A New Extended Fréchet Distribution: Properties, Characterizations, Modeling, and Risk Analysis for Medical Data

Measures like Value-at-Risk (VaR), Tail Value-at-Risk (TVaR), Mean-of-Order-P (MOO-P), and Peaks Over a Random Threshold VaR (PORT-VaR) are important in risk analysis, particularly when it comes to medical data. In order to better capture extreme occurrences in medical applications, this work presents the Extended Rayleigh-Fréchet (ER-Fr) model, a novel extreme value distribution. We derive and discuss several of their key mathematical and statistical properties that are particularly useful for risk assessment. A comprehensive simulation study is conducted to evaluate the performance of the maximum likelihood estimators under various sample sizes, confirming the model’s reliability. The practical utility of the ER-Fr distribution is demonstrated through the analysis of two real medical datasets: one on relief times for arthritic patients and another on survival times of guinea pigs exposed to tuberculosis. The model is compared with several existing Fréchet-type distributions using standard goodness-of-fit criteria, and it consistently outperforms its competitors. Based on this analysis, we provide actionable insights and recommendations for healthcare practitioners and risk analysts. The results highlight the ER-Fr model as a powerful and flexible tool for modeling extreme values and assessing risk in medical studies.