Modeling bounded data with a new unit distribution: regression analysis and applications

Ahmed M. T. Abd El-Bar; Ahmed R. El-Saeed; Kadir Karakaya; Ahmed M. Gemeay; Ahmed M. T. Abd El-Bar; Ahmed R. El-Saeed; Kadir Karakaya; Ahmed M. Gemeay

doi:10.3934/math.2025790

AIMS Mathematics

2025, Volume 10, Issue 8: 17672-17704. doi: 10.3934/math.2025790

Previous Article Next Article

Research article

Modeling bounded data with a new unit distribution: regression analysis and applications

1.
Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt
2.
Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
3.
Department of Statistics, Faculty of Sciences, Selçuk University, Konya, Türkiye

Received: 18 April 2025 Revised: 05 July 2025 Accepted: 28 July 2025 Published: 05 August 2025
MSC : 60E05, 60E99, 62E15

This study proposes a new flexible unit distribution derived from the generalized exponential geometric model, which is intended to model data confined to the unit interval $ (0, 1) $. The introduced distribution density shapes are extremely versatile, accommodating right-skewed, left-skewed, approximately symmetric, decreasing, U-shaped, increasing, and J-shaped. Furthermore, its hazard rate function is adaptable to bathtub, U-shaped, increasing, and J-shaped shapes, making it applicable to a wide range of real-world datasets. Essential statistical properties, such as moments, quantiles, and entropy measures, are closely derived and analyzed. Based on this distribution, a new regression model is developed to link bounded response variables to linear predictors, increasing its practical applicability. The maximum likelihood approach is used to estimate the parameters of the regression model and the suggested distribution. The performance of the maximum likelihood approach based on the suggested distribution and regression model is examined using a Monte Carlo simulation. The applicability of the regression model and the new distribution is demonstrated through real data analysis. The proposed distribution exhibits strong modeling capabilities for bounded data in the unit interval, making it highly applicable in fields such as reliability analysis, survival studies, and modeling of proportions or rates. Its superior performance over existing models, as demonstrated through simulation studies and real data applications, highlights its potential as a practical and flexible tool for applied statisticians.
- generalized exponential geometric,
- entropy,
- exponential transformation,
- simulation,
- quantile regression
Citation: Ahmed M. T. Abd El-Bar, Ahmed R. El-Saeed, Kadir Karakaya, Ahmed M. Gemeay. Modeling bounded data with a new unit distribution: regression analysis and applications[J]. AIMS Mathematics, 2025, 10(8): 17672-17704. doi: 10.3934/math.2025790

Related Papers:

Abstract

This study proposes a new flexible unit distribution derived from the generalized exponential geometric model, which is intended to model data confined to the unit interval $ (0, 1) $. The introduced distribution density shapes are extremely versatile, accommodating right-skewed, left-skewed, approximately symmetric, decreasing, U-shaped, increasing, and J-shaped. Furthermore, its hazard rate function is adaptable to bathtub, U-shaped, increasing, and J-shaped shapes, making it applicable to a wide range of real-world datasets. Essential statistical properties, such as moments, quantiles, and entropy measures, are closely derived and analyzed. Based on this distribution, a new regression model is developed to link bounded response variables to linear predictors, increasing its practical applicability. The maximum likelihood approach is used to estimate the parameters of the regression model and the suggested distribution. The performance of the maximum likelihood approach based on the suggested distribution and regression model is examined using a Monte Carlo simulation. The applicability of the regression model and the new distribution is demonstrated through real data analysis. The proposed distribution exhibits strong modeling capabilities for bounded data in the unit interval, making it highly applicable in fields such as reliability analysis, survival studies, and modeling of proportions or rates. Its superior performance over existing models, as demonstrated through simulation studies and real data applications, highlights its potential as a practical and flexible tool for applied statisticians.

