A flexible extension of the log-logistic model with diverse failure rate shapes and applications

In this article, we introduce an enhanced version of the log-logistic model, termed the Kumaraswamy alpha-power log-logistic (KAPLL) distribution. The KAPLL model expands upon the traditional log-logistic distribution and several well-established distributions. We investigate the mathematical properties of the KAPLL model, highlighting its ability to effectively model various aging and failure criteria. The KAPLL distribution exhibits remarkable flexibility in modeling various types of hazard rate behaviors. It is capable of accommodating a wide range of shapes, including increasing, decreasing, J-shaped, reversed J-shaped, bathtub-shaped, inverted bathtub-shaped, and even more complex forms such as decreasing-increasing-decreasing failure rates. The KAPLL distribution is characterized by its capacity to exhibit both symmetric and asymmetric shapes in its density function. The proposed KAPLL model overcomes key limitations of existing LL-based generalizations by offering enhanced flexibility in modeling diverse hazard rate shapes and tail behaviors. We estimate the KAPLL parameters using eight classical estimation methods. Comprehensive simulation results are presented and ranked to identify the most effective approach for estimating KAPLL parameters, which we believe will be of great interest to engineers and applied statisticians. To further demonstrate the versatility of the KAPLL distribution, we analyze five real-world datasets from reliability, engineering, biomedical, and environmental sciences, highlighting its flexibility relative to other extensions of the log-logistic model. Likelihood ratio tests conducted across five real datasets confirm that the KAPLL model provides a statistically significant improvement over the baseline log-logistic distribution.