Fractional-linear-transformation-based probability distributions: Properties, examples, Hausdorff saturation, and applications

Source abstract

Probability distributions defined on a bounded interval are essential tools in data science, engineering, and risk management. The most important practical reason for their existence is the need to model real-world phenomena that possess strict physical, logical, or economic boundaries. This paper introduces a new class of probability distributions based on the fractional linear transform applied to a distribution defined on the positive real half-line. Consequently, the distribution’s infinite support is mapped onto a bounded interval. We can assume without loss of generality that this interval is (0, 1). We first derive some general theoretical results for the probability density functions, quantile functions, and other probability properties. A duality between distributions is established. Consequently, we investigate several particular cases based on the gamma, inverse gamma, Rayleigh, and generalized inverse Gaussian distributions. The so-called saturation in the Hausdorff sense is discussed. It provides a first-order approximation of the cumulative distribution function to the Γ-shaped curve {(0, 0) → (0, 1) → (1, 1)}. Thus, it can be viewed as a break-even point and can be useful in economics, finance, risk management, and other natural sciences, such as chemistry, physics, meteorology, etc. We also investigate the maximum likelihood estimator and check its asymptotic properties, consistency, and stability. We apply this method, as well as the least-squares approach, to two empirical datasets: one for yield curves and another for unemployment insurance issues. The statistical results strongly confirm the usefulness of the proposed model.