A new generalization of the beta distribution

TL;DR

This paper introduces a novel generalization of the beta distribution using a cubic transformation, enhancing modeling flexibility for bounded data and enabling better data fitting and mean regression applications.

Contribution

It proposes a new cubic-transformed beta distribution with rational moments, discusses its properties, and explores modifications for improved modal behavior and broader distributional utility.

Findings

Better fit to unimodal bounded data than traditional beta distribution

Provides rational expressions for moments facilitating mean regression

Introduces a quadratic distribution derived from the Jacobian of the transformation

Abstract

The beta distribution is the best-known distribution for modelling doubly-bounded data, \eg percentage data or probabilities. A new generalization of the beta distribution is proposed, which uses a cubic transformation of the beta random variable. The new distribution is label-invariant like the beta distribution and has rational expressions for the moments. This facilitates its use in mean regression. The properties are discussed, and two examples of fitting to data are given. A modification is also explored in which the Jacobian of the transformation is omitted. This gives rise to messier expressions for the moments but better modal behaviour. In addition, the Jacobian alone gives rise to a general quadratic distribution that is of interest. The new distributions allow good fitting of unimodal data that fit poorly to the beta distribution, and could also be useful as prior distributions.