Power function pseudo-convex mixtures in reliability

Pseudo-convex mixtures, derived from shape-extended stable distributions for extremes, are a flexible class of distributions. Previous research has established the foundational framework for these shape-extended stable distributions and outlined their key properties. Within this class, the power function distribution emerges as particularly noteworthy due to its capacity to generate pseudo-convex mixtures well-suited for modeling data with a bathtub-shaped (U-shape) hazard rate. In reliability theory, such a hazard rate curve is characterized by an initial decline, followed by a relatively stable period, and finally a subsequent increase. To model this commonly observed behavior in the real-world, a wide range of statistical distributions has been employed, frequently involving inverse Weibull distributions or extensions and transformations of the classic Weibull model. This study proposes a pseudo-convex mixture derived from the power function distribution to model data characterized by a bathtub-shaped hazard rate. Our model represents an original contribution to this important problem, since a theoretical study concerning the behavior of hazard rate is presented. Besides, estimation procedure is detailed and validated through simulation and real data adjustment. Empirical results demonstrate that the proposed model provides a credible and competitive alternative, outperforming several commonly used models in capturing the distinct features of bathtub-shaped hazard rate behavior. For the dataset concerning the time in hours to electrical breakdown of insulating fluid, our adjustment is better than all the others already presented in previous works.