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An upper bound for the probability of a union

Published online by Cambridge University Press:  14 July 2016

David Hunter
David Hunter*
Affiliation:
IBM Research Laboratory, San Jose, California
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Abstract

The problem of bounding P(∪ Ai) given P(Ai) and P(AiAj) for i ≠ j = 1, …, k goes back to Boole (1854) and Bonferroni (1936). In this paper a new family of upper bounds is derived using results in graph theory. This family contains the bound of Kounias (1968), and the smallest upper bound in the family for a given application is easily derivable via the minimal spanning tree algorithm of Kruskal (1956). The properties of the algorithm and of the multivariate normal and t distributions are shown to provide considerable simplifications when approximating tail probabilities of maxima from these distributions.

Keywords

BONFERRONI BOUNDBOOLE'S INEQUALITYSPANNING TREEMINIMAL SPANNING TREE ALGORITHMDISTRIBUTIONS OF MAXIMAMULTIVARIATE NORMAL DISTRIBUTIONMULTIVARIATE T DISTRIBUTION
Type
Short Communications
Information
Journal of Applied Probability , Volume 13 , Issue 3 , September 1976 , pp. 597 - 603
Copyright
Copyright © Applied Probability Trust 1976 

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References

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