Hostname: page-component-7479d7b7d-t6hkb Total loading time: 0 Render date: 2024-07-11T00:19:03.358Z Has data issue: false hasContentIssue false
  • English
  • Français

Bonferroni bounds revisited

Published online by Cambridge University Press:  14 July 2016

Stratis Kounias  and
Kiki Sotirakoglou
Stratis Kounias*
Affiliation:
University of Thessaloniki
Kiki Sotirakoglou*
Affiliation:
University of Thessaloniki
*
∗ Postal address: Department of Mathematics, University of Thessaloniki, 54006 Thessaloniki, Greece.
∗ Postal address: Department of Mathematics, University of Thessaloniki, 54006 Thessaloniki, Greece.
Get access Rights & Permissions [Opens in a new window]

Abstract

Lower and upper bounds of degree m for the probability of the union of n not necessarily exchangeable events are established. These bounds may be constructed to improve the Bonferroni and the Sobel–Uppuluri bounds.

An application to equi-correlated multivariate normal distribution is given.

Keywords

OPTIMALPOLYNOMIALLOWER BOUNDSUPPER BOUNDS
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 2 , June 1989 , pp. 233 - 241
Copyright
Copyright © Applied Probability Trust 1989 

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

Bauer, P. and Hackl, P. (1985) The application of Hunter's inequality in simultaneous testing. Biom. J. 27, 25–38.Google Scholar
Dawson, D. and Sankoff, D. (1967) An inequality for probabilities. Proc. Amer. Math. Soc. 18, 504–507.Google Scholar
Galambos, J. (1972) On the distribution of the maximum of random variables. Ann. Math. Statist. 43, 516–521.CrossRefGoogle Scholar
Galambos, J. (1975) Methods for proving Bonferroni-type inequalities. J. London Math. Soc. (2) 9, 561–564.CrossRefGoogle Scholar
Galambos, J. (1977) Bonferroni inequalities. Ann. Prob. 5, 577–581.CrossRefGoogle Scholar
Gupta, S. S. (1963) Probability integrals of multivariate normal and multivariate t. Ann. Math. Statist. 34, 1792–1828.Google Scholar
Hunter, D. (1976) An upper bound for the probability of a union. J. Appl. Prob. 13, 597–603.Google Scholar
Kounias, E. G. (1968) Bounds for the probability of a union, with applications. Ann. Math. Statist. 39, 2154–2158.CrossRefGoogle Scholar
Kwerel, S. M. (1975) Most stringent bounds on aggregated probabilities of partially specified dependent probability systems. J. Amer. Statist. Assoc. 70, 472–479.Google Scholar
Margolin, B. J. and Maurer, W. (1976) Tests of the Kolmogorov–Smimov type for exponential data with unknown scale and related problems. Biometrika 63, 149–160.CrossRefGoogle Scholar
Maurer, W. (1980) ‘Bivalent trees and forests’ or ‘upper bounds for the probability of a union revisited.’ Berichte des Forschungsinstitutes für Mathematik, ETH-Zürich. Google Scholar
Sobel, M. and Uppuluri, V. R. R. (1972) On Bonferroni-type inequalities of the same degree for the probability of unions and intersections. Ann. Math. Statist. 43, 1549–1558.Google Scholar
Tomescu, I. (1986) Hypertrees and Bonferroni inequalities. J. Combinatorial Theory B41, 209–217.Google Scholar
Worsley, K. J. (1982) An improved Bonferroni inequality and applications. Biometrika 69, 297–302.CrossRefGoogle Scholar