The Sine–Fréchet G Family of Continuous Distributions: Sub-models, Properties, Simulation and Applications

Many real-world datasets exhibit skewness, heavy tails, and complex hazard-rate shapes that classical lifetime distributions and several existing generalized families fail to capture adequately. To address these limitations, this paper introduces the Sine–Fréchet-G family of continuous distributions, constructed by compounding the Sine-G generator with the Fréchet-G transformation. The proposed family provides enhanced flexibility for modeling asymmetric and heavy-tailed data while maintaining analytical tractability. Explicit expressions are derived for the cumulative distribution function, probability density function, survival and hazard rate functions, and quantile function. Series representations are obtained to facilitate the derivation of moments, the moment generating function, the characteristic function, and order statistics. Parameter estimation is carried out using the method of maximum likelihood, and the corresponding score equations are presented for a general baseline distribution. Several important sub-models, including the Sine–Fréchet Exponential, Weibull, Rayleigh, and Gompertz distributions, are developed and their validity is rigorously established. A Monte Carlo simulation study investigates the finite-sample behavior of the maximum likelihood estimators, demonstrating consistency and decreasing bias and mean squared error as sample size increases. The practical usefulness of the proposed family is illustrated through applications to three real-life datasets, where the Sine–Fréchet Exponential model outperforms competing models based on likelihood-based information criteria and goodness-of-fit measures. These results indicate that the Sine–Fréchet-G family constitutes a flexible and effective framework for modeling lifetime and reliability data with complex distributional features.