Marshall–Olkin Alpha Power Transformed Extended-X Distribution and Its Applications

This paper introduces the Marshall–Olkin Alpha Power Transformed Extended-X (MOAPTE-X) family of distributions, a highly flexible class designed to address limitations of existing models. We develop a novel four-parameter distribution, called the Marshall–Olkin Alpha Power Transformed Extended-Inverted Kumaraswamy (MOAPTE-IK), with one scale parameter and three shape parameters, by extending the inverted Kumaraswamy distribution through the integration of the Marshall–Olkin method and alpha power transformations, thereby enhancing its capacity to model complex hazard rate functions. Some of the fundamental mathematical properties of the MOAPTE-IK, including its quantile function, moments, generating functions, mode, order statistics, and Rényi entropy, are derived, establishing its theoretical robustness. Parameter estimation is performed using the maximum likelihood approach, and the behavior of the resulting estimates is assessed using Monte Carlo simulation, demonstrating consistency, accuracy, and efficiency. Comparative analyses using two real datasets reveal that the MOAPTE-IK distribution consistently outperforms competing models such as Marshall–Olkin Extended Inverted Kumaraswamy (MOEIK) distribution, Alpha Power Inverted Kumaraswamy (APIK) distribution, and Inverted Kumaraswamy (IK) distribution. The suggested model consistently performs better than its competitors in terms of selection criteria (AIC, CAIC, BIC, HQIC) and goodness-of-fit tests (K-S, A∗, w∗). The proposed distribution offers a powerful, flexible tool for modeling complex data across multiple disciplines, including engineering, healthcare, and social sciences.