Power-Transformed Bradford Distribution: Properties, Quantile Regression, and Empirical Illustrations

The representation of uncertainty that defines random processes constrained by the unit support, like rates and proportions, is a necessary undertaking in various fields of study, including economics, psychology, and public health. Although the beta distribution is the default option, it can be unsuitable for large-scale practical uses because of some problems with observations that are bound, stiff variance–mean functions, and lack of a closed-form quantile. Moreover, despite the unit-support data desirable characteristics of the Bradford distribution, the single-parameter form prevents widespread adaptability to model the data patterns in complex situations. To overcome these shortcomings, this paper introduces a new power-transformed Bradford (PTB) distribution derived as a result of the transformation of the Bradford distribution to increase its flexibility with the inclusion of an extra shape parameter. The approach includes the derivation of major statistical properties and parameter estimation assessment based on six different procedures, which are evaluated using Monte Carlo simulations. Using the tractability property of the quantile function of the PTB, a novel quantile regression (QR) model is formulated, which is used to estimate a relationship between the conditional quantile of the regressors and the dependent variables. Empirical examples on three real-life datasets show that the PTB distribution has a better parametric fit than extant unit-support distributions in terms of Akaike, Bayesian, and Kolmogorov–Smirnov statistics. Moreover, the PTB QR model also has good performance in biomedical applications unlike the competing median regression models. The primary value of this research is the presentation of resilient unit distribution, which has flexible density and hazard rate forms, and the formulation of a sound QR model, which can be utilized for inference of limited responses, especially where the common mean-based regression is suppressed by asymmetry or extreme values.