Multicollinearity poses significant challenges to parameter estimation in regression models, often undermining the reliability of traditional methods like maximum likelihood estimation (MLE). This study addresses the issue by evaluating and enhancing regularization techniques, specifically the ridge-based regression (RBR), the Liu regression estimator (LRE), and a modified two-parameter ridge estimator (MTPRE) within the context of the Beta regression model (BRM). However, the selection of appropriate shrinkage parameters remains a persistent challenge. To address this limitation, we propose an MTPRE that eliminates the need for shrinkage parameter tuning, thereby improving estimation stability and accuracy. Through extensive simulation studies, the MTPRE consistently outperformed the MLE, RBR, and LRE under severe multicollinearity based on the mean squared error (MSE). The effectiveness of proposed estimators was further validated using a real-world gasoline yield dataset having multicollinearity issues, where the MTPRE demonstrated superior predictive accuracy and estimation precision. These results highlight the potential of the MTPRE as a practical and efficient method for handling multicollinearity in regression analysis.
Citation: Asma Ahmad Alzahrani. Modified two-parameter ridge estimator for the Beta logistic model to mitigate multicollinearity[J]. AIMS Mathematics, 2026, 11(1): 543-557. doi: 10.3934/math.2026023
Multicollinearity poses significant challenges to parameter estimation in regression models, often undermining the reliability of traditional methods like maximum likelihood estimation (MLE). This study addresses the issue by evaluating and enhancing regularization techniques, specifically the ridge-based regression (RBR), the Liu regression estimator (LRE), and a modified two-parameter ridge estimator (MTPRE) within the context of the Beta regression model (BRM). However, the selection of appropriate shrinkage parameters remains a persistent challenge. To address this limitation, we propose an MTPRE that eliminates the need for shrinkage parameter tuning, thereby improving estimation stability and accuracy. Through extensive simulation studies, the MTPRE consistently outperformed the MLE, RBR, and LRE under severe multicollinearity based on the mean squared error (MSE). The effectiveness of proposed estimators was further validated using a real-world gasoline yield dataset having multicollinearity issues, where the MTPRE demonstrated superior predictive accuracy and estimation precision. These results highlight the potential of the MTPRE as a practical and efficient method for handling multicollinearity in regression analysis.
| [1] | L. Fahrmeir, G. Tutz, Multivariate statistical modelling based on generalized linear models, 2 Eds., New York: Springer, 2001. |
| [2] |
S. Ferrari, F. Cribari-Neto, Beta regression for modelling rates and proportions, J. Appl. Stat., 31 (2004), 799–815. https://doi.org/10.1080/0266476042000214501 doi: 10.1080/0266476042000214501
|
| [3] |
Z. W. Xiao, F. X. Zhu, L. F. Wang, R. K. Liu, F. Yu, An evaluation method of construction reliability of cable system of cable-stayed bridge based on Bayesian network, Int. J. Struct. Integrity, 15 (2024), 1027–1050. https://doi.org/10.1108/IJSI-06-2024-0079 doi: 10.1108/IJSI-06-2024-0079
|
| [4] |
H. Ma, S. P. Zhu, Y. Q. Guo, L. D. Pan, S. Y. Yang, D. B. Meng, Residual useful life prediction of the vehicle isolator based on Bayesian inference, Structures, 58 (2023), 105518. https://doi.org/10.1016/j.istruc.2023.105518 doi: 10.1016/j.istruc.2023.105518
|
| [5] | D. B. Meng, S. P. Zhu, Multidisciplinary design optimization of complex structures under uncertainty, Boca Raton: CRC Press, 2024. https://doi.org/10.1201/9781003464792 |
| [6] |
A. Zeileis, C. Kleiber, S. Jackman, Regression models for count data in R, J. Stat. Softw., 27 (2008), 1–25. https://doi.org/10.18637/jss.v027.i08 doi: 10.18637/jss.v027.i08
|
| [7] | N. R. Draper, H. Smith, Applied regression analysis, John Wiley & Sons, 1998. |
| [8] |
