A new quantile regression model for bounded responses: mathematical derivation, estimation, and fitting bounded datasets

Abstract

Bounded response variables, such as rates, proportions, and indices, frequently arise in the applied sciences and require regression models that respect the unit interval support, rather than relying on Gaussian regression. In this work, we propose a new regression model for continuous bounded data based on the unit Gumbel–logistic distribution as a flexible alternative to the existing models available in the literature. By reparameterizing the underlying distribution in terms of a location-type parameter directly linked to the covariate structure, our proposal enables the effect of explanatory variables on the response to be modeled in a natural and interpretable way. The proposed regression is derived from the reparameterization of the unit Gumbel–logistic distribution, which possesses a simple probability density function and a closed-form cumulative distribution function, which facilitates the derivation of its main properties. Parameter estimation is carried out using the maximum likelihood estimation method, and the finite-sample performance of the estimators is evaluated through an extensive Monte Carlo simulation study, comprising simulations for residual analysis and for the model. The proposed regression model is further compared with nine established competitors, including the beta, unit Kumaraswamy, unit Lindley, and unit log-log regression models, using two real-data applications. In both datasets, the unit Gumbel–logistic regression model achieved the highest log-likelihood and the lowest AIC, AICc, and BIC, outperforming its closest competitor, the beta regression model. Residual diagnostics and quantile-process analysis confirmed stable, well-behaved parameter estimates in both applications, supporting the unit Gumbel–logistic regression model as a valuable addition to the toolkit of regression models for bounded data.