Quantile ridge beta regression model and Bayesian regularized quantile beta regression for solving the skewness and improving the accuracy of the model selection in bounded data

Fedaa Noeel Abdulahad; Majid Khan Majahar Ali; Raja Aqib Shamim

doi:10.46481/jnsps.2026.3629

Quantile ridge beta regression model and Bayesian regularized quantile beta regression for solving the skewness and improving the accuracy of the model selection in bounded data

Authors

Fedaa Noeel Abdulahad
School of Mathematical Sciences, Universiti Sains Malaysia, Gelugor 11800, Pulau Pinang, Malaysia;

Department of Mathematics, College of Education for Pure Science, University of Al-Hamdaniya, Bartella 41006, Mosul, Iraq
Majid Khan Majahar Ali
[email protected]

School of Mathematical Sciences, Universiti Sains Malaysia, Gelugor 11800, Pulau Pinang, Malaysia
Raja Aqib Shamim
School of Mathematical Sciences, Universiti Sains Malaysia, Gelugor 11800, Pulau Pinang, Malaysia;

Department of Mathematics, University of Kotli, Kotli 11100, Azad Jammu and Kashmir, Pakistan

Keywords:

Skewed data, Bayesian quantile regression, Beta distribution, Penalized quantile regression, Bounded data

Abstract

Skewed distributions have an additional shape parameter that represents the direction of the asymmetry in the density. If skewness in observations is ignored, statistical inferences based on symmetric distributions may yield biased or even misleading conclusions. Bayesian regularized quantile regression (BRQR) is effective in quantile regression for addressing skewness. Additionally, quantile ridge regression (QRR) has been developed for variable selection and to address multicollinearity among predictor variables. Most studies have used the asymmetric Laplace distribution (ALD) to address skewness, but this solution is not useful for bounded data because it has unbounded support, allowing values far in either direction and potentially violating the bounded-data assumption. The hybrid Bayesian regularized quantile beta regression--quantile ridge beta regression (BRQBR--QRBR) model is proposed for analyzing bounded data within the interval (0,1) with skewness while improving model-selection accuracy. The proposed model applies BRQBR to the skewed, bounded data to address skewness and then uses QRBR to improve model accuracy. Simulation studies and real-dataset applications were conducted. The results show that the proposed BRQBR--QRBR method, in most cases, outperforms the original method by improving prediction accuracy through skewness correction.

Dimensions

REFERENCES

[1] S. M. Ahmed, M. K. M. Ali & A. H. Hasan, ``Evaluating feature selection methods in a hybrid Weibull Freund-Cox proportional hazards model for renal cell carcinoma'', Nigerian Society of Physical Sciences 7 (2025) 2812. https://doi.org/10.46481/jnsps.2025.2812.

[2] Q. Li, N. Lin & R. Xi, ``Bayesian regularized quantile regression'', Bayesian Analysis 5 (2010) 533. https://doi.org/10.1214/10-BA521.

[3] P. Pérez-Rodríguez, O. A. Montesinos-López, A. Montesinos-López & J. Crossa, ``Bayesian regularized quantile regression: A robust alternative for genome-based prediction of skewed data'', The Crop Journal 8 (2020) 713. https://doi.org/10.1016/j.cj.2020.04.009.

[4] R. Koenker & G. Bassett Jr., ``Regression quantiles'', Econometrica 46 (1978) 33. https://doi.org/10.2307/1913643.

[5] Y. Tian & X. Song, ``Bayesian bridge-randomized penalized quantile regression'', Computational Statistics & Data Analysis 144 (2020) 106876. https://doi.org/10.1016/j.csda.2019.106876.

[6] Y. Wu & Y. Liu, ``Variable selection in quantile regression'', Statistica Sinica 19 (2009) 801. https://www.jstor.org/stable/24308857.

[7] K. Yu & R. A. Moyeed, ``Bayesian quantile regression'', Statistics & Probability Letters 54 (2001) 437. https://doi.org/10.1016/S0167-7152(01)00124-9.

[8] E. G. Tsionas, ``Bayesian quantile inference'', Journal of Statistical Computation and Simulation 73 (2003) 659. https://doi.org/10.1080/0094965031000064463.

[9] H. Kozumi & G. Kobayashi, ``Gibbs sampling methods for Bayesian quantile regression'', Journal of Statistical Computation and Simulation 81 (2011) 1565. https://doi.org/10.1080/00949655.2010.496117.

[10] A. Montesinos-Lopez, O. A. Montesinos-Lopez, E. R. Villa-Diharce, D. Gianola & J. Crossa, ``A robust Bayesian genome-based median regression model'', Theoretical and Applied Genetics 132 (2019) 1587. https://doi.org/10.1007/s00122-019-03303-6.

