Power Length-Biased New XLindley Distribution: Properties and Modeling of Real Data

The increasing complexity of modern lifetime data necessitates the development of more flexible probability models. To address this need, we propose the power lengthbiased new XLindley (PLNXL) distribution, a novel two-parameter model tailored to model a wide range of lifetime datasets. Characterized by both shape and scale parameters, the PLNXL distribution effectively captures diverse hazard rate functions, including increasing, decreasing, and inverted bathtub-shaped forms. Additionally, its mean residual life function is capable of exhibiting decreasing, increasing, and bathtub-shaped behaviors, thereby enhancing its practical relevance. We derive key mathematical properties of the distribution, including moments, reliability measures, and entropy. The parameters are estimated using the maximum likelihood method, and simulation studies confirm the consistency and efficiency of the estimators. The applicability of the proposed model is illustrated using real-world datasets, where it consistently outperforms the existing models. These results highlight the robustness and adaptability of the PLNXL distribution for lifetime data analysis across a wide array of applications.