Characterizing mutual exclusivity as the strongest negative multivariate dependence structure

Abstract

Mutual exclusivity is an extreme negative dependence structure that was first proposed and studied in Dhaene and Denuit (1999) (The safest dependence structure among risks. Insurance: Mathematics and Economics 25, 11-21) in the context of insurance risks. In this article, we revisit this notion and present versatile characterizations of mutually exclusive random vectors via their pairwise counter-monotonic behaviour, minimal convex sum property, distributional representation and the characteristic function of the sum of their components. These characterizations highlight the role of mutual exclusivity in generalizing counter-monotonicity as the strongest negative dependence structure in a multi-dimensional setting.

On Minimal Copulas under the Concordance Order

“…hold for every continuous and strictly concordance order preserving measure of concordance κ. Indeed, for d = 3, consider the minimal copula ν 1 (M) and the minimal copula C discussed in Example 3.3 (3). Then, by [27,Example 7], we obtain…”

Two Sufficient Conditions for Convex Ordering on Risk Aggregation

“…More details about comonotonicity and mutual exclusivity can be found in Dhaene and Denuit [10], Cheung and Lo [15,16], Mesfioui and Denuit [17], and Puccetti and Wang [18]. …”

Sharp Convex Bounds on the Aggregate Sums–An Alternative Proof

“…To prove Theorem 2, as in the proof to Lemma 3.6 in [27], there are three cases to consider. Recall that l 1 , .…”

“…, F n are two-point distributions and in Dhaene and Denuit [25] in a more general setting. A revisited and further characterized treatment of mutual exclusivity can be found in [27]. A recent overview paper by Puccetti and Wang [19] introduced the concept of pairwise countermonotonicity, which is more general than that of mutual exclusivity.…”

“…Moreover, several equivalent conditions on pairwise countermonotonicity have been provided (see Theorem 3.3 in [19]). [27]) Let X 1 , . .…”