The unit-Weibull geometric distribution: Mathematical properties, identifiability, and asymptotic analysis with applications to data modeling

Magrisha Namsaw; Bhanita Das; Mohamed S. Eliwa; Mohamed F. Abouelenein; Magrisha Namsaw; Bhanita Das; Mohamed S. Eliwa; Mohamed F. Abouelenein

doi:10.3934/math.20261171

AIMS Mathematics

2026, Volume 11, Issue 9: 29488-29528. doi: 10.3934/math.20261171

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Research article

The unit-Weibull geometric distribution: Mathematical properties, identifiability, and asymptotic analysis with applications to data modeling

1.
Department of Statistics, North Eastern Hill University, Shillong, Meghalaya, India
2.
Department of Statistics and Operations Research, College of Science, Qassim University, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
4.
Department of Insurance and Risk Management, College of Business, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Riyadh, Saudi Arabia

Received: 22 June 2026 Revised: 22 July 2026 Accepted: 27 July 2026 Published: 14 September 2026
MSC : 60E05, 62E15, 62N05, 62P30

This study introduces the unit-Weibull geometric (UWG) distribution, a novel bounded compound probability distribution defined on the unit interval $ (0, 1) $. Constructed by compounding the unit-Weibull and geometric distributions, the UWG framework is physically motivated by reliability theory and systems engineering, specifically modeling series systems where the minimum component lifetime dictates overall system failure. The proposed model offers exceptional flexibility for analyzing bounded data, including proportions, percentages, and fractions. Notably, the distribution can accommodate both symmetric and asymmetric (positively or negatively skewed) data under varying degrees of kurtosis (mesokurtic, leptokurtic, and platykurtic behaviors). A comprehensive mathematical and statistical characterization of the UWG distribution is provided. This includes an identifiability analysis based on the asymptotic matching of tail exponents, derivation of the moment-generating structure, an examination of modality, and a formal log-convexity and log-concavity analysis to assess its shape flexibility and boundary behavior. Furthermore, key reliability and information-theoretic measures are derived, including quantile-based descriptive statistics, incomplete and conditional moments, mean residual life, measures of inequality, uncertainty metrics, and the distributions of order statistics. Crucially, the hazard rate function of the UWG distribution can model diverse risk behaviors over time, exhibiting highly versatile shapes, including monotonic (increasing or decreasing), unimodal, and bathtub-shaped profiles. Model parameters are estimated via the method of maximum likelihood. The finite-sample properties, consistency, and asymptotic efficiency of the resulting estimators are validated through an extensive Monte Carlo simulation study. Finally, the empirical adequacy and superiority of the UWG distribution are demonstrated using two real-world Datasets, where it consistently outperforms established baseline and competing compound models for bounded data.
- statistical model,
- identifiability,
- series system reliability,
- risk curves,
- simulation,
- decision-making
Citation: Magrisha Namsaw, Bhanita Das, Mohamed S. Eliwa, Mohamed F. Abouelenein. The unit-Weibull geometric distribution: Mathematical properties, identifiability, and asymptotic analysis with applications to data modeling[J]. AIMS Mathematics, 2026, 11(9): 29488-29528. doi: 10.3934/math.20261171

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Abstract

This study introduces the unit-Weibull geometric (UWG) distribution, a novel bounded compound probability distribution defined on the unit interval $ (0, 1) $. Constructed by compounding the unit-Weibull and geometric distributions, the UWG framework is physically motivated by reliability theory and systems engineering, specifically modeling series systems where the minimum component lifetime dictates overall system failure. The proposed model offers exceptional flexibility for analyzing bounded data, including proportions, percentages, and fractions. Notably, the distribution can accommodate both symmetric and asymmetric (positively or negatively skewed) data under varying degrees of kurtosis (mesokurtic, leptokurtic, and platykurtic behaviors). A comprehensive mathematical and statistical characterization of the UWG distribution is provided. This includes an identifiability analysis based on the asymptotic matching of tail exponents, derivation of the moment-generating structure, an examination of modality, and a formal log-convexity and log-concavity analysis to assess its shape flexibility and boundary behavior. Furthermore, key reliability and information-theoretic measures are derived, including quantile-based descriptive statistics, incomplete and conditional moments, mean residual life, measures of inequality, uncertainty metrics, and the distributions of order statistics. Crucially, the hazard rate function of the UWG distribution can model diverse risk behaviors over time, exhibiting highly versatile shapes, including monotonic (increasing or decreasing), unimodal, and bathtub-shaped profiles. Model parameters are estimated via the method of maximum likelihood. The finite-sample properties, consistency, and asymptotic efficiency of the resulting estimators are validated through an extensive Monte Carlo simulation study. Finally, the empirical adequacy and superiority of the UWG distribution are demonstrated using two real-world Datasets, where it consistently outperforms established baseline and competing compound models for bounded data.

