## The Annals of Statistics

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

#### 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

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