Hostname: page-component-cb9f654ff-h4f6x Total loading time: 0 Render date: 2025-08-08T21:54:06.165Z Has data issue: false hasContentIssue false
  • English
  • Français

Poisson approximation for a sum of dependent indicators: an alternative approach

Part of: Limit theorems

Published online by Cambridge University Press:  01 July 2016

N. Papadatos  and
V. Papathanasiou
N. Papadatos*
Affiliation:
University of Athens
V. Papathanasiou*
Affiliation:
University of Athens
*
∗ Postal address: Section of Statistics and Operational Research, Department of Mathematics, University of Athens, Panepistemiopolis, 157 84 Athens, Greece.
∗ Postal address: Section of Statistics and Operational Research, Department of Mathematics, University of Athens, Panepistemiopolis, 157 84 Athens, Greece.
Get access Rights & Permissions [Opens in a new window]

Abstract

The random variables X 1, X 2, …, X n are said to be totally negatively dependent (TND) if and only if the random variables X i and ∑j≠i X j are negatively quadrant dependent for all i. Our main result provides, for TND 0-1 indicators X 1, x 2, …, X n with P[X i = 1] = p i = 1 - P[X i = 0], an upper bound for the total variation distance between ∑n i=1 X i and a Poisson random variable with mean λ ≥ ∑n i=1 p i . An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

Keywords

Totally negatively dependent indicatorChen-Stein equationnegatively related indicatorPoisson approximationtotal variation distancebirthday problemmonotone coupling

MSC classification

Primary: 60F05: Central limit and other weak theorems

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 3 , September 2002 , pp. 609 - 625
Copyright
Copyright © Applied Probability Trust 2002 

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.)

Article purchase

Temporarily unavailable

Footnotes

Research partially supported by the research foundation of the University of Athens.

References

Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: the Chen–Stein method. Ann. Prob. 17, 9–25.Google Scholar
Arratia, R., Goldstein, L. and Gordon, L. (1990). Poisson approximation and the Chen–Stein method. Statist. Sci. 5, 403–434.Google Scholar
Barbour, A. D. and Eagleson, G. K. (1983). Poisson approximation for some statistics based on exchangeable trials. Adv. Appl. Prob. 15, 585–600.Google Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation (Oxford Studies Prob. 2). Oxford University Press.Google Scholar
Boutsikas, M. V. and Koutras, M. V. (2000). A bound for the distribution of the sum of discrete associated or negatively associated random variables. Ann. Appl. Prob. 10, 1137–1150.Google Scholar
Cacoullos, T. and Papathanasiou, V. (1989). Characterizations of distributions by variance bounds. Statist. Prob. Lett. 7, 351–356.Google Scholar
Cacoullos, T., Papadatos, N. and Papathanasiou, V. (1997). Variance inequalities for covariance kernels and applications to central limit theorems. Theory Prob. Appl. 42, 195–201.Google Scholar
Cacoullos, T., Papadatos, N. and Papathanasiou, V. (2002). An application of a density transform and the local limit theorem. Theory Prob. Appl. 46, 803–810.Google Scholar
Cacoullos, T., Papathanasiou, V. and Utev, S. (1994). Variational inequalities with examples and an application to the central limit theorem. Ann. Prob. 22, 1607–1618.Google Scholar
Chen, L. H. Y. (1975). Poisson approximation for dependent trials. Ann. Prob. 3, 534–545.Google Scholar
Chen, L. H. Y. (1998). Stein's method: some perspectives with applications. In Probability Towards 2000, eds Accardi, L. and Heyde, C. C. (Lecture Notes Statist. 128), Springer, New York, pp. 97–122.CrossRefGoogle Scholar
Goldstein, L. and Reinert, G. (1997). Stein's method and the zero bias transformation with application to simple random sampling. Ann. Appl. Prob. 7, 935–952.CrossRefGoogle Scholar
Henze, N. (1998). A Poisson limit law for a generalized birthday problem. Statist. Prob. Lett. 39, 333–336.Google Scholar
Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables, with applications. Ann. Statist. 11, 286–295.Google Scholar
Kolchin, V. F., Sevastyanov, B. A. and Chistyakov, V. P. (1978). Random Allocations. Winston, Washington, DC.Google Scholar
Lehmann, E. L. (1966). Some concepts of dependence. Ann. Math. Statist. 37, 1137–1153.CrossRefGoogle Scholar
Majsnerowska, M. (1998). A note on Poisson approximation by w-functions. Appl. Math. (Warsaw) 25, 387–392.Google Scholar
Olkin, I. and Sobel, M. (1965). Integral expressions for tail probabilities of the multinomial and the negative multinomial distribution. Biometrika 52, 167–179.CrossRefGoogle Scholar
Papadatos, N. and Papathanasiou, V. (1995). Distance in variation between two arbitrary distributions via the associated w-functions. Theory Prob. Appl. 40, 685–694.Google Scholar
Papadatos, N. and Papathanasiou, V. (2001). Unified variance bounds and a Stein-type identity. In Probability and Statistical Models with Applications, eds Charalambides, Ch. A., Koutras, M. V. and Balakrishnan, N., Chapman and Hall/CRC, New York, pp. 87–100.Google Scholar
Papathanasiou, V. and Utev, S. A. (1995). Integro-differential inequalities and the Poisson approximation. Siberian Adv. Math. 5, 138–150.Google Scholar
Preston, C. J. (1974). A generalization of the FKG inequalities. Commun. Math. Phys. 36, 233–241.Google Scholar
Serfling, R. J. (1975). A general Poisson approximation theorem. Ann. Prob. 3, 726–731.Google Scholar
Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. 6th Berkeley Symp. Math. Statist. Prob., Vol. II, University of California Press, Berkeley, pp. 583–602.Google Scholar
Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423–439.Google Scholar