Versatile Extension of the Unit Gompertz: Efficient Estimation and Application

Year 2025, Volume: 38 Issue: 3, 1540 - 1564

Amal Hassan , Asma Khalil , Heba Nagy

https://doi.org/10.35378/gujs.1541941

Abstract

Despite the availability of numerous statistical models for describing real-world data, the need remains for flexible distributions capable of accurately capturing diverse spread patterns, particularly within the unit interval. This study introduces the Kavya-Manoharan (KM)-unit Gompertz (KM-UGo) distribution, a novel model tailored for data confined to the unit interval. By combining the unit Gompertz distribution and the KM transformation, the KM-UGo distribution is an improved version of the existing unit-Gompertz distribution, offering more adaptability and the possibility of better model fit in a a wider range of data with diverse spread patterns. This enhances its ability to model various hazard rate shapes, including J-shaped, bathtub, increasing, inverted bathtub, and decreasing. The paper delves into the mathematical properties of the KM-UGo distribution, deriving key characteristics such as moments, probability-weighted moments, incomplete moments, residual and reversed residual life, quantile function, and entropy measures. Classical estimation techniques, including maximum likelihood, least squares, maximum product spacing, Cramér-von Mises, Anderson-Darling, and weighted least squares are employed to determine the distribution's parameters and the results are assessed using a Monte Carlo method. The study's findings showed that the maximum likelihood and maximum product spacing estimation methods offer more accurate and reliable parameter estimates. Furthermore, as demonstrated in simulation studies, larger sample sizes produce better parameter estimates, which are characterized by lower bias and higher accuracy. To illustrate its practical application, the KM-UGo distribution is applied to two real-world datasets residing within the unit interval.

Keywords

Kavya and Manoharan, Unit Gompertz, Entropy measures, Parameter estimation, Goodness of fit tests

