Improved Bonferroni Inequalities via Abstract Tubes

2003 ◽  
Author(s):  
Klaus Dohmen
Keyword(s):  
Bonferroni Inequalities

Bonferroni inequalities and deviations of discrete distributions

1996 ◽  
Vol 33 (01) ◽  
pp. 115-121 ◽  
Author(s):  
Tamás F. Móri
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Probability Distributions ◽  
Type Inequality ◽  
Discrete Distributions ◽  
Maximum Distance ◽  
Discrete Probability ◽  
Bonferroni Inequalities ◽  
Discrete Probability Distributions

In the paper we first show how to convert a generalized Bonferroni-type inequality into an estimation for the generating function of the number of occurring events, then we give estimates for the deviation of two discrete probability distributions in terms of the maximum distance between their generating functions over the interval [0, 1].

On the classical Bonferroni inequalities and the corresponding Galambos inequalities

1981 ◽  
Vol 18 (03) ◽  
pp. 757-763 ◽  
Author(s):  
A. M. Walker
Keyword(s):  
Lower Bounds ◽  
Probability Space ◽  
Upper And Lower Bounds ◽  
Partial Sums ◽  
Bonferroni Inequalities ◽  
Indicator Functions

Let (A 1 A 2, · ··, An ) be a set of n events on a probability space. Let be the sum of the probabilities of all intersections of r events, and Mn the number of events in the set which occur. The classical Bonferroni inequalities provide upper and lower bounds for the probabilities P(Mn = m), and equal to partial sums of series of the form which give the exact probabilities. These inequalities have recently been extended by J. Galambos to give sharper bounds. Here we present straightforward proofs of the Bonferroni inequalities, using indicator functions, and show how they lead naturally to new simple proofs of the Galambos inequalities.

Lower Bounds for the Probability of Intersection of Several Unions of Events

1998 ◽  
Vol 7 (4) ◽  
pp. 353-364 ◽  
Author(s):  
TUHAO CHEN ◽  
E. SENETA
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Numerical Example ◽  
Bonferroni Inequalities ◽  
Indicator Functions ◽  
Single Set ◽  
Better Than

To bound the probability of a union of n events from a single set of events, Bonferroni inequalities are sometimes used. There are sharper bounds which are called Sobel–Uppuluri–Galambos inequalities. When two (or more) sets of events are involved, bounds are considered on the probability of intersection of several such unions, one union from each set. We present a method for unified treatment of bivariate lower and upper bounds in this note. The lower bounds obtained are new and at least as good as lower bounds appearing in the literature so far. The upper bounds coincide with existing bivariate Sobel–Uppuluri–Galambos type upper bounds derived by the method of indicator functions. A numerical example is given to illustrate that the new lower bounds can be strictly better than existing ones.

scholarly journals Bonferroni-Galambos Inequalities for Partition Lattices

10.37236/1838 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Klaus Dohmen ◽  
Peter Tittmann
Keyword(s):  
Network Reliability ◽  
Partition Lattice ◽  
Bonferroni Inequalities ◽  
Uniform Hypergraphs ◽  
Negative Function ◽  
Finite Set ◽  
Partition Lattices

In this paper, we establish a new analogue of the classical Bonferroni inequalities and their improvements by Galambos for sums of type $\sum_{\pi\in {\Bbb P}(U)} (-1)^{|\pi|-1} (|\pi|-1)! f(\pi)$ where $U$ is a finite set, ${\Bbb P}(U)$ is the partition lattice of $U$ and $f:{\Bbb P}(U)\rightarrow{\Bbb R}$ is some suitable non-negative function. Applications of this new analogue are given to counting connected $k$-uniform hypergraphs, network reliability, and cumulants.

scholarly journals Bonferroni Inequalities

1977 ◽  
Vol 5 (4) ◽  
pp. 577-581 ◽  
Author(s):  
Janos Galambos
Keyword(s):  
Bonferroni Inequalities
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