Probability inequalities for multivariate distributions with dependence structures. (English) Zbl 0546.62024

Glaz, Joseph; Johnson, Bruce McK.

J. Am. Stat. Assoc. 79, 436-440 (1984).

Summary: Let \(X_ 1,...,X_ n\) be a sequence of random variables with a given positive or negative dependence structure. In this article we exploit the assumed dependence structure to construct a sequence of bounds for the \(P(X_ i\in C_ i\); \(i=1,...,n)\), where \(C_ i\) are infinite intervals of the same type. These bounds are superior to the well-known product bounds that are based solely on the marginal probabilities. Moreover, the new bounds can serve as respectable approximations for the \(P(X_ i\in C_ i\); \(i=1,...,n)\).

MSC:

62H05	Characterization and structure theory for multivariate probability distributions; copulas
60E15	Inequalities; stochastic orderings

Keywords:

approximations; dependence structure; bounds; marginal probabilities