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In this paper, we propose a new family of continuous probability distributions, referred to as the flexible exponential-G (FEx-G) family, which generalizes and extends several well-known distributions. This family is highly flexible and possesses desirable properties. We introduce a variety of new distributions as special cases within the FEx-G family, including the well-known flexible-Weibull distribution. A key special case, the flexible exponential-inverse Lomax-Lomax (FExILL) distribution, is studied in detail. We present characterizations of the FExILL distribution based on its hazard function, and derive the probability density function for the order statistics of this distribution. Additionally, we discuss seven methods for estimating the parameters of the FExILL distribution. To evaluate the performance of these estimation methods, we conduct a simulation study. Finally, we demonstrate the practical application and flexibility of the FExILL distribution by modeling three real-world datasets from applied fields such as environmental, mechanical engineering, and reliability engineering. The results show that the FExILL distribution outperforms other competing distributions, making it a robust choice for modeling real-world data.
Abbasi et al. (2025) studied this question.