A Novel Chen Extension for Risk Analysis with MOOP and PORT-VAR Assessments under Hydrological Flow Data and Financial Case Study

This paper introduces a new extension of the Chen distribution, designed to better model extreme low-flow events in hydrology and rare events in the medical field. The proposed model incorporates asymmetrical and heavy-tailed behavior, making it particularly useful for analyzing extreme values in complex real datasets. We derive the mathematical properties of the BGC distribution and apply two advanced analytical techniques: the Mean-of-Order-P (MOOP) method to determine the optimal value of P (referred to as Opt-P), and the Peaks Over Threshold Value-at-Risk (PORT-VaR) approach to identify and assess critical extreme events. These methods are applied to real datasets including relief times, minimum river flow data from the Cuiabá River, and U.S. indemnity losses from general liability claims. The MOOP analysis shows that increasing the order P leads to reduced Mean Squared Error (MSE) and Bias, indicating improved estimation accuracy. For example, in the relief times dataset, MSE decreases from 0.64 at P=1 to 0.3844 at P=5. Similarly, for the minimum flow data, MSE drops from 4402.88 to 3684.27 with increasing P, highlighting the benefits of higher-order statistics in capturing central tendencies. Using PORT-VaR, we analyze extreme peaks under varying confidence levels (50%, 70%, 90%, and 99%) and compute key risk indicators such as Value-at-Risk (VaR) , Tail Value-at-Risk (TVaR) , Mean Excess Loss (MEXL) , Tail Variance (TV) , and Tail Mean Variance (TMV) . In the relief times dataset, VaR increases from 1.70 at 50% confidence to 3.055 at 99% confidence, demonstrating growing risk exposure at higher confidence levels. For the minimum flow data, VaR rises from 115.925 at 50% to 157.169 at 99%, underscoring the importance of adaptive risk thresholds in managing water scarcity and dam safety. A financial case study using U.S. indemnity loss data further validates the robustness of the BGC model in capturing tail behavior and estimating extreme risks. At the 99% confidence level, VaR reaches 170400 (in thousands of USD), and MEXL is 203411, illustrating the nonlinear growth of risk in heavy-tailed insurance claims. Finally, a comparative study under a historical financial claims data through an application.