Stochastic dominance of sums of risks under dependence conditions

TL;DR

The paper addresses when the sum $Y=X+Z$ is riskier than $X$ under potential dependence between $X$ and $Z$. It adopts a copula-based framework, leveraging recent results by Guan et al. to translate risk-order relations into conditions on the joint copula $C$ and the distribution of $Z$, with symmetry or skew constraints playing a key role. Main contributions include distribution-free sufficiency results for $X \leq_{ICX} X+Z$, $X \leq_{CX} X+Z$, and $X \leq_{ICV} X+Z$ under weak dependence notions ($wPQD$, $sPQD$) and $PQDE$, the introduction of new dependence concepts, and explicit counterexamples showing the hierarchy and limitations of these notions. The findings provide robust tools for risk comparisons in actuarial and reliability settings without fixing marginals, highlighting how dependence structure and environment-driven noise influence risk ordering.

Abstract

We provide conditions for the stochastic dominance comparisons of a risk $X$ and an associated risk $X+Z$, where $Z$ represents the uncertainty due to the environment and where $X$ and $Z$ can be dependent. The comparisons depend on both the copula $C$ between the distributions of $X$ and $Z$ and on the distribution of $Z$. We provide two different conditions for $C$ which represents new positive dependence properties. Regarding $Z$, we need some symmetry or asymmetry (skew) properties. Some illustrative examples are provided.

Figures (2)

• Figure 1: Plot and contour plot of the PDF for the copula in Example \ref{['ex1']}.
• Figure 2: Supports of the (uniform) copula functions in Examples \ref{['ex2']} (left) and \ref{['ex3']} (right) for $\delta=0.1$.

Theorems & Definitions (38)

• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4: Theorem 1 in GHW24
• Proposition 2.5: Corollary 4 in GHW24
• Proposition 2.6: Corollary 5 in GHW24
• Definition 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4