The Annals of Statistics

Inclusion-Exclusion-Bonferroni Identities and Inequalities for Discrete Tube-Like Problems via Euler Characteristics

Daniel Q. Naiman and Henry P. Wynn

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Abstract

Improvements to the classical inclusion-exclusion identity are developed. There are two main results: an abstract combinatoric result and a concrete geometric result. In the abstract result conditions are given which guarantee the existence of a depth $d + 1$ identity or inequality for the indicator function of a union of a finite collection of events, that is, an expression which is a linear combination of indicator functions of at most $(d + 1)$-fold intersections of the events. Such an identity or inequality can be integrated with respect to any probability measure to yield a probability identity or inequality. Connections are given to previous work on Bonferroni-type inequalities. The concrete result says that there is a depth $d + 1$ identity for the union of finitely many balls in $d$-dimensional Euclidean space. With a single correction term this result also holds in the $d$-dimensional sphere. These results form the basis for a discrete theory of tubes, which up to now has been continuous in nature. The spherical result is used to give a simulation method for finding critical probabilities for multiple-comparisons procedures, and a computer program implementing the method is described. Numerical results are presented which demonstrate that in the tails of the distribution probability estimates based on the method tend to exhibit less variability than estimates based on naive simulation.

Article information

Source
Ann. Statist. Volume 20, Number 1 (1992), 43-76.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176348512

Digital Object Identifier
doi:10.1214/aos/1176348512

Mathematical Reviews number (MathSciNet)
MR1150334

Zentralblatt MATH identifier
0752.62028

JSTOR
links.jstor.org

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62J01 62J05: Linear regression 62F25: Tolerance and confidence regions 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Multiple comparisons simultaneous confidence interrals inclusion-exclusion identities discrete tubes

Citation

Naiman, Daniel Q.; Wynn, Henry P. Inclusion-Exclusion-Bonferroni Identities and Inequalities for Discrete Tube-Like Problems via Euler Characteristics. Ann. Statist. 20 (1992), no. 1, 43--76. doi:10.1214/aos/1176348512. https://projecteuclid.org/euclid.aos/1176348512


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