An extension of the bivariate method of polynomials and a reduction formula for Bonferroni-type inequalities
type: Item
instance of: scholarly article
title: An extension of the bivariate method of polynomials and a reduction formula for Bonferroni-type inequalities
zbMATH Open document ID: 0937.60009
author: Italo Simonelli
published in: Journal of Multivariate Analysis
publication date: 2000-06-07T00:00:00Z
review text: Let \(A_1\), \(A_2,\dots,A_N\) and \(B_1\), \(B_2,\dots,B_M\) be two sequences of events, and let \(\nu_N(A)\) and \(\nu_M(B)\) be the number of those \(A_j\) and \(B_i\), respectively, that occur. Let \(S_{k,t}\) be the \((k,t)\)th bivariate binomial moment of the sequences \(A_j\) and \(B_i\). Set \(p(u,v)=P(\nu_N(A)=u\), \(\nu_M(B)=v)\) and \(q(u,v)=P(\nu_N(A)\geq u\), \(\nu_M(B)\geq v)\). In the main result the author establishes that Bonferroni-type bounds on \(q(u,v)\), that is, bounds by means of linear combinations of the moments \(S_{k,t}\), are equivalent to specific bivariate polynomial inequalities. The author then deduces from the main result that several Bonferroni-type inequalities on \(q(u,v)\) are equivalent to Bonferroni-type inequalities on \(p(0,0)\). This brings together the inequalities on \(q(u,v)\) and \(p(u,v)\), since from inequalities on \(p(0,0)\) general inequalities on \(p(u,v)\) can be generated. With this new method, the parallel theory on \(p(u,v)\) and \(q(u,v)\) is unified which is a remarkable result on its own. The method is also used for the establishment of new inequalities. For earlier results on Bonferroni-type inequalities, see the book of the reviewer and the author [``Bonferroni-type inequalities with applications'' (1996; Zbl 0869.60014)].
reviewed by: Janos Galambos
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 62E17
zbMATH DE Number: 1295233
zbMATH Keywords: Bonferroni-type inequalities
zbMATH Keywords: bivariate
zbMATH Keywords: reduction formula
zbMATH Keywords: bivariate method of polynomials
MaRDI profile type: Publication
full work available at URL: jmva.1998.1798
OpenAlex ID: W2088357869
cites work: Q3903766
cites work: Q4217897
cites work: An extension of the method of polynomials and a new reduction formula for Bonferroni-type inequalities
cites work: Some Optimal Bivariate Bonferroni-Type Bounds
cites work: Two Sets of Multivariate Bonferroni—type Inequalities
cites work: Bivariate extension of the method of polynomials for Bonferroni-type inequalities
cites work: Q3764949
DOI: 10.1006/JMVA.1998.1798
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description: scientific article; zbMATH DE number 1295233