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

Electronic Journal of Probability

Volumn 20, Issue , 2015, Pages

  Döbler, Christian ^a  

^a UNIVERSITY OF LUXEMBOURG   (Luxembourg)

Author keywords

Beta distribution; Exchangeable pairs; P lya urn model; Stein s method

Indexed keywords

EID: 84944754768     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v20-3933     Document Type: Article

References (32)

1. R. Arratia, L. Goldstein, and L. Gordon, Two moments suffice for Poisson approximations: the Chen-Stein method, Ann. Probab. 17 (1989), no. 1, 9-25. MR-972770
2. A. D. Barbour, Stein's method and Poisson process convergence, J. Appl. Probab. (1988), no. Special Vol. 25A, 175-184, A celebration of applied probability. MR-974580
3. A. D. Barbour, L. Holst, and S. Janson, Poisson approximation, Oxford Studies in Probability, vol. 2, The Clarendon Press Oxford University Press, New York, 1992, Oxford Science Publications. MR-1163825
4. S. Chatterjee, J. Fulman, and A. Röllin, Exponential approximation by Stein's method and spectral graph theory, ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011), 197-223. MR-2802856
5. S. Chatterjee and Q.-M. Shao, 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. MR-2807964
6. L. H. Y. Chen, Poisson approximation for dependent trials, Ann. Probab. 3 (1975), no. 3, 534-545. MR-0428387
7. L. H. Y. Chen, L. Goldstein, and Q.-M. Shao, Normal approximation by Stein's method, Probability and its Applications (New York), Springer, Heidelberg, 2011. MR-2732624
8. L. H. Y. Chen and Q.-M. Shao, Stein's method for normal approximation, An introduction to Stein's method, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 4, Singapore Univ. Press, Singapore, 2005, pp. 1-59. MR-2235448
9. F. Daly, Upper bounds for Stein-type operators, Electron. J. Probab. 13 (2008), no. 20, 566-587. MR-2399291
10. C. Döbler, New developments in Stein's method with applications, (2012), (Ph.D.)-Thesis Ruhr-Universität Bochum.
11. C. Döbler, Stein's method of exchangeable pairs for absolutely continuous, univariate distributions with applications to the Polya urn model, arXiv:1207.0533 (2012).
12. R. Eden and J. Víquez, Nourdin-Peccati analysis on Wiener and Wiener-Poisson space for general distributions, Stochastic Process. Appl. 125 (2015), no. 1, 182-216. MR-3274696
13. P. Eichelsbacher and M. Löwe, Stein's method for dependent random variables occurring in statistical mechanics, Electron. J. Probab. 15 (2010), no. 30, 962-988. MR-2659754
14. J. Fulman and N. Ross, Exponential approximation and Stein's method of exchangeable pairs, ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), no. 1, 1-13. MR-3083916
15. 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).
16. R. E. Gaunt, Variance-gamma approximation via stein's method, Electron. J. Probab. 19 (2014).
17. 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).
18. L. Goldstein and G. Reinert, Stein's method for the beta distribution and the Pólya-Eggenberger urn, J. Appl. Probab. 50 (2013), no. 4, 1187-1205. MR-3161381
19. L. Goldstein and Y. Rinott, Multivariate normal approximations by Stein's method and size bias couplings, J. Appl. Probab. 33 (1996), no. 1, 1-17. MR-1371949
20. S. Kusuoka and C. A. Tudor, Stein's method for invariant measures of diffusions via Malliavin calculus, Stochastic Process. Appl. 122 (2012), no. 4, 1627-1651. MR-2914766
21. C. Ley, G. Reinert, and Y. Swan, Approximate computation of expectations: a canonical Stein operator, arXiv:1408.2998 (2014).
22. H. M. Luk, Stein's method for the Gamma distribution and related statistical applications, ProQuest LLC, Ann Arbor, MI, 1994, Thesis (Ph.D.)-University of Southern California. MR-2693204
23. I. Nourdin and G. Peccati, Stein's method on Wiener chaos, Probab. Theory Related Fields 145 (2009), no. 1-2, 75-118. MR-2520122
24. I. Nourdin and F. G. Viens, Density formula and concentration inequalities with Malliavin calculus, Electron. J. Probab. 14 (2009), no. 78, 2287-2309. MR-2556018
25. E. A. Peköz and A. Röllin, New rates for exponential approximation and the theorems of Rényi and Yaglom, Ann. Probab. 39 (2011), no. 2, 587-608. MR-2789507
26. E. A. Peköz, A. Röllin, and N. Ross, Degree asymptotics with rates for preferential attachment random graphs, Ann. Appl. Probab. 23 (2013), no. 3, 1188-1218. MR-3076682
27. J. Pike and H. Ren, Stein's method and the Laplace distribution, ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014), no. 1, 571-587. MR-3283586
28. G. Reinert, Three general approaches to Stein's method, An introduction to Stein's method, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 4, Singapore Univ. Press, Singapore, 2005, pp. 183-221. MR-2235451
29. A. Röllin, A note on the exchangeability condition in Stein's method, Statist. Probab. Lett. 78 (2008), no. 13, 1800-1806. MR-2453918
30. W. Schoutens, Orthogonal polynomials in Stein's method, J. Math. Anal. Appl. 253 (2001), no. 2, 515-531. MR-1808151
31. C. Stein, 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 (Berkeley, Calif.), Univ. California Press, 1972, pp. 583-602. MR-0402873
32. C. Stein, Approximate computation of expectations, Institute of Mathematical Statistics Lecture Notes-Monograph Series, 7, Institute of Mathematical Statistics, Hayward, CA, 1986. MR-882007