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.

翻译：厚尾概率分布在建模各类金融数据集方面极为有用且发挥着关键作用。本研究提出了一种双重方法论。首先，通过结合高斯分布与瑞利分布并采用反正切变换，构建了一个混合概率分布，旨在产生更厚重的尾部特征并提升与真实市场数据的契合度。文中亦讨论了所提模型的部分统计性质。此外，研究采用了关键的精算风险评估工具，如风险价值（VaR）、尾部风险价值（TVaR）与尾部方差（TV），以支持有效的风险管理实践。最后，通过一个保险数据集的应用案例展示了所提模型的适用性。相较于现有基准分布，所提模型展现出更优的拟合性能，彰显了其在金融风险评估中的实用价值。通过反正切变换结合高斯分布与瑞利分布的方法，在表征现实金融数据中常见的极端市场行为与尾部依赖性方面尤为成功。