UNIT PROBABILITY DISTRIBUTIONS: A COMPREHENSIVE REVIEW OF MODELS, PROPERTIES, AND APPLICATIONS
Abstract
Unit probability distributions defined on the standard interval (0, 1) serve as foundational tools for modeling data constrained within this bounded domain. Such data frequently emerge in disciplines where proportions, rates, and probabilities are analyzed, including economics, finance, hydrology, environmental sciences, biomedical research, and reliability engineering. Traditional models such as the Beta and Kumaraswamy distributions have long provided flexible frameworks for these applications. However, the increasing complexity of real-world phenomena has spurred the development of more versatile and specialized unit distributions. This article presents a comprehensive review of the literature on unit probability distributions, encompassing both classical models and recent innovations. Emphasis is placed on transformation techniques used to generate new families, key analytical properties, and a comparative evaluation of estimation methods. A diverse array of real-world applications is examined, highlighting the practical relevance and empirical performance of modern unit distributions across multiple domains. By synthesizing these developments, the review offers a structured resource to support further methodological advancement and informed model selection for bounded data analysis.
References
Afify, A. Z., Nassar, M., Kumar, D., & Cordeiro, G. M. (2022). A new unit distribution: Properties, inference, and applications. Electronic Journal of Applied Statistical Analysis, 15(2), 438462.
Ahmad, A., Ahmad, A., Dar, I. H., & Tripathi, R. (2022). The Log-Hamza distribution with statistical properties and application: An alternative for distributions having domain (0,1). Reliability: Theory & Applications, 17(3[69]), 306314.
Alsadat, N., Tani, C., Sapkota, L. P., Rajitha, C. S., Bahloul, M. M., & Gemeay, A. M. (2024). Power unit exponential probability distribution: Statistical inference and applications. Alexandria Engineering Journal, 107, 332346. https://doi.org/10.1016/j.aej.2024.07.038
Bashiru, S. O., Kayid, M., Sayed, R. M., Balogun, O. S., Abd El-Raouf, M. M., & Gemeay, A. M. (2025). Introducing the unit Zeghdoudi distribution as a novel statistical model for analyzing proportional data. Journal of Radiation Research and Applied Sciences, 18(1), 101204. https://doi.org/10.1016/j.jrras.2024.101204
Altun, E., Chesneau, C., & Alqifari, H. N. (2025). Two parameter log-Lindley distribution with LTPL web-tool. AIMS Mathematics, 10(4), 83068321. https://doi.org/10.3934/math.2025382
Ehiwario, J., Igabari, J., and Ezimadu, P. (2023). The alpha power topp-leone distribution: Properties, simulations and applications. Journal of Applied Mathematics and Physics, 11:316331.
Eldessouky, E. A., Hassan, O. H. M., Aloraini, B., & Elbatal, I. (2025). Modeling medical and economic data using the transmuted power unit inverse Lindley distribution. Alexandria Engineering Journal, 113, 633647. https://doi.org/10.1016/j.aej.2024.11.008
Fayomi, A., Hassan, A. S., & Almetwally, E. M. (2023). Inference and quantile regression for the unit-exponentiated Lomax distribution. PLOS ONE, 18(7), e0288635. https://doi.org/10.1371/journal.pone.0288635
Gemeay, A. M., Sapkota, L. P., Tashkandy, Y. A., Bakr, M. E., Balogun, O. S., & Hussam, E. (2024a). New bounded probability model: Properties, estimation, and applications. Heliyon, 10(23), e38965. https://doi.org/10.1016/j.heliyon.2024.e38965
Gemeay, A. M., Alsadat, N., Chesneau, C., & Elgarhy, M. (2024b). Power unit inverse Lindley distribution with different measures of uncertainty, estimation, and applications. AIMS Mathematics, 9(8), 2097621024. https://doi.org/10.3934/math.20241021
Gen, M., & zbilen, . (2023). Exponentiated UEHL distribution: Properties and applications. Recep Tayyip Erdogan University Journal of Science and Engineering, 4(2), 232241. https://doi.org/10.53501/rteufemud.1388416
Gen, M., & zbilen, . (2024). Transmuted Unit Exponentiated Half-Logistic distribution and its applications. Ordu niversitesi Bilim ve Teknoloji Dergisi, 14(2), 249260. https://doi.org/10.54370/ordubtd.1512101
Gong, Q., Luo, L., & Ren, H. (2024). Unit compound Rayleigh model: Statistical characteristics, estimation and application. AIMS Mathematics, 9(8), 2281322841. https://doi.org/10.3934/math.20241110
Hashmi, S., Ahsan-ul-Haq, M., Zafar, J., & Khalee, M. A. (2022). Unit Xgamma distribution: Its properties, estimation and application. Proceedings of the Pakistan Academy of Sciences: A. Physical and Computational Sciences, 59(1), 1528. https://doi.org/10.53560/PPASA(59-1)636
Jodr, P. (2020). A bounded distribution derived from the shifted Gompertz law. Journal of King Saud University - Science, 32(1), 523536. https://doi.org/10.1016/j.jksus.2018.08.001
Johnson, N. L., Kotz, S., and Balakrishnan, N. (1955). Continuous Univariate Distributions, volume 2. Wiley, Hoboken, NJ, USA, 2nd edition.
