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Stein's Method and Stochastic Orderings

Part of: Distribution theory Limit theorems Markov processes

Published online by Cambridge University Press:  04 January 2016

Fraser Daly ,
Claude Lefèvre  and
Sergey Utev
Fraser Daly*
Affiliation:
Universität Zürich
Claude Lefèvre*
Affiliation:
Université Libre de Bruxelles
Sergey Utev*
Affiliation:
University of Nottingham
*
∗ Current address: Heilbronn Institute for Mathematical Research, Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK. Email address: fraser.daly@bristol.ac.uk
∗∗ Postal address: Département de Mathématique, Université Libre de Bruxelles, Campus de la Plaine, CP 210, B-1050 Bruxelles, Belgium. Email address: clefevre@ulb.ac.be
∗∗∗ Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK. Email address: sergey.utev@nottingham.ac.uk
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Abstract

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A stochastic ordering approach is applied with Stein's method for approximation by the equilibrium distribution of a birth-death process. The usual stochastic order and the more general s-convex orders are discussed. Attention is focused on Poisson and translated Poisson approximations of a sum of dependent Bernoulli random variables, for example, k-runs in independent and identically distributed Bernoulli trials. Other applications include approximation by polynomial birth-death distributions.

Keywords

Stein's methodbirth-death processstochastic orderingtotal variation distance(in)dependent indicators(translated) Poisson approximationtotal negative and positive dependence(approximate) local dependencepolynomial birth-death approximationk-runs

MSC classification

Primary: 62E17: Approximations to distributions (nonasymptotic)
Secondary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 2 , June 2012 , pp. 343 - 372
Copyright
© Applied Probability Trust 

References

Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: the Chen–Stein method. Ann. Prob. 17, 9–25.CrossRefGoogle Scholar
Barbour, A. D. and Chen, L. H. Y. (eds) (2005). An Introduction to Stein's Method (Lecture Notes Ser., Inst. Math. Sci. 4), Singapore University Press.CrossRefGoogle Scholar
Barbour, A. D. and Jensen, J. L. (1989). Local and tail approximations near the Poisson limit. Scand. J. Statist. 16, 75–87.Google Scholar
Barbour, A. D. and Pugliese, A. (2000). On the variance-to-mean ratio in models of parasite distributions. Adv. Appl. Prob. 32, 701–719.CrossRefGoogle Scholar
Barbour, A. D. and Xia, A. (1999). Poisson perturbations. ESAIM Prob. Statist. 3, 131–150.Google Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford University Press.CrossRefGoogle Scholar
Brown, T. C. and Phillips, M. J. (1999). Negative binomial approximation with Stein's method. Methodology Comput. Appl. Prob. 1, 407–421.CrossRefGoogle Scholar
Brown, T. C. and Xia, A. (2001). Stein's method and birth–death processes. Ann. Prob. 29, 1373–1403.CrossRefGoogle Scholar
Čekanavičius, V. and Vaıˇtkus, P. (2001). Centered Poisson approximation via Stein's method. Lithuanian Math. J. 41, 319–329.Google Scholar
Chatterjee, S., Diaconis, P. and Meckes, E. (2005). Exchangeable pairs and Poisson approximation. Prob. Surveys 2, 64–106.Google Scholar
Chen, L. H. Y. (1975). Poisson approximation for dependent trials. Ann. Prob. 3, 534–545.Google Scholar
Chen, L. H. Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein's Method. Springer, Heidelberg.Google Scholar
Denuit, M. and Lefèvre, C. (1997). Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences. Insurance Math. Econom. 20, 197–213.Google Scholar
Ehm, W. (1991). Binomial approximation to the Poisson binomial distribution. Statist. Prob. Lett. 11, 7–16.CrossRefGoogle Scholar
Eichelsbacher, P. and Reinert, G. (2008). Stein's method for discrete Gibbs measures. Ann. Appl. Prob. 18, 1588–1618.CrossRefGoogle Scholar
Erhardsson, T. (2005). Stein's method for Poisson and compound Poisson approximation. In An Introduction to Stein's Method (Lecture Notes Ser., Inst. Math. Sci. 4), eds Barbour, A. D. and Chen, L. H. Y., Singapore University Press, pp. 61–113.CrossRefGoogle Scholar
Goldstein, L. and Rinott, Y. (1996). Multivariate normal approximations by Stein's method and size bias couplings. J. Appl. Prob. 33, 1–17.Google Scholar
Holmes, S. (2004). Stein's method for birth and death chains. In Stein's Method: Expository Lectures and Applications (IMS Lecture Notes Monogr. Ser. 46), eds Diaconis, P. and Holmes, S., Beachwood, OH, pp. 45–67.Google Scholar
Johnson, N. L., Kotz, S. and Kemp, A. W. (1992). Univariate Discrete Distributions. John Wiley, New York.Google Scholar
Kamae, T., Krengel, U. and O'Brien, G. L. (1977). Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.Google Scholar
Lefèvre, C. and Utev, S. (1996). Comparing sums of exchangeable Bernoulli random variables. J. Appl. Prob. 33, 285–310.Google Scholar
Papadatos, N. and Papathanasiou, V. (2002). Poisson approximation for a sum of dependent indicators: an alternative approach. Adv. Appl. Prob. 34, 609–625.Google Scholar
Peköz, E. A. (1996). Stein's method for geometric approximation. J. Appl. Prob. 33, 707–713.Google Scholar
Phillips, M. J. and Weinberg, G. V. (2000). Non-uniform bounds for geometric approximation. Statist. Prob. Lett. 49, 305–311.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. 4), eds Barbour, A. D. and Chen, L. H. Y., Singapore University Press, pp. 183–221.Google Scholar
Röllin, A. (2005). Approximation of sums of conditionally independent variables by the translated Poisson distribution. Bernoulli 11, 1115–1128.Google Scholar
Röllin, A. (2007). Translated Poisson approximation using exchangeable pair couplings. Ann. Appl. Prob. 17, 1596–1614.Google Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.Google Scholar
Stein, C. (1986). Approximate Computation of Expectations (IMS Lecture Notes Monogr. Ser. 7). Institute of Mathematical Statistics, Hayward, CA.Google Scholar
Wang, X. and Xia, A. (2008). On negative binomial approximation to k-runs. J. Appl. Prob. 45, 456–471.CrossRefGoogle Scholar