a given random variable $W$ to a Beta distribution in terms of a given exchangeable pair $(W,W')$ and provide new bounds on the solution to the Stein equation for the Beta distribution, which complement the existing bounds from previous works. The abstract plug-in result is then applied to derive bounds of order $n^{-1}$ for the distance between the distribution of the relative number of drawn red balls after $n$ drawings in a Pólya urn model and the limiting Beta distribution measured by a certain class of smooth test functions.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 109, 34 pp.

Dates
Accepted: 19 October 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067215

Digital Object Identifier
doi:10.1214/EJP.v20-3933

Mathematical Reviews number (MathSciNet)
MR3418541

Zentralblatt MATH identifier
1328.60064

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60E99: None of the above, but in this section

Keywords
Stein's method exchangeable pairs Beta distribution Pólya urn model

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Döbler, Christian. Stein's method of exchangeable pairs for the Beta distribution and generalizations. Electron. J. Probab. 20 (2015), paper no. 109, 34 pp. doi:10.1214/EJP.v20-3933. https://projecteuclid.org/euclid.ejp/1465067215


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References

  • Arratia, R.; Goldstein, L.; Gordon, L. Two moments suffice for Poisson approximations: the Chen-Stein method. Ann. Probab. 17 (1989), no. 1, 9–25.
  • Barbour, A. D. Stein's method and Poisson process convergence. A celebration of applied probability. J. Appl. Probab. 1988, Special Vol. 25A, 175–184.
  • Barbour, A. D.; Holst, Lars; Janson, Svante. Poisson approximation. Oxford Studies in Probability, 2. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992. x+277 pp. ISBN: 0-19-852235-5.
  • Chatterjee, Sourav; Fulman, Jason; Roellin, Adrian. Exponential approximation by Stein's method and spectral graph theory. ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011), 197–223.
  • Chatterjee, Sourav; Shao, Qi-Man. Nonnormal approximation by Stein's method of exchangeable pairs with application to the Curie-Weiss model. Ann. Appl. Probab. 21 (2011), no. 2, 464–483.
  • Chen, Louis H. Y. Poisson approximation for dependent trials. Ann. Probability 3 (1975), no. 3, 534–545.
  • Chen, Louis H. Y.; Goldstein, Larry; Shao, Qi-Man. Normal approximation by Stein's method. Probability and its Applications (New York). Springer, Heidelberg, 2011. xii+405 pp. ISBN: 978-3-642-15006-7
  • Chen, Louis H. Y.; Shao, Qi-Man. Stein's method for normal approximation. An introduction to Stein's method, 1–59, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 4, Singapore Univ. Press, Singapore, 2005.
  • Daly, Fraser. Upper bounds for Stein-type operators. Electron. J. Probab. 13 (2008), no. 20, 566–587.
  • C. Döbler, New developments in Stein's method with applications, (2012), (Ph.D.)-Thesis Ruhr-Universität Bochum.
  • C. Döbler, Stein's method of exchangeable pairs for absolutely continuous, univariate distributions with applications to the Polya urn model, arXiv:1207.0533 (2012).
  • Eden, Richard; Viquez, Juan. Nourdin-Peccati analysis on Wiener and Wiener-Poisson space for general distributions. Stochastic Process. Appl. 125 (2015), no. 1, 182–216.
  • Eichelsbacher, Peter; Loewe, Matthias. Stein's method for dependent random variables occurring in statistical mechanics. Electron. J. Probab. 15 (2010), no. 30, 962–988.
  • Fulman, Jason; Ross, Nathan. Exponential approximation and Stein's method of exchangeable pairs. ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), no. 1, 1–13.
  • R. E. Gaunt, Rates of convergence in normal approximation under moment conditions via new bounds on solutions of the stein equation, to apper in J. Theoret. Probab. (2014).
  • R. E. Gaunt, Variance-gamma approximation via stein's method, Electron. J. Probab. 19 (2014).
  • R. E. Gaunt, A. Pickett, and G. Reinert, Chi-square approximation by Stein's method with application to Pearson's statistic, arXiv:1507.01707 (2015).
  • Goldstein, Larry; Reinert, Gesine. Stein's method for the beta distribution and the Polya-Eggenberger urn. J. Appl. Probab. 50 (2013), no. 4, 1187–1205.
  • Goldstein, Larry; Rinott, Yosef. Multivariate normal approximations by Stein's method and size bias couplings. J. Appl. Probab. 33 (1996), no. 1, 1–17.
  • Kusuoka, Seiichiro; Tudor, Ciprian A. Stein's method for invariant measures of diffusions via Malliavin calculus. Stochastic Process. Appl. 122 (2012), no. 4, 1627–1651.
  • C. Ley, G. Reinert, and Y. Swan, Approximate computation of expectations: a canonical Stein operator, arXiv:1408.2998 (2014).
  • Luk, Ho Ming. Stein's method for the Gamma distribution and related statistical applications. Thesis (Ph.D.) - University of Southern California. ProQuest LLC, Ann Arbor, MI, 1994. 74 pp.
  • Nourdin, Ivan; Peccati, Giovanni. Stein's method on Wiener chaos. Probab. Theory Related Fields 145 (2009), no. 1-2, 75–118.
  • Nourdin, Ivan; Viens, Frederi G. Density formula and concentration inequalities with Malliavin calculus. Electron. J. Probab. 14 (2009), no. 78, 2287–2309.
  • Pekoez, Erol A.; Roellin, Adrian. New rates for exponential approximation and the theorems of Renyi and Yaglom. Ann. Probab. 39 (2011), no. 2, 587–608.
  • Pekoez, Erol A.; Roellin, Adrian; Ross, Nathan. Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Probab. 23 (2013), no. 3, 1188–1218.
  • Pike, John; Ren, Haining. Stein's method and the Laplace distribution. ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014), no. 1, 571–587.
  • Reinert, Gesine. Three general approaches to Stein's method. An introduction to Stein's method, 183–221, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 4, Singapore Univ. Press, Singapore, 2005.
  • Roellin, Adrian. A note on the exchangeability condition in Stein's method. Statist. Probab. Lett. 78 (2008), no. 13, 1800–1806.
  • Schoutens, Wim. Orthogonal polynomials in Stein's method. J. Math. Anal. Appl. 253 (2001), no. 2, 515–531.
  • Stein, Charles. A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, pp. 583–602. Univ. California Press, Berkeley, Calif., 1972.
  • Stein, Charles. Approximate computation of expectations. Institute of Mathematical Statistics Lecture Notes-Monograph Series, 7. Institute of Mathematical Statistics, Hayward, CA, 1986. iv+164 pp. ISBN: 0-940600-08-0.