A Modified Fréchet–Gumbel Distribution for Modeling Lifetime and Extreme Value Data

Accurate modeling of lifetime and extreme value data is crucial in environmental, engineering, and biomedical applications, where skewed or heavy-tailed behavior is common. However, many existing models, including the classical Fréchet and Gumbel families, lack the flexibility to simultaneously capture both lower-tail and upper-tail extremes, particularly for data defined over the entire real line. To address this limitation, we develop a modified Fréchet–Gumbel (MFG) distribution, a novel one-parameter extension that integrates the flexibility of the modified Fréchet generator with the wide applicability of the Gumbel model. The MFG distribution introduces a shape parameter that enhances tail adaptability and asymmetry control while maintaining analytical simplicity. We derive its fundamental properties—including the probability density, cumulative distribution, survival, and hazard functions—and estimate parameters using the maximum likelihood method. A Monte Carlo simulation study evaluates estimator performance under varying sample sizes and parameter settings. The proposed model’s practical relevance is demonstrated through three real datasets (annual maximum precipitation, flood data, and cancer survival times). Goodness-of-fit statistics (log-likelihood, AIC, BIC, and KS tests) confirm that the MFG model provides superior or comparable fit to benchmark distributions, particularly for extreme observations. Overall, the MFG distribution offers a theoretically sound and empirically flexible alternative for modeling heavy-tailed and asymmetric data in environmental, reliability, and biomedical studies.