A new quantile regression model for bounded responses: mathematical derivation, estimation, and fitting bounded datasets
7 affiliations
- 1Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University
- 2Department of Mathematics, Faculty of Science, Tanta University
- 3Department of Statistics, Faculty of Science, University of Tabuk
- 4Department of Mathematics, College of Science, Umm Al-Qura University
- 5Department of Mathematics, College of Science, Northern Border University
- 6Department of Accounting, College of Business Administration in Hawtat Bani Tamim, Prince Sattam bin Abdulaziz University
- 7Department of Mathematics, College of Natural Science, Debre Tabor University
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.
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BibTeX
@article{alqasem2026quantile,
title = {{A new quantile regression model for bounded responses: mathematical derivation, estimation, and fitting bounded datasets}},
author = {Ohud A. Alqasem and Ahmed M. Gemeay and Dalia Kamal Alnagar and Hazar A. Khogeer and Amani Alrumayh and Eslam Hussam and Gizachew Tirite Gellow and Ali T. Hammad},
journal = {Boundary Value Problems},
year = {2026},
month = oct,
doi = {10.1186/s13661-026-02369-8},
url = {https://doi.org/10.1186/s13661-026-02369-8},
note = {CC BY},
}
RIS
TY - JOUR
TI - A new quantile regression model for bounded responses: mathematical derivation, estimation, and fitting bounded datasets
AU - Ohud A. Alqasem
AU - Ahmed M. Gemeay
AU - Dalia Kamal Alnagar
AU - Hazar A. Khogeer
AU - Amani Alrumayh
AU - Eslam Hussam
AU - Gizachew Tirite Gellow
AU - Ali T. Hammad
PY - 2026
DA - 2026/10/25
JO - Boundary Value Problems
DO - 10.1186/s13661-026-02369-8
UR - https://doi.org/10.1186/s13661-026-02369-8
AB - 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.
SN - 1687-2770
SN - 1687-2762
ER -