Risk aggregation and stochastic dominance for a class of heavy-tailed distributions

Paper: arXiv 2408.15033

Authors: Yuyu Chen, Seva Shneer

Abstract

We introduce a new class of heavy-tailed distributions for which any weighted average of independent and identically distributed random variables is larger than one such random variable in (usual) stochastic order. We show that many commonly used extremely heavy-tailed (i.e., infinite-mean) distributions, such as the Pareto, Fréchet, and Burr distributions, belong to this class. The established stochastic dominance relation can be further generalized to allow negatively dependent or non-identically distributed random variables. In particular, the weighted average of non-identically distributed random variables dominates their distribution mixtures in stochastic order.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a strong theoretical framework introducing a new class of heavy-tailed, infinite-mean distributions for stochastic dominance under negative dependence. While mathematically rigorous and novel in probability theory and operational risk modeling, it is entirely theoretical with no empirical data or backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Analyze risk aggregation & stochastic dominance for infinite-mean heavy-tailed distributions"] --> B["Input Distributions: Pareto, Fréchet, Burr (IID & Non-IID)"]
    B --> C["Methodology: Formulate weighted averages of heavy-tailed RVs"]
    C --> D["Theoretical Analysis: Evaluate usual stochastic order & allow negative dependence"]
    D --> E["Key Outcome 1: Weighted average > single RV in stochastic order"]
    D --> F["Key Outcome 2: Weighted average of non-IID RVs dominates their distribution mixtures"]