An Extension of the Bivariate Method of Polynomials and a Reduction Formula for Bonferroni-Type Inequalities

✍ Scribed by Italo Simonelli

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 102 KB
• Volume: 69
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1998.1798

✦ Synopsis

be two sequences of events, and let & N (A) and & M (B) be the number of those A i and B j , respectively, that occur. We prove that Bonferroni-type inequalities for P(& N (A) u, & M (B) v), where u and v are positive integers, are valid if and only if they are valid for a two dimensional triangular array of independent events A i and B j , with P(A i )= p 1 and P(B j )= p 2 for all i and j. This result allows to derive a formula from which arbitrary Bonferronitype inequalities of the above type are reduced to the special case of no events occurring. Such methods for proof and similar reduction formula were so far available only for the case of exactly u and v events occurring. Several new inequalities are obtained by using our results.