Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-06T02:42:45.398Z Has data issue: false hasContentIssue false
  • English
  • Français

An improved Poisson limit theorem for sums of dissociated random variables

Published online by Cambridge University Press:  14 July 2016

A. D. Barbour  and
G. K. Eagleson
A. D. Barbour*
Affiliation:
Universität Zürich
G. K. Eagleson*
Affiliation:
CSIRO Division of Mathematics and Statistics
*
∗Postal address: Institut für Angewandte Mathematik, Universität Zürich, CH-8001, Zürich, Switzerland.
∗∗Postal address: CSIRO Division of Mathematics and Statistics, P.O. Box 218, Lindfield, NSW 2070, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

A Poisson limit theorem for sums of dissociated 0–1 random variables is refined by deriving the first terms in an asymptotic expansion. The most natural refinement does not remove all the first-order error in a number of applications to tests of clustering, and a further approximation is derived which gives excellent results in practice. The proofs are based on the technique of Stein and Chen.

Keywords

CORRELATIONSINTERPOINT DISTANCESPOISSON APPROXIMATIONSRATES OF CONVERGENCE
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 3 , September 1987 , pp. 586 - 599
Copyright
Copyright © Applied Probability Trust 1987 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barbour, A. D. and Eagleson, G. K. (1983) Poisson approximation for some statistics based on exchangeable trials. Adv. Appl. Prob. 15, 585–600.CrossRefGoogle Scholar
Barbour, A. D. and Eagleson, G. K. (1985) Poisson convergence for dissociated statistics, J.R. Statist. Soc. B 46, 397–402.Google Scholar
Barbour, A. D. and Hall, P. (1984) On the rate of Poisson convergence. Math. Proc. Camb. Phil. Soc. 95, 473–480.CrossRefGoogle Scholar
Brown, T. C. and Eagleson, G. K. (1984) A useful property of some symmetric statistics. Amer. Statistician 38, 63–65.Google Scholar
Brown, T. C. and Silverman, B. W. (1979) Rates of Poisson convergence for U-statistics. J. Appl. Prob. 16, 428–432.CrossRefGoogle Scholar
Chen, L. H. Y. (1975) Poisson approximation for dependent trials. Ann. Prob. 3, 534–545.CrossRefGoogle Scholar
Geisser, S. and Mantel, N. (1962) Pairwise independence of jointly dependent variables. Ann. Math. Statist. 33, 290–291.CrossRefGoogle Scholar
Mantel, N. and Bailar, J. C. (1970) A class of permutational and multinomial tests arising in epidemiological research. Biometrics 26, 687–700.CrossRefGoogle ScholarPubMed
Mcginley, W. G. and Sibson, R. (1975) Dissociated random variables. Math. Proc. Camb. Phil. Soc. 77, 185–188.CrossRefGoogle Scholar
Silverman, B. W. and Brown, T. C. (1978) Short distances, flat triangles and Poisson limits. J. Appl. Prob. 15, 815–825.CrossRefGoogle Scholar
Stein, C. (1970) A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proc. 6th Berkeley Symp. Math. Statist. Prob. 2, 583–602.Google Scholar