Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-09T13:44:22.868Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal lower bounds for bivariate probabilities

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Tuhao Chen
Tuhao Chen*
Affiliation:
University of Sydney, Australia
*
∗ Postal address: Dept. of Mathematics and Statistics, McMaster University, 1280 Main St. West, Hamilton, Ontario, Canada L8S 4K1.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper deals with existence of bivariate Fréchet optimal lower bounds for two sets of events, and provides a practical approach to find this kind of bound. The main device used is linear programming ideas, coupled with construction of probability spaces. The highlight of this paper is that perturbation terms in the optimization process, even when a tie occurs, are not necessary in this practical implementation.

Keywords

Fréchet optimalitylinear programmingprobability spaceiterationoptimizationcyclingperturbation

MSC classification

Primary: 60E15: Inequalities; stochastic orderings

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 2 , June 1998 , pp. 476 - 492
Copyright
Copyright © Applied Probability Trust 1998 

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

References

[1] Barnett, V. and Lewis, T. (1984). Outliers in Statistical Data. Wiley, New York.Google Scholar
[2] Chen, T. and Seneta, E. (1995). A note on bivariate Dawson–Sankoff-type bounds. Statist. Prob. Lett. 24, 99–104.Google Scholar
[3] Chen, T. (1995). Direct product Bonferroni-type optimality. Bull. Int. Statist. Instit. Contributed Papers, Proc. 50th session, Beijing. Two Books, Book 1, pp. 189–190.Google Scholar
[4] Dawson, D. A. and Sankoff, D. (1967). An inequality for probabilities. Proc. Amer. Math. Soc. 18, 504–507.Google Scholar
[5] Galambos, J. and Xu, Y. (1993). Some optimal bivariate Bonferroni-type bounds. Proc. Amer. Math. Soc. 117, 523–528.Google Scholar
[6] Hadley, G. (1962). Linear Programming. Addison-Wesley, Reading, Massachusetts.Google Scholar
[7] Hailperin, T. (1965). Best possible inequalities for the probability of a logical function of events. Amer. Math. Monthly 72, 343–359.CrossRefGoogle Scholar
[8] Hochberg, Y. and Tamhane, A. C. (1987). Multiple Comparison Procedures. Wiley, New York.Google Scholar
[9] Kounias, S. and Marin, J. (1976). Best linear Bonferroni bounds. SIAM J. Appl. Math. 30, 307–323.Google Scholar
[10] 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
[11] Meyer, R. M. (1969). Note on a ‘multivariate’ form of Bonferroni's inequalities. Ann. Math. Stat. 40, 692–693.Google Scholar
[12] Meyer, R. M. (1973). A Poisson-type limit theorem for mixing sequences of dependent ‘rare’ events. Ann. Prob. 1, 480–483.Google Scholar
[13] Prékopa, A., (1988). Boole–Bonferroni inequalities and linear programming. Operat. Res. 36, 145–162.Google Scholar
[14] Seneta, E. (1992). On the history of the strong law of large numbers and Boole's inequality. Historia Mathematica 19, 24–39.CrossRefGoogle Scholar
[15] Seneta, E. and Chen, T. (1995). Fréchet optimality of upper bivariate Bonferroni-type bounds. Teoriia Imovirnostei ta Matematychna Statystyka 52, 137–142. Kiev. (English version published by American Math. Soc. (1996). In Theory of Probability and Mathematical Statistics 52, 147–152.)Google Scholar