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A new method for generating Bonferroni-type inequalities by iteration

Published online by Cambridge University Press:  24 October 2008

Janos Galambos  and
Yuan Xu
Janos Galambos
Affiliation:
Department of Mathematics, TU 038-16, Temple University, Philadelphia, PA 19122, U.S.A.
Yuan Xu
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, Texas 78712, U.S.A.
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Extract

Let A1,A2,…,An be events on a given probability space, and let mn be the number of those Aj which occur. Put S0 = S0,n = 1, and

where the summation is over all subscripts satisfying 1 ≤ i1 < i2 < … < ik ≤ n. For convenience in some formulae we adopt the convention Sk, n = 0 if k > n. By turning to indicator variables one immediately finds that

Inequalities of the form

where ck = ck(r, n) and dk = dk(r, n) are constants (not dependent on the events Aj, 1 ≤ j ≤ n), possibly zero, are called Bonferroni-type inequalities. This same name applies if P(mn = r) is replaced by P(mn ≥ r) in the middle. The best known such inequalities are the method of inclusion and exclusion

where j ≥ 0 is an arbitrary integer. An extension of (4) to arbitrary r, called Jordan's inequalities (see Takács [16]), is as follows: for 0 ≤ r ≤ n and for any integer j ≥ 0,

It is observed by Galambos and Mucci[9] that (5) follows from (4), and indeed, one can always generate inequalities of the form (3) for arbitrary r from the special case r = 0 if one utilizes only Sr,Sr+1… in the case of P(mn = r). As a matter of fact, from the instructions of Galambos and Mucci one has that the inequalities

hold for an arbitrary sequence A1,A2,…,An of events if, and only if, for an arbitrary sequence A1,A2,…,An−r of events,

where

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 3 , May 1990 , pp. 601 - 607
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

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