M. Qasim, K. Månsson, B. M. G. Kibria, On some Beta ridge regression estimators: method, simulation and application, J. Stat. Comput. Simul., 91 (2021), 1699–1712. https://doi.org/10.1080/00949655.2020.1867549 doi: 10.1080/00949655.2020.1867549
|
| [9] | S. Seifollahi, H. Bevrani, James-Stein type estimators in Beta regression model: simulation and application, Hacet. J. Math. Stat., 52 (2023), 1046–1065. |
| [10] | A. Erkoç, E. Ertan, Z. Y. Algamal, K. U. Akay, The Beta Liu-type estimator: simulation and application, Hacet. J. Math. Stat., 52 (2023), 828–840. |
| [11] |
A. E. Hoerl, R. W. Kennard, Ridge regression: biased estimation for nonorthogonal problems, Technometrics, 12 (1970), 55–67. https://doi.org/10.1080/00401706.1970.10488634 doi: 10.1080/00401706.1970.10488634
|
| [12] |
M. Amin, H. Ashraf, H. S. Bakouch, N. Qarmalah, James Stein estimator for the Beta regression model with application to heat-treating test and body fat datasets, Axioms, 12 (2023), 526. https://doi.org/10.3390/axioms12060526 doi: 10.3390/axioms12060526
|
| [13] |
Y. Asar, A. Genç, Two-parameter ridge estimator in the binary logistic regression, Commun. Stat. Simul. Comput., 46 (2017), 7088–7099. https://doi.org/10.1080/03610918.2016.1224348 doi: 10.1080/03610918.2016.1224348
|
| [14] |
D. Y. Dai, D. Wang, A generalized Liu-type estimator for logistic partial linear regression model with multicollinearity, AIMS Math., 8 (2023), 11851–11874. https://doi.org/10.3934/math.2023600 doi: 10.3934/math.2023600
|
| [15] |
W. B. Altukhaes, M. Roozbeh, N. A. Mohamed, Feasible robust Liu estimator to combat outliers and multicollinearity effects in restricted semiparametric regression model, AIMS Math., 9 (2024), 31581–31606. https://doi.org/10.3934/math.20241519 doi: 10.3934/math.20241519
|
| [16] |
A. T. Hammad, I. Elbatal, E. M. Almetwally, M. M. A. El-Raouf, M. A. El-Qurashi, A. M. Gemeay, A novel robust estimator for addressing multicollinearity and outliers in Beta regression: simulation and application, AIMS Math., 10 (2025), 21549–21580. https://doi.org/10.3934/math.2025958 doi: 10.3934/math.2025958
|
| [17] |
M. R. Abonazel, I. M. Taha, Beta ridge regression estimators: simulation and application, Commun. Stat. Simul. Comput., 52 (2023), 4280–4292. https://doi.org/10.1080/03610918.2021.1960373 doi: 10.1080/03610918.2021.1960373
|
| [18] |
K. J. Liu, A new class of blased estimate in linear regression, Commun. Stat. Theory Methods, 22 (1993), 393–402. https://doi.org/10.1080/03610929308831027 doi: 10.1080/03610929308831027
|
| [19] |
K. Månsson, B. M. G. Kibria, G. Shukur, On Liu estimators for the logit regression model, Econ. Model., 29 (2012), 1483–1488. https://doi.org/10.1016/j.econmod.2011.11.015 doi: 10.1016/j.econmod.2011.11.015
|
| [20] |
Z. Y. Algamal, M. R. Abonazel, Developing a Liu-type estimator in Beta regression model, Concurr. Comput. Pract. Exp., 34 (2022), e6685. https://doi.org/10.1002/cpe.6685 doi: 10.1002/cpe.6685
|
| [21] |
A. F. Lukman, B. Aladeitan, K. Ayinde, M. R. Abonazel, Modified ridge-type for the Poisson regression model: simulation and application, J. Appl. Stat., 49 (2022), 2124–2136. https://doi.org/10.1080/02664763.2021.1889998 doi: 10.1080/02664763.2021.1889998
|
| [22] |
N. Akhtar, M. F. Alharthi, Enhancing accuracy in modelling highly multicollinear data using alternative shrinkage parameters for ridge regression methods, Sci. Rep., 15 (2025), 10774. https://doi.org/10.1038/s41598-025-94857-7 doi: 10.1038/s41598-025-94857-7
|
| [23] |
S. Chand, B. M. G. Kibria, A new ridge type estimator and its performance for the linear regression model: simulation and application, Hacet. J. Math. Stat., 53 (2024), 837–850. https://doi.org/10.15672/hujms.1359446 doi: 10.15672/hujms.1359446
|
| [24] | N. H. Prater, Estimate gasoline yields from crudes, Petroleum Refiner, 35 (1956), 236–238. |
Figures(2) / Tables(5)
Asma Ahmad Alzahrani. Modified two-parameter ridge estimator for the Beta logistic model to mitigate multicollinearity[J]. AIMS Mathematics, 2026, 11(1): 543-557. doi: 10.3934/math.2026023
DownLoad: