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Upper bounds for the probability of a union by multitrees

Part of: Combinatorial probability

Published online by Cambridge University Press:  01 July 2016

József Bukszár
József Bukszár*
Affiliation:
University of Miskolc
*
∗ Postal address: Institute of Mathematics, University of Miskolc, Miskolc, H-3515, Hungary. Email address: matbuk@silver.uni-miskolc.hu
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Abstract

The problem of finding bounds for P(A 1 ∪ ⋯ ∪ A n ) based on P(A k1 ∩ ⋯ ∩ A k i ) (1 ≤ k 1 < ⋯ < k i ≤ n, i = 1,…,d) goes back to Boole (1854), (1868) and Bonferroni (1937). In this paper upper bounds are presented using methods in graph theory. The main theorem is a common generalization of the earlier results of Hunter, Worsley and recent results of Prékopa and the author. Algorithms are given to compute bounds. Examples for bounding values of multivariate normal distribution functions are presented.

Keywords

Bonferroni-type boundhypergraphmultivariate normal distribution

MSC classification

Primary: 60C05: Combinatorial probability

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 2 , June 2001 , pp. 437 - 452
Copyright
Copyright © Applied Probability Trust 2001 

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Footnotes

Partly supported by OTKA T032369.

References

[1] Bonferroni, C. E. (1937). Teoria Statistica Delle Classi e Calcolo Delle Probabilitá. In Volume in onore di Riccardo Dalla Volta. Universitá di Firenze, pp. 1–62.Google Scholar
[2] Boole, G. (1957). Laws of Thought, reprint of 1854 edn, Dover, New York.Google Scholar
[3] Boole, G. (1868). Of propositions numerically definite. Trans. Camb. Phil. Soc. 11, 396–411.Google Scholar
[4] Boros, E. and Prékopa, A. (1989). Closed form two-sided bounds for probabilities that exactly r and at least r out of n events occur. Math. Operat. Res. 14, 317–342.CrossRefGoogle Scholar
[5] Bukszár, J. and Prékopa, A. (2001). Probability bounds with cherry trees. Math. Operat. Res. 26, 174–192.Google Scholar
[6] Galambos, J. (1975). Methods for proving Bonferroni-type inequalities. J. London Math. Soc. 2, 561–564.Google Scholar
[7] Galambos, J. and Simonelli, I. (1996). Bonferroni-type Inequalities with Applications. Springer, New York.Google Scholar
[8] Genz, A. (1992). Numerical computation of the multivariate normal probabilities. J. Comput. Graph. Statist. 1, 141–150.Google Scholar
[9] Grable, D. A. (1991). Two packing problems on k-matroid trees. Eur. J. Combin. 12, 309–316.Google Scholar
[10] Hunter, D. (1976). An upper bound for the probability of a union. J. Appl. Prob. 13, 597–603.Google Scholar
[11] Prékopa, A., (1988). Boole–Bonferroni inequalities and linear programming. Operat. Res. 36, 145–162.CrossRefGoogle Scholar
[12] Prékopa, A., (1990). Sharp bounds on probabilities using linear programming. Operat. Res. 38, 227–239.Google Scholar
[13] Prékopa, A., (1995). Stochastic Programming. Kluwer, Dordrecht.Google Scholar
[14] Seneta, E. (1988). Degree, iteration and permutation in improving Bonferroni-type bounds. Austral. J. Statist. 30A, 27–38.Google Scholar
[15] Tomescu, I. (1986). Hypertrees and Bonferroni inequalities. J. Combin. Theory B 41, 209–217.Google Scholar
[16] Worsley, K. J. (1982). An improved Bonferroni inequality and applications. Biometrika 69, 297–302.Google Scholar