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.