A New Two-Parameter Model: Bayesian and Non- Bayesian Risk Actuarial Analysis with Applications and Two Case Studies Under the Peaks over Random Threshold Analysis in Economy and Insurance

This study introduces a new two-parameter exponential (TPEX) model for modeling skewed phenomena and risk analysis, motivated by the need for flexible yet tractable models capturing asymmetric behavior in actuarial, financial, and reliability data. An extensive simulation study evaluated seven estimation procedures: maximum likelihood estimation (MLE), ordinary least squares (OrLS), weighted least squares (WLSQ), Cramér–von Mises (CVM), Anderson–Darling estimation (ADE), Kolmogorov estimation (KE), L-moments, and Bayesian estimation, comparing bias, efficiency, and stability across sample sizes and parameter settings. Four real-data applications were conducted: two comparing estimation methods on relief and survival datasets and two assessing competitive performance against exponential-type models. Key risk indicators (KRIs), including the Value at Risk (VaR), Tail Value at Risk (TVaR), Tail Variance (TV), Tail Mean–Variance (TMV), and expected loss (EL), were computed using UK motor non-comprehensive claims and US house price data, illustrating the model’s relevance for insurance reserving and market risk assessment.