Weak Dependence Notions and Their Mutual Relationships

Abstract

New weak notions of positive dependence between the components X and Y of a random pair (X,Y) have been considered in recent papers that deal with the effects of dependence on conditional residual lifetimes and conditional inactivity times. The purpose of this paper is to provide a structured framework for the definition and description of these notions, and other new ones, and to describe their mutual relationships. An exhaustive review of some well-know notions of dependence, with a complete description of the equivalent definitions and reciprocal relationships, some of them expressed in terms of the properties of the copula or survival copula of (X,Y), is also provided.

Dependence among order statistics for time-transformed exponential models

“…In this respect, it might be interesting to combine those characterizations with the more general comparison results presented in [22]. It might also be interesting to check if the condition in item 5, it being weaker than PQD, could be seen as a special property of weak dependence for (X 1 , X 2 ) according to the theory developed by Navarro et al [23].…”

Directional Dependence Orders of Random Vectors

“…Thus, copulas link joint distribution functions to their one-dimensional margins. For a survey on copulas, see [4,21] and some results about positive dependence properties and ordering by using copulas can be seen, for instance, in [4,5,9,[22][23][24][25].…”

Stochastic Comparisons of Some Distances between Random Variables

“…Intuitively, if X = (X 1 , X 2 ) is PDS, then its components are more likely simultaneously to have large values, compared with a vector of independent random variables with the same marginal distributions. For relationships between this and other dependence notions see, for example, Table 2 in [25]. The negative dependence analog of Definition 5 is as follows (see [24]).…”