Novel family of probability generating distributions: Properties and data analysis

Current probability distributions often fall short when modeling lifespan data with non-monotonic hazard rate shapes, highlighting a gap in probability theory. To address this issue, this study introduces the novel family of probability-generating distributions (NP-G) as a solution to capture complex data patterns better. This research aims to develop and validate the NP-G distributions, which include a two-parameter continuous model based on the Weibull distribution. The study extensively explores the mathematical properties of the NP-G distributions, including series expansions of probability density functions, and evaluates various probabilistic measures such as moments, stress strength, mean deviations, entropy measures, and order statistics. Using the maximum likelihood method for parameter estimation and conducting a numerical simulation analysis, the study assesses the performance of the proposed extended exponentiated Weibull model. The findings demonstrate that this new model offers superior fit and accuracy compared to existing distributions when applied to two referenced datasets. This advancement is significant as it provides a more effective tool for analyzing lifespan data with non-monotonic hazard rates, enhancing both theoretical understanding and practical applications in fields such as reliability engineering and survival analysis.