Dynamic Linkages for Multivariate Distributions with Given Nonoverlapping Multivariate Marginals

Abstract

One of the most useful tools for handling multivariate distributions with givenunivariatemarginals is the copula function. Using it, any multivariate distribution function can be represented in a way that emphasizes the separate roles of the marginals and of the dependence structure. Liet al.(1996) introduced an analogous tool, called linkage, which is useful for handling multivariate distributions with givenmultivariatemarginals. The goal of the present paper is to introduce a new kind of linkage, called thedynamic linkage, which can usefully handle multivariate life distributions (that is, distributions of non-negative random variables) by taking advantage of the time dynamics of the underlying lifetimes. Like the linkages of Liet al.(1996), the new dynamic linkage can be used for the study of multivariate distributions with given multivariate marginals by emphasizing the separate roles of the dependence structureamongthe given multivariate marginals and the dependence structurewithineach of the nonoverlapping marginals. Preservation of some setwise positive dependence properties, from the dynamic linkage functionLto the joint distributionFand vice versa, are studied. When two different distribution functions are associated with the same dynamic linkage (that is, have the same setwise dependence structure) we show that the cumulative hazard order among the corresponding multivariate marginal distributions implies an overall stochastic dominance between the two underlying distribution functions.