References

• [1] Kavya, P., Manoharan, M., “Some parsimonious models for lifetimes and applications”, Journal of Statistical Computation and Simulation, 91(18): 3693–3708, (2021). DOI: https://doi.org/10.1080/00949655.2021.1946064.
• [2] Kumar, D., Singh, U., Singh, S.K., “A method of proposing new distribution and its application to bladder cancer patients data”, Journal of Statistics Applications & Probability Letters, 2(3): 235–245, (2015).
• [3] Thomas, B., Chacko, V.M., “Power generalized DUS transformation of exponential distribution”, arXiv preprint arXiv:2111.14627, (2021). DOI: https://doi.org/10.48550/arXiv.2111.14627.
• [4] Mazucheli, J., Menezes, A.F.B., Dey, S., “Unit-Gompertz distribution with applications”, Statistica, 79(1): 25–43, (2019). DOI: https://doi.org/10.6092/issn.1973-2201/8497.
• [5] Abdelall, Y.Y., Ismail, G., Nagy, H., “A New Bounded Distribution: Covid-19 Application”, The Egyptian Statistical Journal, 69(1): 1–29, (2025). DOI: http://doi.org/10.21608/esju.2025.330520.1047.
• [6] Mazucheli, J., Bapat, S.R., Menezes, A.F.B., “A new one-parameter unit-Lindley distribution”, Chilean Journal of Statistics (ChJS), 11(1): 53–67, (2020).
• [7] Karakuş, H., Doğru, F.Z., Akgül, F.G., “Unit power Lindley distribution: properties and estimation”, Gazi University Journal of Science, 38(1): 502–526, (2025). DOI: http://doi.org/10.35378/gujs.1432128.
• [8] Mazucheli, J., Menezes, A.F.B., Fernandes, L.B., De Oliveira, R.P., Ghitany, M.E., “The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates”, Journal of Applied Statistics, 47(6): 954–974, (2020). DOI: https://doi.org/10.1080/02664763.2019.1657813.
• [9] Korkmaz, M.Ç., Chesneau, C., “On the unit Burr-XII distribution with the quantile regression modeling and applications”, Computational and Applied Mathematics, 40(1): 29, (2021). DOI: https://doi.org/10.1007/s40314-021-01418-5.
• [10] Ramadan, A.T., Tolba, A.H., El-Desouky, B.S., “A unit half-logistic geometric distribution and its application in insurance”, Axioms, 11(12): 676, (2022). DOI: https://doi.org/10.3390/axioms11120676.
• [11] Hassan, A.S., Fayomi, A., Algarni, A., Almetwally, E.M., “Bayesian and non-Bayesian inference for unit-exponentiated half-logistic distribution with data analysis”, Applied Sciences, 12(21): 11253, (2022). DOI: https://doi.org/10.3390/app122111253.
• [12] Hashmi, S., Ahsan-ul-Haq, M., Zafar, J., Khaleel, M.A., “Unit Xgamma distribution: its properties, estimation and application: Unit-Xgamma distribution”, Proceedings of the Pakistan Academy of Sciences: A. Physical and Computational Sciences, 59(1): 15–28, (2022). DOI: https://doi.org/10.53560/PPASA(59-1)636.
• [13] Krishna, A., Maya, R., Chesneau, C., Irshad, M.R., “The unit Teissier distribution and its applications”, Mathematical and Computational Applications, 27(1): 12, (2022). DOI: https://doi.org/10.3390/mca27010012.
• [14] Fayomi, A., Hassan, A.S., Almetwally, E.M., “Inference and quantile regression for the unit-exponentiated Lomax distribution”, Plos One, 18(7): e0288635, (2023). DOI: https://doi.org/10.1371/journal.pone.0288635.
• [15] Hassan, A.S., Alharbi, R.S., “Different estimation methods for the unit inverse exponentiated Weibull distribution”, Communications for Statistical Applications and Methods, 30(2): 191–213, (2023). DOI: https://doi.org/10.29220/CSAM.2023.30.2.191.
• [16] Yıldırım, E., Ilıkkan, E.S., Gemeay, A.M., Makumi, N., Bakr, M.E., Balogun, O.S., “Power unit Burr-XII distribution: Statistical inference with applications”, AIP Advances, 13(10): 1–24, (2023). DOI: https://doi.org/10.1063/5.0171077.
• [17] Fayomi, A., Hassan, A.S., Baaqeel, H., Almetwally, E.M., “Bayesian inference and data analysis of the unit–power Burr X distribution”, Axioms, 12(3): 297, (2023). DOI: https://doi.org/10.3390/axioms12030297.
• [18] Haj Ahmad, H., Almetwally, E.M., Elgarhy, M., Ramadan, D.A., “On unit exponential Pareto distribution for modeling the recovery rate of COVID-19”, Processes, 11(1): 232, (2023). DOI: https://doi.org/10.3390/pr11010232.
• [19] Karakaya, K., Sağlam, Ş., “Unit Gamma-Lindley distribution: properties, estimation, regression analysis, and practical applications”, Gazi University Journal of Science, 38(2): 1021-1040, (2025). DOI: https://doi.org/10.35378/gujs.1549073.
• [20] Hassan, A.S., Khalil, A.M., Nagy, H.F., “Data analysis and classical estimation methods of the bounded power Lomax distribution”, Reliability: Theory & Applications, 19(1 (77)): 770–789, (2024).
• [21] Hamedani, G.G., Korkmaz, M.Ç., Butt, N.S., Yousof, H.M., “The Type I quasi Lambert family: properties, characterizations and different estimation methods”, Pakistan Journal of Statistics and Operation Research, 17(3): 545–558, (2021). DOI: https://doi.org/10.18187/pjsor.v17i3.3562.
