Mitigating multicollinearity issues to improve the two-parameter estimators for a Beta regression model

Muteb Faraj Alharthi; Nadeem Akhtar; Muteb Faraj Alharthi; Nadeem Akhtar

doi:10.3934/math.2026785

AIMS Mathematics

2026, Volume 11, Issue 6: 19321-19337. doi: 10.3934/math.2026785

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Mitigating multicollinearity issues to improve the two-parameter estimators for a Beta regression model

Muteb Faraj Alharthi ^{1
,},
Nadeem Akhtar ^{2
,
,}

1.
Department of Mathematics and Statistics, College of Science, Taif University, Taif 21944, Saudi Arabia; Email: muteb.faraj@tu.edu.sa
2.
Government Degree College Achini Payan, Higher Education, Archives and Libraries Department Peshawar 25000, Khyber Pakhtunkhwa, Pakistan; Email: nadeemscholar@icp.edu.pk

Received: 10 March 2026 Revised: 30 May 2026 Accepted: 15 June 2026 Published: 30 June 2026
MSC : 62J07, 62F10, 62G05

The Maximum Likelihood Estimation (MLE) method is commonly used for parameter estimation in statistical models. However, it can be significantly affected by multicollinearity issues, which undermine the reliability of parameter estimates. In Beta regression models (BRMs) for continuous proportions in (0, 1), multicollinearity complicates estimation and produces unstable results when using MLE. To address this challenge, ridge-based and alternative regression estimators have been introduced for BRMs with logit links. Selecting a two-parameter estimator remains a challenge for obtaining reliable parameter estimates for the BRM. In this study, we proposed new two-parameter ridge estimators for the BRM that effectively mitigate multicollinearity. We evaluated the performance of our newly proposed estimators and existing methods based on the mean squared error (MSE) criterion as well as bias, variance, and predictive accuracy. Simulation results showed that the proposed estimators generally performed better than the existing methods across most scenarios. Furthermore, a real-world data application confirmed the simulation results, demonstrating that our proposed estimators provide reliable and robust parameter estimates compared to MLE and the other existing methods.
- maximum likelihood estimation,
- Beta regression model,
- Ridge parameter,
- Monte Carlo simulation,
- mean squared error
Citation: Muteb Faraj Alharthi, Nadeem Akhtar. Mitigating multicollinearity issues to improve the two-parameter estimators for a Beta regression model[J]. AIMS Mathematics, 2026, 11(6): 19321-19337. doi: 10.3934/math.2026785

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Abstract

The Maximum Likelihood Estimation (MLE) method is commonly used for parameter estimation in statistical models. However, it can be significantly affected by multicollinearity issues, which undermine the reliability of parameter estimates. In Beta regression models (BRMs) for continuous proportions in (0, 1), multicollinearity complicates estimation and produces unstable results when using MLE. To address this challenge, ridge-based and alternative regression estimators have been introduced for BRMs with logit links. Selecting a two-parameter estimator remains a challenge for obtaining reliable parameter estimates for the BRM. In this study, we proposed new two-parameter ridge estimators for the BRM that effectively mitigate multicollinearity. We evaluated the performance of our newly proposed estimators and existing methods based on the mean squared error (MSE) criterion as well as bias, variance, and predictive accuracy. Simulation results showed that the proposed estimators generally performed better than the existing methods across most scenarios. Furthermore, a real-world data application confirmed the simulation results, demonstrating that our proposed estimators provide reliable and robust parameter estimates compared to MLE and the other existing methods.

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