References

[1]	P. Kumaraswamy, A generalized probability density function for double-bounded random processes, J. Hydrol., 46 (1980), 79–88. https://doi.org/10.1016/0022-1694(80)90036-0 doi: 10.1016/0022-1694(80)90036-0
[2]	M. E. Ghitany, J. Mazucheli, A. F. B. Menezes, F. Alqallaf, The unit-inverse Gaussian distribution: a new alternative to two-parameter distributions on the unit interval, Commun. Stat.-Theor. M., 48 (2019), 3423–3438. https://doi.org/10.1080/03610926.2018.1476717 doi: 10.1080/03610926.2018.1476717
[3]	A. M. T. Abd El-Bar, H. S. Bakouch, S. Chowdhury, A new trigonometric distribution with bounded support and an application, Rev. Union Mat. Argent., 62 (2021), 459–473. https://doi.org/10.33044/revuma.1872 doi: 10.33044/revuma.1872
[4]	A. M. T. Abd El-Bar, M. Lima, M. Ahsanullah, Some inferences based on a mixture of power function and continuous logarithmic distribution, J. Taibah Univ. Sci., 14 (2020), 1116–1126. https://doi.org/10.1080/16583655.2020.1804140 doi: 10.1080/16583655.2020.1804140
[5]	R. A. R. Bantan, C. Chesneau, F. Jamal, M. Elgarhy, M. H. Tahir, A. Ali, et al., Some new facts about the unit-Rayleigh distribution with applications, Mathematics, 8 (2020), 1954. https://doi.org/10.3390/math8111954 doi: 10.3390/math8111954
[6]	J. Mazucheli, A. F. Menezes, S. Dey, Unit-Gompertz distribution with applications, Statistica, 79 (2019), 25–43. https://doi.org/10.6092/issn.1973-2201/8497 doi: 10.6092/issn.1973-2201/8497
[7]	M. A. Haq, S. Hashmi, K. Aidi, P. L. Ramos, F. Louzada, Unit modified Burr-III distribution: Estimation, characterizations and validation test, Ann. Data Sci., 10 (2023), 415–440. https://doi.org/10.1007/s40745-020-00298-6 doi: 10.1007/s40745-020-00298-6
[8]	J. Mazucheli, A. Menezes, L. Fernandes, R. De Oliveira, M. Ghitany, The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates, J. Appl. Stat., 47 (2020), 954–974. https://doi.org/10.1080/02664763.2019.1657813 doi: 10.1080/02664763.2019.1657813
[9]	E. Altun, The log-weighted exponential regression model: alternative to the beta regression model, Commun. Stat.-Theor. M., 50 (2021), 2306–2321. https://doi.org/10.1080/03610926.2019.1664586 doi: 10.1080/03610926.2019.1664586
[10]	E. Altun, G. Cordeiro, The unit-improved second-degree Lindley distribution: inference and regression modeling, Comput. Stat., 35 (2020), 259–279. https://doi.org/10.1007/s00180-019-00921-y doi: 10.1007/s00180-019-00921-y
[11]	E. Gómez-Déniz, M. Sordo, E. Calderín-Ojeda, The Log-Lindley distribution as an alternative to the beta regression model with applications in insurance, Insurance: Mathematics and Economics, 54 (2014), 49–57. https://doi.org/10.1016/j.insmatheco.2013.10.017 doi: 10.1016/j.insmatheco.2013.10.017
[12]	M. Korkmaz, E. Altun, M. Alizadeh, M. El-Morshedy, The log exponential-power distribution: properties, estimations and quantile regression model, Mathematics, 9 (2021), 2634. https://doi.org/10.3390/math9212634 doi: 10.3390/math9212634
[13]	M. Korkmaz, Z. Korkmaz, The unit log-log distribution: a new unit distribution with alternative quantile regression modeling and educational measurements applications, J. Appl. Stat., 50 (2023), 889–908. https://doi.org/10.1080/02664763.2021.2001442 doi: 10.1080/02664763.2021.2001442
[14]	M. Korkmaz, C. Chesneau, Z. Korkmaz, A new alternative quantile regression model for the bounded response with educational measurements applications of OECD countries, J. Appl. Stat., 50 (2023), 131–154. https://doi.org/10.1080/02664763.2021.1981834 doi: 10.1080/02664763.2021.1981834
[15]	M. Korkmaz, C. Chesneau, Z. Korkmaz, The unit folded normal distribution: a new unit probability distribution with the estimation procedures, quantile regression modeling and educational attainment applications, J. Reliab. Stat. Stud., 15 (2022), 261–298. https://doi.org/10.13052/jrss0974-8024.15111 doi: 10.13052/jrss0974-8024.15111