Karakaya, K., Rajitha, C. S., Salam, ., & et al. (2024). A new unit distribution: Properties, estimation, and regression analysis. Scientific Reports, 14, 7214. https://doi.org/10.1038/s41598-024-57390-7
Korkmaz, M. ., Chesneau, C., & Korkmaz, Z. S. (2021). Transmuted unit Rayleigh quantile regression model: Alternative to beta and Kumaraswamy quantile regression models. U.P.B. Scientific Bulletin, Series A, 83(3), 149158.
Krishna, A., Maya, R., Chesneau, C., & Irshad, M. R. (2022). The Unit Teissier Distribution and Its Applications. Mathematical and Computational Applications, 27(1), 12. https://doi.org/10.3390/mca27010012
Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(12):7988.
Mazucheli, J., Bapat, S. R., & Menezes, A. F. B. (2020). A new one-parameter unit-Lindley distribution. Chilean Journal of Statistics, 11(1), 5367.
Mazucheli, J., Menezes, A. F. B., & Dey, S. (2019). Unit-Gompertz distribution with applications. Statistica, 79(1), 2543.
Mazucheli, J., Menezes, A. F. B., & Ghitany, M. E. (2018). The unit-Weibull distribution and associated inference. Journal of Applied Probability and Statistics, 13(2), 122.
Muhammad, M. (2017). A new lifetime model with a bounded support. Asian Research Journal of Mathematics, 7(3), 111.
Nasiru, S., Abubakari, A. G., & Chesneau, C. (2022). New Lifetime Distribution for Modeling Data on the Unit Interval:
Properties, Applications and Quantile Regression. Mathematical and Computational Applications, 27(6), 105. https://doi.org/10.3390/mca27060105
Nasiru, S., Chesneau, C., & Ocloo, S. K. (2024). The log-cosine-power unit distribution: A new unit distribution for proportion data analysis. Decision Analytics Journal, 10, 100397. https://doi.org/10.1016/j.dajour.2024.100397
Ragab, I. E., Alsadat, N., Balogun, O. S., & Elgarhy, M. (2024). Unit extended exponential distribution with applications. Journal of Radiation Research and Applied Sciences, 17(4), 101118. https://doi.org/10.1016/j.jrras.2024.101118
Sindhu, T. N., Shafiq, A., Riaz, M. B., & Abushal, T. A. (2024). A statistical framework for a new two-parameter unit Bilal distribution with application to model asymmetric data. Heliyon, 10, e38340. https://doi.org/10.1016/j.heliyon.2024.e38340
Sarhan, A. M. (2025). Unit-Chen distribution and its quantile regression model with applications. Scientific African, 27, e02555. https://doi.org/10.1016/j.sciaf.2025.e02555
Suleiman, A. A., Daud, H., Ishaq, A. I., Othman, M., Alshanbari, H. M., & Alaziz, S. N. (2024). A novel extended Kumaraswamy distribution and its application to COVID-19 data. Engineering Reports, 6(12), e12967.
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