Nonuniform bounds in probability approximations using stein’s method

Probability and Statistical Models with Applications

Volumn , Issue , 2000, Pages 3-13

  Chen, Louis H Y ^a  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Binary expansion of a random integer; Binomial approximation; Concentration inequality approach; Non uniform bounds; Normal approximation; Poisson approximation; Probability approximations; Stein s method

Indexed keywords

EID: 0039396076     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Chapter

References (34)

1. Arratia, R., Goldstein, L., and Gordon, L. (1990). Poisson approximation and the Chen-Stein method. Statistical Science5, 403-434.
2. Barbour, A. D. (1988). Stein’s method and Poisson process convergence. Journal of Applied Probability 25 (A), 175 184.
3. Barbour, A. D., Chen, L. H. Y., and Choi, K. P. (1995). Poisson approximation for unbounded functions, I: independent summands. Statistica Sinica5, 749-766.
4. Barbour, A. D., Chen, L. H. Y., and Loh, W. L. (1992). Compound Poisson approximation for nonnegative random variables via Stein’s method. Annals of Probability20, 1843-1866.
5. Barbour, A. D. and Eagleson, G. (1983). Poisson approximation for some statistics based on exchangeable trials. Advances in Applied Probability15, 585-600.
6. Barbour, A. D. and Hall, P. (1984a). On the rate of Poisson convergence. Mathematical Proceedings of the Cambridge Philosophical Society95, 473-480.
7. Barbour, A. D. and Hall, P. (1984b). Stein’s method and the Berry-Esseen theorem. The Australian Journal Statistics26, 8-15.
8. Barbour, A. D., Holst, L., and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability 2, Clarendon Press, Oxford.
9. Bikelis, A. (1966). Estimates of the remainder in the central limit theorem. Litovsk. Mat. Sb. 6(3), 323-346 (in Russian).
10. Bolthauscn, E. (1984). An estimate of the remainder in a combinatorial central limit theorem. Zeitschrift Wahrscheinlichkeilstheorie und Verumndtc Gebiete66, 379-386.
11. Cacoullos, T., Papathanasiou, V. and Utev, S. A. (1994). Variational inequalities with examples and an application to the central limit theorem. Annals of Probability 22, 1607-1618.
12. Chen, L. H. Y. (1975a). Poisson approximation for dependent trials. Annals of Probability 3, 534-545.
13. Chen, L. H. Y. (1975b). An approximation theorem for convolutions of probability measures. Annals of Probability3, 992-999.
14. Chen, L. H. Y. (1986). The rate of convergence in a central limit theorem for dependent random variables with arbitrary index set. IMA Preprint Series ff 243, University of Minnesota.
15. Chen, L. H. Y. (1993). Extending the Poisson approximation. Science262, 379-380.
16. Chen, L. H. Y. (1998). Stein’s method: some perspectives with applications. Probability Towards 2000 (Eds., L. Accardi and C. Heyde), pp. 97-122. Lecture Notes in Statistics No. 128. Springer Verlag.
17. Chen, L. H. Y. and Choi, K. P. (1992). Some asymptotic and large deviation results in Poisson approximation. Annals of Probability20,1867-1876.
18. Chen, L. H. Y. and Roos, M. (1995). Compound Poisson approximation for unbounded functions on a group, with application to large deviations. Probability Theory and Related, Fields103, 515-528.
19. Chen, L. H. Y. and Shao, Q. M. (2000). A non-uniform Berry-Esseen bound via Stein’s method. Preprint.
20. Chen, L. H. Y. and Soon, S. Y. T. (1994). On the number of ones in the binary expansion of a random integer. Unpublished manuscript.
21. Deheuvels, P. and Pfeifer, D. (1986). A semigroup approach to Poisson approximation. Annals of Probability14, 663-676.
22. Diaconis, P. (1977). The distribution of leading digits and uniform distribution mod 1. Annals of Probability5, 72-81.
23. Esseen, C.-G. (1945). Fourier analysis of distribution functions: a mathematical study of the Laplace-Gaussian law. Acta Mathematica 771-125.
24. p bound for the remainder in a combinatorial central limit theorem. Annals of Probability6, 231-249.
25. Loh, W. L. (1992). Stein’s method and multinomial approximation. Annals of Applied Probability2, 536-554.
26. Michel, R. (1981). On the constant in the non-uniform version of the Berry-Esseen Theorem. Zeitschrift, WahrscJiemliehkeitsthcorie und Verwandte Gebiete 55, 109-117.
27. Nagaev, S. V. (1965). Some limit theorems for large deviations. Theory of Probability and Us Applications10, 214-235.
28. Paditz, L. (1977). Uber die Annäherung der Verteilungsfunktionen von Summen unabhängiger Zufallsgröben gegen unberrenzt teilbare Verteilungsfunktionen unter besonderer berchtung der Verteilungsfunktion der standardisierten Normalverteilung. Dissertation, A.TU Dresden.
29. Soon, S. Y. T. (1993). Some Problems in Binomial and Compound Poisson Approximations. Ph.D. dissertation, National University of Singapore.
30. Stein, C. (1972). 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 Mathematics, Statistics and Probability 2, 583 602, University California Press. Berkeley, California.
31. Stein, C. (1986). Approximation Computation of expectations. Lecture Notes 7, Institute of Mathematics arid Statistics, Hayward, California.
32. Stroock, D. W. (1993). Probability theory: An Analytic View. Cambridge University Press, Cambridge, U.K.
33. Utev, S. A. (1989). Probability problems connected with a certain integrodifferential inequality. Siberian Mathematics Journal30, 490-493.
34. Van Beek, P. (1972). An approximation of Fourier methods to the problem of sharpening the Berry-Esseen inequality. Zeitschrift Wahrscheinlichkeitstheorie und Verwandte Gebiete23, 187-196.