Inverse Lomax Frechet Distribution with Applications

In the world of engineering, medicine, and insurance, predicting how long something will last is a constant challenge. Whether it is a bridge, a human organ, or a piece of glass, everything eventually wears out or fails. To make sense of this inevitable decline, scientists rely on mathematical maps called probability distributions. These are not maps of land, but of time and chance, describing how likely an object is to survive up to a certain point and then break. For decades, researchers have used a handful of standard maps, such as the Frechet distribution, to chart these lifetimes. These tools work well for simple cases where the risk of failure either steadily increases or steadily decreases. However, the real world is rarely so simple. Many things face complex risks that change in unpredictable ways, bending and twisting in patterns that old, rigid maps cannot capture. When the standard tools fail to fit the messy reality of data, scientists must build new, more flexible models that can bend to the shape of the truth.

This is the territory explored by Sadiq Abubakar Ismaila and Taiwo Yakubu Hassan, who have crafted a new mathematical model called the Inverse Lomax Frechet distribution. Their work begins with the Frechet distribution, a classic tool used to describe extreme events like wind speeds or material strength. While useful, the original Frechet model has a fixed shape that limits its ability to describe complex survival patterns. To solve this, the researchers combined the Frechet distribution with a newer, more adaptable family of models known as the Inverse Lomax Generalized family. By weaving these two together, they created a four-parameter model that acts like a highly adjustable lens. Unlike its simpler predecessors, this new distribution can stretch and shrink to fit a wide variety of data shapes, including those with complex, non-linear patterns of failure. The researchers did not just propose this new shape; they rigorously tested its mathematical validity, proving that it is a sound probability model, and then derived the specific formulas needed to calculate its average behavior, its median, and how likely it is to fail at any given moment.

To ensure this new model works in practice, the team put it through a series of computer simulations. They generated thousands of fake data sets with known properties and asked the model to find the correct settings. The results were clear: as the amount of data grew, the model's estimates became increasingly accurate, settling precisely on the true values. This confirmed that the model is stable and reliable for real-world use. The researchers then took the next step, applying their new distribution to two very different sets of real-life data. The first set tracked the remission times of 128 bladder cancer patients, measuring how long they remained free of the disease. The second set recorded the breaking strength of 64 glass fibers, a critical measure for materials science. In both cases, the researchers compared their new Inverse Lomax Frechet model against three established competitors: the Marshall-Olkin Frechet, the Inverse Lomax, and the standard Frechet distributions.

The comparison revealed a distinct advantage for the new model. When fitted to the cancer remission data, the Inverse Lomax Frechet distribution provided a significantly better match than the other three options, capturing the nuances of the patient recovery times more accurately. Similarly, when applied to the glass fiber strength data, it outperformed the competition, offering a tighter and more precise description of how the material failed. The researchers used several standard statistical measures to judge these fits, and in every instance, the new model scored the best, indicating it was the most efficient and accurate tool for the job. The study concludes that this new distribution is not just a theoretical exercise but a practical improvement for analyzing lifetime data. By offering a more flexible way to model how things break down or recover, it provides scientists and engineers with a sharper tool for understanding the complex, often unpredictable nature of survival and failure in the physical world.