[16]	J. Mazucheli, A. Menezes, S. Chakraborty, On the one parameter unit-Lindley distribution and its associated regression model for proportion data, J. Appl. Stat., 46 (2019), 700–714. https://doi.org/10.1080/02664763.2018.1511774 doi: 10.1080/02664763.2018.1511774
[17]	M. Ozcalci, E. Kaya, Retail products price forecasting with empirical mode decomposition and auto regressive integrated moving average model using web-scraped price microdata, Spectrum of Decision Making and Applications, 2 (2025), 315–355. https://doi.org/10.31181/sdmap21202519 doi: 10.31181/sdmap21202519
[18]	R. Silva, W. Barreto-Souza, G. Cordeiro, A new distribution with decreasing, increasing and upside-down bathtub failure rate, Comput. Stat. Data Anal., 54 (2010), 935–944. https://doi.org/10.1016/j.csda.2009.10.006 doi: 10.1016/j.csda.2009.10.006
[19]	R. Gupta, D. Kundu, Generalized exponential distribution: existing results and some recent developments, J. Stat. Plan. Infer., 137 (2007), 3537–3547. https://doi.org/10.1016/j.jspi.2007.03.030 doi: 10.1016/j.jspi.2007.03.030
[20]	A. Rényi, On measures of entropy and information, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 1961,547–561.
[21]	A. Mathai, H. Haubold, On generalized distributions and pathways, Phys. Lett. A, 372 (2008), 2109–2113. https://doi.org/10.1016/j.physleta.2007.10.084 doi: 10.1016/j.physleta.2007.10.084
[22]	C. Shannon, A mathematical theory of communication, The Bell System Technical Journal, 27 (1948), 379–423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x doi: 10.1002/j.1538-7305.1948.tb01338.x
[23]	R. Dasgupta, On the distribution of Burr with applications, Sankhya B, 73 (2011), 1–19. https://doi.org/10.1007/s13571-011-0015-y doi: 10.1007/s13571-011-0015-y
[24]	A. Abd El-Bar, W. da Silva, A. Nascimento, An extended log-Lindley-G family: properties and experiments in repairable data, Mathematics, 9 (2021), 3108. https://doi.org/10.3390/math9233108 doi: 10.3390/math9233108
[25]	C. Kumar, S. Dharmaja, On reduced Kies distribution, In: Collection of recent statistical methods and applications, Trivandrum: University of Kerala Publisher, 2013,111–123.
[26]	A. Grassia, On a family of distributions with argument between 0 and 1 obtained by transformation of the gamma and derived compound distributions, Aust. J. Stat., 19 (1977), 108–114. https://doi.org/10.1111/j.1467-842X.1977.tb01277.x doi: 10.1111/j.1467-842X.1977.tb01277.x
[27]	M. Amini, S. MirMostafaee, J. Ahmadi, Log-gamma-generated families of distributions, Statistics, 48 (2014), 913–932. https://doi.org/10.1080/02331888.2012.748775 doi: 10.1080/02331888.2012.748775
[28]	C. Chesneau, On a logarithmic weighted power distribution: theory, modelling and applications, Journal of Mathematical Sciences: Advances and Applications, 67 (2021), 1–59. https://doi.org/10.18642/jmsaa_7100122214 doi: 10.18642/jmsaa_7100122214
[29]	M. Korkmaz, C. Chesneau, Z. Korkmaz, Transmuted unit Rayleigh quantile regression model: alternative to beta and Kumaraswamy quantile regression models, UPB Sci. Bull., Series A, 83 (2021), 149–158.
[30]	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
[31]	P. Mitnik, S. Baek, The Kumaraswamy distribution: median-dispersion re-parameterizations for regression modeling and simulation-based estimation, Stat. Papers, 54 (2013), 177–192. https://doi.org/10.1007/s00362-011-0417-y doi: 10.1007/s00362-011-0417-y
[32]	P. Jodra, M. Jimenez-Gamero, A quantile regression model for bounded responses based on the exponential-geometric distribution, REVSTAT Stat. J., 18 (2020), 415–436. https://doi.org/10.57805/revstat.v18i4.309 doi: 10.57805/revstat.v18i4.309

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)