The Annals of Statistics

Monomial ideals and the Scarf complex for coherent systems in reliability theory

Beatrice Giglio and Henry P. Wynn

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A certain type of integer grid, called here an echelon grid, is an object found both in coherent systems whose components have a finite or countable number of levels and in algebraic geometry. If α=(α1,…,αd) is an integer vector representing the state of a system, then the corresponding algebraic object is a monomial x1α1xdαd in the indeterminates x1,…,xd. The idea is to relate a coherent system to monomial ideals, so that the so-called Scarf complex of the monomial ideal yields an inclusion–exclusion identity for the probability of failure, which uses many fewer terms than the classical identity. Moreover in the “general position” case we obtain via the Scarf complex the tube bounds given by Naiman and Wynn [J. Inequal. Pure Appl. Math. (2001) 2 1–16]. Examples are given for the binary case but the full utility is for general multistate coherent systems and a comprehensive example is given.

Article information

Ann. Statist. Volume 32, Number 3 (2004), 1289-1311.

First available in Project Euclid: 24 May 2004

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Zentralblatt MATH identifier

Primary: 90B25: Reliability, availability, maintenance, inspection [See also 60K10, 62N05] 06A06: Partial order, general

Network reliability inclusion–exclusion coherent systems multistate systems monomial ideals Scarf complex


Giglio, Beatrice; Wynn, Henry P. Monomial ideals and the Scarf complex for coherent systems in reliability theory. Ann. Statist. 32 (2004), no. 3, 1289--1311. doi:10.1214/009053604000000373.

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