A New Class of Probability Distributions via Half-Elliptical Functions

In this paper, we develop a new family of distributions supported on a bounded interval with a probability density function that is constructed from two elliptical arcs. The distribution can take on a variety of shapes and has three basic parameters: minimum, maximum, and mode. Compared to classical bounded distributions such as the beta and triangular distributions, the proposed semi-elliptical family offers greater flexibility in capturing diverse shapes of distributions, in symmetric and asymmetric settings. Its construction from elliptical arcs enables smoother transitions and more natural tail behaviors, making it suitable for applications where classical models may exhibit rigidity or over-simplicity. We give general expression for the density and distribution function of the new distribution. Properties of this distribution are studied and parameter estimation is discussed. Monte Carlo simulation results show the performance of our estimators under many sets of situations. Furthermore, we show the advantages of our distribution over the commonly used triangular distribution in approximating beta distributions.