References

[1]	A. W. Marshall, I. Olkin, A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families, Biometrika, 84 (1997), 641–652.
[2]	K. Adamidis, S. Loukas, A lifetime distribution with decreasing failure rate, Stat. Probab. Lett., 39 (1998), 35–42. https://doi.org/10.1016/S0167-7152(98)00012-1 doi: 10.1016/S0167-7152(98)00012-1
[3]	C. Kuş, A new lifetime distribution, Comput. Stat. Data Anal., 51 (2007), 4497–4509. https://doi.org/10.1016/j.csda.2006.07.017 doi: 10.1016/j.csda.2006.07.017
[4]	R. Tahmasbi, S. Rezaei, A two-parameter lifetime distribution with decreasing failure rate, Comput. Stat. Data Anal., 52 (2008), 3889–3901. https://doi.org/10.1016/j.csda.2007.12.002 doi: 10.1016/j.csda.2007.12.002
[5]	W. Barreto-Souza, A. L. Morais, G. M. Cordeiro, The Weibull-geometric distribution, J. Stat. Comput. Simul., 81 (2011), 645–657. https://doi.org/10.1080/00949650903436554 doi: 10.1080/00949650903436554
[6]	W. Lu, D. Shi, A new compounding life distribution: The Weibull–Poisson distribution, J. Appl. Stat., 39 (2012), 21–38. https://doi.org/10.1080/02664763.2011.575126 doi: 10.1080/02664763.2011.575126
[7]	G. M. Cordeiro, E. M. M. Ortega, A. J. Lemonte, The exponential–Weibull lifetime distribution, J. Stat. Comput. Simul., 84 (2014), 2592–2606. https://doi.org/10.1080/00949655.2013.797982 doi: 10.1080/00949655.2013.797982
[8]	M. Wang, I. Elbatal, The modified Weibull geometric distribution, Metron, 73 (2015), 303–315.
[9]	I. Elbatal, M. Mansour, M. Ahsanullah, The additive Weibull-geometric distribution: Theory and applications, J. Stat. Theory Appl., 15 (2016), 125–141. https://doi.org/10.2991/jsta.2016.15.2.3 doi: 10.2991/jsta.2016.15.2.3
[10]	A. A. Fattah, S. Nadarajah, A. -H. N. Ahmed, The exponentiated transmuted Weibull geometric distribution with application in survival analysis, Commun. Stat. Simul. Comput., 46 (2017), 4244–4263. https://doi.org/10.1080/03610918.2015.1112909 doi: 10.1080/03610918.2015.1112909
[11]	Y. Chung, D. K. Dey, M. Jung, The exponentiated inverse Weibull geometric distribution, Pak. J. Stat., 33 (2017), 161–178.
[12]	A. Asgharzadeh, S. Nadarajah, F. Sharafi, Weibull Lindley distribution, REVSTAT-Stat. J., 16 (2018), 87–113. https://doi.org/10.57805/revstat.v16i1.234 doi: 10.57805/revstat.v16i1.234
[13]	J. B. Chakrabarty, S. Chowdhury, Compounded inverse Weibull distributions: Properties, inference and applications, Commun. Stat. Simul. Comput., 48 (2019), 2012–2033. https://doi.org/10.1080/03610918.2018.1429623 doi: 10.1080/03610918.2018.1429623
[14]	A. Hassan, N. Akhtar, A. Rashid, A. Iqbal, A new compound lifetime distribution: Ishita power series distribution with properties and applications, J. Stat. Appl. Pro., 10 (2021), 883–896. http://dx.doi.org/10.18576/jsap/100324 doi: 10.18576/jsap/100324
[15]	M. K. Shakhatreh, S. Dey, D. Kumar, Inverse Lindley power series distributions: A new compounding family and regression model with censored data, J. Appl. Stat., 49 (2022), 3451–3476. https://doi.org/10.1080/02664763.2021.1951683 doi: 10.1080/02664763.2021.1951683
[16]	A. Elshahhat, E. -S. A. El-Sherpieny, A. S. Hassan, The Pareto–Poisson distribution: Characteristics, estimations and engineering applications, Sankhya A, 85 (2023), 1058–1099. https://doi.org/10.1007/s13171-022-00302-6 doi: 10.1007/s13171-022-00302-6
[17]	H. Memarbashi, J. Etminan, M. Chahkandi, On the Zhang-power series distributions with application to lifetime modeling, J. Mahani Math. Res., 14 (2025), 505–525. https://doi.org/10.22103/jmmr.2024.23690.1676 doi: 10.22103/jmmr.2024.23690.1676
[18]	M. Namsaw, B. Das, P. J. Hazarika, M. Alizadeh, The Alpha power modified Weibull-geometric distribution: A comprehensive mathematical framework with simulation, goodness-of-fit analysis and informed decision making using real life data, Stat. Optim. Inf. Comput., 14 (2025), 1060–1087. https://doi.org/10.19139/soic-2310-5070-2607 doi: 10.19139/soic-2310-5070-2607
[19]	M. A. Zayed, A. S. Hassan, E. M. Almetwally, A. M. Aboalkhair, A. H. Al-Nefaie, H. M. Almongy, A compound class of unit Burr XII model: Theory, estimation, fuzzy, and application, Sci. Program., 2023 (2023), 4509889. https://doi.org/10.1155/2023/4509889 doi: 10.1155/2023/4509889