• [22] Korkmaz, M.Ç., Altun, E., Alizadeh, M., El-Morshedy, M., “The log exponential-power distribution: Properties, estimations and quantile regression model”, Mathematics, 9(21): 2634, (2021). DOI: https://doi.org/10.3390/math9212634.
• [23] Korkmaz, M.C., Altun, E., Chesneau, C., Yousof, H.M., “On the unit-Chen distribution with associated quantile regression and applications”, Mathematica Slovaca, 72(3): 765–786, (2022). DOI: http://doi.org/10.1515/ms-2022-0052.
• [24] Korkmaz, M.Ç., Leiva, V., Martin-Barreiro, C., “The continuous Bernoulli distribution: Mathematical characterization, fractile regression, computational simulations, and applications”, Fractal and Fractional, 7(5): 386, (2023). DOI: https://doi.org/10.3390/fractalfract7050386.
• [25] Mazucheli, J., Alves, B., Korkmaz, M.Ç., Leiva, V., “Vasicek quantile and mean regression models for bounded data: New formulation, mathematical derivations, and numerical applications”, Mathematics, 10(9): 1389, (2022). DOI: https://doi.org/10.3390/math10091389.
• [26] Mazucheli, J., Korkmaz, M.Ç., Menezes, A.F., Leiva, V., “The unit generalized half-normal quantile regression model: formulation, estimation, diagnostics, and numerical applications”, Soft Computing, 27(1): 279–295, (2023). DOI: http://doi.org/10.1007/s00500-022-07278-3.
• [27] Bhatti, F.A., Ali, A., Hamedani, G., Korkmaz, M.Ç., Ahmad, M., “The unit generalized log Burr XII distribution: Properties and application”, AIMS Mathematics, 6(9): 10222–10252, (2021). DOI: https://doi.org/10.3934/math.2021592.
• [28] Hassan, O.H.M., Elbatal, I., Al-Nefaie, A.H., Elgarhy, M., “On the Kavya–Manoharan–Burr X model: Estimations under ranked set sampling and applications”, Journal of Risk and Financial Management, 16(1): 19, (2022). DOI: https://doi.org/10.3390/jrfm16010019.
• [29] Al-Nefaie, A.H., “Applications to bio-medical data and statistical inference for a Kavya-Manoharan log-logistic model”, Journal of Radiation Research and Applied Sciences, 16(1): 100523, (2023). DOI: https://doi.org/10.1016/j.jrras.2023.100523.
• [30] Alsadat, N., Hassan, A.S., Elgarhy, M., Chesneau, C., El-Saeed, A.R., “Sampling plan for the Kavya–Manoharan generalized inverted Kumaraswamy distribution with statistical inference and applications”, Axioms, 12(8): 739, (2023). DOI: https://doi.org/10.3390/axioms12080739.
• [31] Alotaibi, N., Al-Moisheer, A., Hassan, A.S., Elbatal, I., Alyami, S.A., Almetwally, E.M., “Epidemiological modeling of COVID-19 data with advanced statistical inference based on Type-II progressive censoring”, Heliyon, 10(18), (2024). DOI: https://doi.org/10.1016/j.heliyon.2024.e36774.
• [32] Greenwood, J.A., Landwehr, J.M., Matalas, N.C., Wallis, J.R., “Probability weighted moments: definition and relation to parameters of several distributions expressable in inverse form”, Water Resources Research, 15(5): 1049–1054, (1979). DOI: https://doi.org/10.1029/WR015i005p01049.
• [33] Balkema, A.A., De Haan, L., “Residual life time at great age”, The Annals of Probability, 2(5): 792–804, (1974). DOI: https://doi.org/10.1214/aop/1176996548.
• [34] Havrda, J., Charvát, F., “Quantification method of classification processes. Concept of structural a-entropy”, Kybernetika, 3(1): 30–35, (1967).
• [35] Tsallis, C., “Possible generalization of Boltzmann-Gibbs statistics”, Journal of Statistical Physics, 52(1): 479–487, (1988). DOI: http://doi.org/10.1007/BF01016429.
• [36] Arimoto, S., “Information-theoretical considerations on estimation problems”, Information and Control, 19(3): 181–194, (1971). DOI: https://doi.org/10.1016/S0019-9958(71)90065-9.
• [37] Cheng, R.C.H., Amin, N.A.K., “Maximum product-of-spacings estimation with applications to the lognormal distribution”, University of Wales IST: Cardiff, UK, Mathematical Report, 79-1, (1979).
• [38] Bantan, R.A.R., Jamal, F., Chesneau, C., Elgarhy, M., “Theory and Applications of the Unit Gamma/Gompertz Distribution”, Mathematics, 9(16): 1850, (2021). DOI: https://doi.org/10.3390/math9161850.
• [39] Ribeiro-Reis, L.D., “Unit log-logistic distribution and unit log-logistic regression model”, Journal of the Indian Society for Probability and Statistics, 22(2): 375–388, (2021). DOI: https://doi.org/10.1007/s41096-021-00109-y.
• [40] Iqbal, M.Z., Arshad, M.Z., Özel, G., Balogun, O.S., “A better approach to discuss medical science and engineering data with a modified Lehmann Type–II model”, F1000 Research, 10: 823, (2021). DOI: https://doi.org/10.12688/f1000research.54305.1.
• [41] Musa, Y., Muhammad, A., Usman, U., Zakari, Y., “On the properties of Burr X-Topp Leone distribution and its application”, Lapai Journal of Applied and Natural Sciences, 6(1): 114–120, (2021).
• [42] Cordeiro, G.M., Brito, R.d.S., “The beta power distribution”, Brazilian Journal of Probability and Statistics, 26(1): 88–112, (2012). DOI: https://doi.org/10.1214/10-BJPS124.