Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T06:22:18.082Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

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

Bonferroni, C. E. (1936) Teoria statistica classi e calcolo delle probabilita. Pubbl. R. Ist. Super. Sci. Econ. Comm. Firenze 8, 1–62.Google Scholar
Boole, G. (1854) (1916 reprint) The Laws of Thought. Open Court, Chicago, Chapter XIX.Google Scholar
Cayley, A. (1889) A theorem on trees. Quart. J. Math. 23, 376–378.Google Scholar
Dunnett, C. W. and Sobel, M. (1954) A bivariate generalization of Student's t-distribution, with tables for certain special cases. Biometrika 41, 153–169.CrossRefGoogle Scholar
Hailperin, T. (1965) Best possible inequalities for the probability of a logical function of events. Amer. Math. Monthly 72, 343–359.CrossRefGoogle Scholar
Hunter, D. (1975a) Simultaneous t-tests in normal models. Technical Report No. 87, Series 2, Dept. of Statistics, Princeton University.Google Scholar
Hunter, D. (1975b) Approximating percentage points of statistics expressible as maxima. Technical Report No. 88, Series 2, Department of Statistics, Princeton University.Google Scholar
Kounias, E. (1968) Bounds for the probability of a union, with applications. Ann. Math. Statist. 39, 2154–2158.CrossRefGoogle Scholar
Kounias, E. and Marin, J. (1976) Best linear Bonferroni bounds. SIAM J. Appl. Math. 30, 307–323.CrossRefGoogle Scholar
Kruskal, J. B. (1956) On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Amer. Math. Soc. 7, 48–50.CrossRefGoogle Scholar
Slepian, D. (1962) The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41, 463–501.CrossRefGoogle Scholar