[11] T. T. Mai, ``Handling bounded response in high dimensions: A horseshoe prior Bayesian beta regression approach'', Bayesian Analysis (2026) 1. https://arxiv.org/abs/2505.22211.

[12] S. Ferrari & F. Cribari-Neto, ``Beta regression for modelling rates and proportions'', Journal of Applied Statistics 31 (2004) 799. https://doi.org/10.1080/0266476042000214501.

[13] M. Smithson & J. Verkuilen, ``A better lemon squeezer? Maximum-likelihood regression with beta-distributed dependent variables'', Psychological Methods 11 (2006) 54. https://doi.org/10.1037/1082-989X.11.1.54.

[14] A. J. Branscum, W. O. Johnson & M. C. Thurmond, ``Bayesian beta regression: Applications to household expenditure data and genetic distance between foot-and-mouth disease viruses'', Australian & New Zealand Journal of Statistics 49 (2007) 287. https://doi.org/10.1111/j.1467-842X.2007.00481.x.

[15] J. I. Figueroa-Zúñiga, R. B. Arellano-Valle & S. L. P. Ferrari, ``Mixed beta regression: A Bayesian perspective'', Computational Statistics & Data Analysis 61 (2013) 137. https://doi.org/10.1016/j.csda.2012.12.002.

[16] G. James, D. Witten, T. Hastie & R. Tibshirani, An Introduction to Statistical Learning: With Applications in R, 1st ed., Springer, New York, USA, 2013. https://doi.org/10.1007/978-1-4614-7138-7.

[17] A. Junaid, A. Khan, A. Alrumayh, F. M. Alghamdi, E. Hussam, H. M. Aljohani & A. Alrashidi, ``Modified two parameter ridge estimator for beta regression model'', Journal of Radiation Research and Applied Sciences 17 (2024) 100905. https://doi.org/10.1016/j.jrras.2024.100905.

[18] M. Qasim, K. Månsson & B. M. Golam Kibria, ``On some beta ridge regression estimators: Method, simulation and application'', Journal of Statistical Computation and Simulation 91 (2021) 1699. https://doi.org/10.1080/00949655.2020.1867549.

[19] M. N. Akram, M. Amin, A. Elhassanein & M. A. Ullah, ``A new modified ridge-type estimator for the beta regression model: Simulation and application'', AIMS Mathematics 7 (2022) 1035. https://www.aimspress.com/article/doi/10.3934/math.2022062.

[20] A. Erkoç, E. Ertan, Z. Y. Algamal & K. U. Akay, ``The beta Liu-type estimator: Simulation and application'', Hacettepe Journal of Mathematics and Statistics 52 (2023) 828. https://doi.org/10.15672/hujms.1145607.

[21] P. Karlsson, K. Månsson & B. M. Golam Kibria, ``A Liu estimator for the beta regression model and its application to chemical data'', Journal of Chemometrics 34 (2020) e3300. https://doi.org/10.1002/cem.3300.

[22] S. Seifollahi & H. Bevrani, ``James-Stein type estimators in beta regression model: Simulation and application'', Hacettepe Journal of Mathematics and Statistics 52 (2023) 1046. https://doi.org/10.15672/hujms.1122207.

[23] M. R. Abonazel, Z. Y. Algamal, F. A. Awwad & I. M. Taha, ``A new two-parameter estimator for beta regression model: Method, simulation, and application'', Frontiers in Applied Mathematics and Statistics 7 (2022) 780322. https://doi.org/10.3389/fams.2021.780322.

[24] M. R. Abonazel, I. Dawoud, F. A. Awwad & A. F. Lukman, ``Dawoud--Kibria estimator for beta regression model: Simulation and application'', Frontiers in Applied Mathematics and Statistics 8 (2022) 775068. https://doi.org/10.3389/fams.2022.775068.

[25] A. E. Hoerl & R. W. Kennard, ``Ridge regression: Applications to nonorthogonal problems'', Technometrics 12 (1970) 69. https://doi.org/10.1080/00401706.1970.10488635.

[26] R. Tibshirani, ``Regression shrinkage and selection via the lasso'', Journal of the Royal Statistical Society Series B: Statistical Methodology 58 (1996) 267. https://doi.org/10.1111/j.2517-6161.1996.tb02080.x.

[27] H. Zou & T. Hastie, ``Regularization and variable selection via the elastic net'', Journal of the Royal Statistical Society Series B: Statistical Methodology 67 (2005) 301. https://doi.org/10.1111/j.1467-9868.2005.00503.x.

