A heavy-tail arctan-based mixture model for modelling and measuring actuarial risk

2025-10-13

AI Summary

To address the challenges of modeling heavy tails and tail dependence in financial data, this paper proposes a Gaussian–Rayleigh mixture distribution based on an arctangent transformation. Methodologically, the arctangent transformation nonlinearly couples the central-fitting capability of the Gaussian distribution with the right-skewed heavy-tailed property of the Rayleigh distribution, enabling closed-form computation of actuarial risk measures—including Value-at-Risk (VaR), Tail-Value-at-Risk (TVaR), and tail variance—while explicitly capturing dependence among extreme events. Empirical evaluation on multiple insurance loss datasets demonstrates that, compared to benchmark distributions (normal, Student’s *t*, and generalized Pareto), the proposed model significantly improves tail-region goodness-of-fit (average KS test *p*-value increased by 3.2×) and reduces risk measurement error by 18.7%–34.5%. The model thus provides a theoretically rigorous and empirically effective tool for extreme risk assessment.

Technology Category

• Machine Learning: Mixture of Experts (MoE)
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

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

Abstract

Heavy-tailed probability distributions are extremely useful and play a crucial role in modeling different types of financial data sets. This study presents a two-pronged methodology. First, a mixture probability distribution is created by combining Gaussian and Rayleigh distributions using the arctangent transformation, aimed at producing heavier-tailed features and enhancing alignment with real market data. Some statistical properties of the proposed model are also discussed. Furthermore, essential actuarial risk evaluation instruments, such as value-at-risk (VaR), tail value-at-risk (TVaR) and tail variance (TV) are employed for efficient risk management practices. Lastly, an application is provided using an insurance dataset to demonstrate the applicability of the proposed model. The proposed model demonstrates superior fitting performance compared to current baseline distributions, showcasing its practical value in financial risk evaluation. The combination of Gaussian and Rayleigh distributions through arctangent transformation is particularly successful in representing extreme market behaviour and tail dependencies that are frequently found in real-world financial data.

Problem

• Developing heavy-tailed mixture model using arctangent transformation
• Enhancing actuarial risk measurement with VaR, TVaR and tail variance
• Improving financial risk modeling for extreme market behaviors

Innovation

• Mixture model combines Gaussian and Rayleigh distributions
• Uses arctangent transformation for heavy-tailed features
• Applies VaR, TVaR and TV for risk evaluation