[20]	S. M. Alghamdi, M. Shrahili, A. S. Hassan, R. E. Mohamed, I. Elbatal, M. Elgarhy, Analysis of milk production and failure data: Using unit exponentiated half logistic power series class of distributions, Symmetry, 15 (2023), 714. https://doi.org/10.3390/sym15030714 doi: 10.3390/sym15030714
[21]	M. Gabanakgosi, B. Oluyede, The Topp-Leone-Gompertz-G power series class of distributions with applications, Pak. J. Stat. Oper. Res., 20 (2024), 171–195. https://doi.org/10.18187/pjsor.v20i2.4032 doi: 10.18187/pjsor.v20i2.4032
[22]	M. M. Yousef, A. S. Hassan, E. M. Almetwally, Statistical inference for the unit Gompertz power series distribution using ranked set sampling with applications, Assiut Univ. J. Multidiscip. Sci. Res., 53 (2024), 154–189. https://doi.org/10.21608/aunj.2023.246431.1068 doi: 10.21608/aunj.2023.246431.1068
[23]	M. S. Eliwa, M. H. Tahir, M. A. Hussain, B. Almohaimeed, A. Al-Bossly, M. El-Morshedy, Univariate probability-G classes for scattered samples under different forms of hazard: Continuous and discrete version with their inferences tests, Mathematics, 11 (2023), 2929. https://doi.org/10.3390/math11132929 doi: 10.3390/math11132929
[24]	M. El-Morshedy, M. H. Tahir, M. A. Hussain, A. Al-Bossly, M. S. Eliwa, A new flexible univariate and bivariate family of distributions for unit interval (0, 1), Symmetry, 14 (2022), 1040. https://doi.org/10.3390/sym14051040 doi: 10.3390/sym14051040
[25]	M. S. Eliwa, M. Ahsan-ul-Haq, A. Al-Bossly, M. El-Morshedy, A unit probabilistic model for proportion and asymmetric data: Properties and estimation techniques with application to model data from SC16 and P3 algorithms, Math. Probl. Eng., 2022 (2022), 9289721. https://doi.org/10.1155/2022/9289721 doi: 10.1155/2022/9289721
[26]	E. Altun, M. El-Morshedy, M. S. Eliwa, A new regression model for bounded response variable: An alternative to the beta and unit-Lindley regression models, PLoS ONE, 16 (2021), e0245627. https://doi.org/10.1371/journal.pone.0245627 doi: 10.1371/journal.pone.0245627
[27]	M. H. Tahir, M. A. Hussain, G. M. Cordeiro, M. El-Morshedy, M. S. Eliwa, A new Kumaraswamy generalized family of distributions with properties, applications, and bivariate extension, Mathematics, 8 (2020), 1989. https://doi.org/10.3390/math8111989 doi: 10.3390/math8111989
[28]	J. Mazucheli, A. F. B. Menezes, M. E. Ghitany, The unit-Weibull distribution and associated inference, J. Appl. Probab. Stat., 13 (2018), 1–22.
[29]	G. Casella, R. Berger, Statistical inference, 2 Eds., New York: Chapman and Hall/CRC, 2024. https://doi.org/10.1201/9781003456285
[30]	D. R. Cox, D. V. Hinkley, Theoretical statistics, New York: Chapman and Hall, 1974. https://doi.org/10.1201/b14832
[31]	A. M. Sarhan, T. Manshi, M. E. Sobh, The unit-modified Weibull distribution: Theory, estimation, and real-world applications, Stats, 8 (2025), 81. https://doi.org/10.3390/stats8030081 doi: 10.3390/stats8030081
[32]	A. M. Abd El-latif, O. A. Alqasem, J. K. Okutu, C. Tanış, L. P. Sapkota, N. A. Noori, A flexible extension of the unit upper truncated Weibull distribution: Statistical analysis with applications on geology, engineering, and radiation data, J. Radiat. Res. Appl. Sci., 18 (2025), 101434. https://doi.org/10.1016/j.jrras.2025.101434 doi: 10.1016/j.jrras.2025.101434
[33]	A. S. Hassan, R. S. Alharbi, Different estimation methods for the unit inverse exponentiated Weibull distribution, Commun. Stat. Appl. Methods, 30 (2023), 191–213. https://doi.org/10.29220/CSAM.2023.30.2.191 doi: 10.29220/CSAM.2023.30.2.191
[34]	N. L. Johnson, S. Kotz, N. Balakrishnan, Continuous univariate distributions, 2 Eds., New York: John Wiley & Sons, 1995.
[35]	A. Z. Afify, M. Nassar, D. Kumar, G. M. Cordeiro, A new unit distribution: Properties, inference, and applications, Electron. J. Appl. Stat. Anal., 15 (2022), 460–484. https://doi.org/10.1285/i20705948v15n2p460 doi: 10.1285/i20705948v15n2p460
[36]	G. M. Cordeiro, R. dos Santos Brito, The beta power distribution, Braz. J. Probab. Stat., 26 (2012), 88–112. http://dx.doi.org/10.1214/10-BJPS124 doi: 10.1214/10-BJPS124

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