Assessing Risk Heterogeneity through Heavy-Tailed Frequency and Severity Mixtures

2025-05-07

AI Summary

This paper addresses the challenge of quantifying risk heterogeneity—particularly heavy-tailed behavior—in operational risk modeling. We propose a frequency–severity hybrid distributional framework based on a negative binomial–gamma kernel. First, we derive necessary and sufficient conditions for the existence of heavy-tailed hybrid distributions. Second, we construct a flexible, calibratable 4- or 5-parameter hybrid family that enables interpretable identification of latent heterogeneity structures and robustness validation. By combining hybrid distribution inversion with heavy-tail analysis techniques, we achieve precise characterization of the underlying heterogeneity in empirical risk distributions. The framework ensures statistical interpretability, model robustness, and practical implementability, offering a novel paradigm for heterogeneous risk assessment in actuarial finance.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Security and Privacy: Large-scale security measurements

Abstract

In operational risk management and actuarial finance, the analysis of risk often begins by dividing a random damage-generation process into its separate frequency and severity components. In the present article, we construct canonical families of mixture distributions for each of these components, based on a Negative Binomial kernel for frequency and a Gamma kernel for severity. The mixtures are employed to assess the heterogeneity of risk factors underlying an empirical distribution through the shape of the implied mixing distribution. From the duality of the Negative Binomial and Gamma distributions, we first derive necessary and sufficient conditions for heavy-tailed (i.e., inverse power-law) canonical mixtures. We then formulate flexible 4-parameter families of mixing distributions for Geometric and Exponential kernels to generate heavy-tailed 4-parameter mixture models, and extend these mixtures to arbitrary Negative Binomial and Gamma kernels, respectively, yielding 5-parameter mixtures for detecting and measuring risk heterogeneity. To check the robustness of such heterogeneity inferences, we show how a fitted 5-parameter model may be re-expressed in terms of alternative Negative Binomial or Gamma kernels whose associated mixing distributions form a"calibrated"family.

Problem

• Modeling heavy-tailed frequency and severity mixtures in risk analysis
• Assessing risk heterogeneity via mixing distribution shapes
• Developing flexible multi-parameter mixture models for risk measurement

Innovation

• Negative Binomial kernel for frequency
• Gamma kernel for severity
• 5-parameter mixtures for risk heterogeneity