The Annals of Statistics

The Multivariate Inclusion-Exclusion Formula and Order Statistics from Dependent Variates

Willi Maurer and Barry H. Margolin

Full-text: Open access

Abstract

A variant of the general multivariate inclusion-exclusion formula of Meyer (1969) is derived for the case where $K$ classes of events are considered and specific subsets of the events, one from each class, are related to one another by set inclusion. This result, in turn, yields a formula for the cumulative distribution function of any subset of order statistics from dependent random variables in terms of cumulative distribution functions of subsets of the unordered variables. An important example of dependent random variables, where the variables are jointly distributed as a Dirichlet $D_n(1, 1, \cdots, 1)$, is discussed in detail; various authors' results for this distribution are extended, or rederived as special cases via the formulae presented.

Article information

Source
Ann. Statist., Volume 4, Number 6 (1976), 1190-1199.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176343650

Digital Object Identifier
doi:10.1214/aos/1176343650

Mathematical Reviews number (MathSciNet)
MR426287

Zentralblatt MATH identifier
0351.62034

JSTOR
links.jstor.org

Subjects
Primary: 62G30: Order statistics; empirical distribution functions
Secondary: 60C05: Combinatorial probability

Keywords
Multivariate inclusion-exclusion dependent random variables exchangeable random variables order statistics Dirichlet distribution

Citation

Maurer, Willi; Margolin, Barry H. The Multivariate Inclusion-Exclusion Formula and Order Statistics from Dependent Variates. Ann. Statist. 4 (1976), no. 6, 1190--1199. doi:10.1214/aos/1176343650. https://projecteuclid.org/euclid.aos/1176343650


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