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

2024-08-27

AI Summary

This paper investigates risk aggregation properties of weighted averages under heavy-tailed distributions—such as Pareto, Fréchet, and Burr—with infinite means. Methodologically, it integrates stochastic order theory, extreme value analysis, probabilistic inequalities, and dependence modeling. For the i.i.d. case, it rigorously proves that any positively weighted average strictly dominates a single constituent variable in the usual stochastic order; moreover, it constructs a novel class of heavy-tailed distributions satisfying this stochastic dominance property—a first in the literature. The results are extended to negatively dependent and non-identically distributed settings, establishing stochastic dominance of weighted averages over the corresponding mixture distributions. These theoretical advances provide a rigorous foundation for ordinal optimization in financial risk aggregation, reinsurance pricing, and extreme-event assessment.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

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, Fr'echet, 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.

Problem

• Study heavy-tailed distributions for risk aggregation
• Analyze stochastic dominance in weighted averages
• Generalize dominance to non-identical dependent variables

Innovation

• New heavy-tailed distributions class introduced
• Weighted averages dominate single variables
• Generalized to non-identical, dependent variables