[28] Q. Tang, Y. Gu & B. Wang, ``fastkqr: A fast algorithm for kernel quantile regression'', Journal of Computational and Graphical Statistics 35 (2026) 395. https://doi.org/10.1080/10618600.2025.2541004.

[29] K. M. Tan, L. Wang & W.-X. Zhou, ``High-dimensional quantile regression: Convolution smoothing and concave regularization'', Journal of the Royal Statistical Society Series B: Statistical Methodology 84 (2022) 205. https://doi.org/10.1111/rssb.12485.

[30] N. H. A. Afouna & M. K. M. Ali, ``Optimizing precision farming: Enhancing machine learning efficiency with robust regression techniques in high-dimensional data'', Nigerian Society of Physical Sciences 7 (2025) 2314. https://doi.org/10.46481/jnsps.2025.2314.

[31] A. R. M. Alsayed, ``Turkish stock market from pandemic to Russian invasion, evidence from developed machine learning algorithm'', Computational Economics 62 (2023) 1107. https://doi.org/10.1007/s10614-022-10293-z.

[32] A. S. A. Ambark & M. T. Ismail, ``Penalized quantile regression and empirical mode decomposition for improving the accuracy of the model selection'', Pakistan Journal of Statistics 40 (2024) 199. https://www.pakjs.com/wp-content/uploads/2024/04/40204.pdf.

[33] E. S. B. de Oliveira, M. de Castro, C. L. Bayes & J. L. Bazan, ``Bayesian quantile regression models for heavy tailed bounded variables using the No-U-Turn sampler'', Computational Statistics 40 (2025) 3007. https://doi.org/10.1007/s00180-022-01297-2.

[34] M. Ç. Korkmaz, C. Chesneau & Z. S. Korkmaz, ``A new alternative quantile regression model for the bounded response with educational measurements applications of OECD countries'', Journal of Applied Statistics 50 (2023) 131. https://doi.org/10.1080/02664763.2021.1981834.

[35] M. Bourguignon, D. I. Gallardo & H. Saulo, ``Parametric quantile beta regression model'', International Statistical Review 92 (2024) 106. https://doi.org/10.1111/insr.12564.

[36] A. S. A. Ambark, M. T. Ismail, A. S. Al-Jawarneh & S. A. A. Karim, ``Elastic net penalized quantile regression model and empirical mode decomposition for improving the accuracy of the model selection'', IEEE Access 11 (2023) 26152. https://doi.org/10.1109/access.2023.3257032.

[37] S. L. P. Ferrari, P. L. Espinheira & F. Cribari-Neto, ``Diagnostic tools in beta regression with varying dispersion'', Statistica Neerlandica 65 (2011) 337. https://doi.org/10.1111/j.1467-9574.2011.00488.x.

[38] A. B. Simas, W. Barreto-Souza & A. V. Rocha, ``Improved estimators for a general class of beta regression models'', Computational Statistics & Data Analysis 54 (2010) 348. https://doi.org/10.1016/j.csda.2009.08.017.

[39] J. Piironen & A. Vehtari, ``On the hyperprior choice for the global shrinkage parameter in the horseshoe prior'', Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, (2017) 905.https://doi.org/10.48550/arXiv.1610.05559.

[40] A. Bager, M. Roman, M. Algedih & B. Mohammed, Addressing multicollinearity in regression models: A ridge regression application, MPRA Paper 81390, University Library of Munich, Germany, 2017. https://mpra.ub.uni-muenchen.de/81390/.

[41] M. E. Suhaeri, Alimudin, A. Javaid, M. T. Ismail & M. K. M. Ali, ``Evaluation of clustering approach with Euclidean and Manhattan distance for outlier detection'', AIP Conference Proceedings 2423 (2021) 070025. https://doi.org/10.1063/5.0075570.

[42] M. Nascimento, F. F. e Silva, M. D. V. de Resende, C. D. Cruz, A. C. C. Nascimento, J. M. S. Viana, C. F. Azevedo & L. M. A. Barroso, ``Regularized quantile regression applied to genome-enabled prediction of quantitative traits'', Genetics and Molecular Research 16 (2017) gmr16019538. https://pubmed.ncbi.nlm.nih.gov/28340274/.

[43] D. Gianola, A. Cecchinato, H. Naya & C.-C. Schön, ``Prediction of complex traits: Robust alternatives to best linear unbiased prediction'', Frontiers in Genetics 9 (2018) 195. https://doi.org/10.3389/fgene.2018.00195.

Published

2026-09-17

How to Cite

Quantile ridge beta regression model and Bayesian regularized quantile beta regression for solving the skewness and improving the accuracy of the model selection in bounded data. (2026). Journal of the Nigerian Society of Physical Sciences, 8(4), 3629. https://doi.org/10.46481/jnsps.2026.3629