Open Access
March, 1983 Negative Association of Random Variables with Applications
Kumar Joag-Dev, Frank Proschan
Ann. Statist. 11(1): 286-295 (March, 1983). DOI: 10.1214/aos/1176346079


Random variables, $X_1, \cdots, X_k$ are said to be negatively associated (NA) if for every pair of disjoint subsets $A_1, A_2$ of $\{1, 2, \cdots, k\}, \operatorname{Cov}\lbrack f(X_1, i \in A_1), g(X_j, j \in A_2) \rbrack \leq 0$, for all nondecreasing functions $f, g$. The basic properties of negative association are derived. Especially useful is the property that nondecreasing functions of mutually exclusive subsets of NA random variables are NA. This property is shown not to hold for several other types of negative dependence recently proposed. One consequence is the inequality $P(X_i \leq x_i, i = 1, \cdots, k) \leq \prod^k_1P(X_i \leq x_i)$ for NA random variables $X_1, \cdots, X_k$, and the dual inequality resulting from reversing the inequalities inside the square brackets. In another application it is shown that negatively correlated normal random variables are NA. Other NA distributions are the (a) multinomial, (b) convolution of unlike multinomials, (c) multivariate hypergeometric, (d) Dirichlet, and (e) Dirichlet compound multinomial. Negative association is shown to arise in situations where the probability measure is permutation invariant. Applications of this are considered for sampling without replacement as well as for certain multiple ranking and selection procedures. In a somewhat striking example, NA and positive association representing quite strong opposing types of dependence, are shown to exist side by side in models of categorical data analysis.


Download Citation

Kumar Joag-Dev. Frank Proschan. "Negative Association of Random Variables with Applications." Ann. Statist. 11 (1) 286 - 295, March, 1983.


Published: March, 1983
First available in Project Euclid: 12 April 2007

zbMATH: 0508.62041
MathSciNet: MR684886
Digital Object Identifier: 10.1214/aos/1176346079

Primary: 62H05
Secondary: 62H10

Keywords: association , conditionally decreasing in sequence , multivariate inequalities , negative association , Negative dependence , negative orthant dependence , negatively correlated normal random variables , negatively dependent in sequence , reverse rule of order 2

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 1 • March, 1983
Back to Top