## The Annals of Mathematical Statistics

### Association of Random Variables, with Applications

#### Abstract

It is customary to consider that two random variables $S$ and $T$ are associated if $\operatorname{Cov}\lbrack S, T\rbrack = EST - ES\cdot ET$ is nonnegative. If $\operatorname{Cov}\lbrack f(S), g(T)\rbrack \geqq 0$ for all pairs of nondecreasing functions $f, g$, then $S$ and $T$ may be considered more strongly associated. Finally, if $\operatorname{Cov}\lbrack f(S, T), g(S, T)\rbrack \geqq 0$ for all pairs of functions $f, g$ which are nondecreasing in each argument, then $S$ and $T$ may be considered still more strongly associated. The strongest of these three criteria has a natural multivariate generalization which serves as a useful definition of association: DEFINITION 1.1. We say random variables $T_1,\cdots, T_n$ are associated if \begin{equation*}\tag{1.1}\operatorname{Cov}\lbrack f(\mathbf{T}), g(\mathbf{T})\rbrack \geqq 0\end{equation*} for all nondecreasing functions $f$ and $g$ for which $Ef(\mathbf{T}), Eg(\mathbf{T}), Ef(\mathbf{T})g(\mathbf{T})$ exist. (Throughout, we use $\mathbf{T}$ for $(T_1,\cdots, T_n)$; also, without further explicit mention we consider only test functions $f, g$ for which $\operatorname{Cov}\lbrack f(\mathbf{T}), g(\mathbf{T})\rbrack$ exists.) In Section 2 we develop the fundamental properties of association: Association of random variables is preserved under (a) taking subsets, (b) forming unions of independent sets, (c) forming sets of nondecreasing functions, (d) taking limits in distribution. In Section 3 we develop some simpler criteria for association. We show that to establish association it suffices to take in (1.1) nondecreasing test functions $f$ and $g$ which are either (a) binary or (b) bounded and continuous. In Section 4 we develop the special properties of association that hold in the case of binary random variables, i.e., random variables that take only the values 0 or 1. These properties turn out to be quite useful in applications. We also discuss association in the bivariate case. We relate our concept of association in this case to several discussed by Lehmann (1966). Finally, in Section 5 applications in probability and statistics are presented yielding results by Robbins (1954), Marshall-Olkin (1966), and Kimball (1951). An application in reliability which motivated our original interest in association will be presented in a forthcoming paper.

#### Article information

Source
Ann. Math. Statist. Volume 38, Number 5 (1967), 1466-1474.

Dates
First available in Project Euclid: 27 April 2007

http://projecteuclid.org/euclid.aoms/1177698701

Digital Object Identifier
doi:10.1214/aoms/1177698701

Mathematical Reviews number (MathSciNet)
MR217826

Zentralblatt MATH identifier
0183.21502

JSTOR