An Extension of the Gompertz Distribution for Modeling COVID-19 Mortality Dynamics

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DOI: https://doi.org/10.28924/2291-8639-23-2025-206
Published: Aug 25, 2025
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Ahmad Abubakar Suleiman, Hanita Daud, Aliyu Ismail Ishaq, Narinderjit Singh Sawaran Singh, Diaa S. Metwally, H. E. Semary, Mohammed Elgarhy, An Extension of the Gompertz Distribution for Modeling COVID-19 Mortality Dynamics, Int. J. Anal. Appl., 23 (2025), 206.

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Ahmad Abubakar Suleiman, Hanita Daud, Aliyu Ismail Ishaq, Narinderjit Singh Sawaran Singh, Diaa S. Metwally, H. E. Semary, Mohammed Elgarhy

Abstract

The Gompertz distribution is widely used in medical and reliability studies, particularly for modeling mortality rates and failure data. However, it has limitations in capturing complex data behaviors, such as heavy tails and varying hazard rate shapes. This paper introduces the Odd Beta Prime-Gompertz (OBP-Gompertz) distribution, a four-parameter extension of the traditional Gompertz model. The OBP-Gompertz distribution offers flexibility in modeling various shapes of probability density functions, including right-skewed, left-skewed, heavy-tailed, light-tailed, and unimodal distributions. Its hazard function can accommodate multiple forms, such as increasing, decreasing, bathtub-shaped, and inverted bathtub-shaped curves, making it well-suited for mortality rate data. The paper investigates key statistical properties, including moments, moment generating function, quantile function, Rényi and Tsallis entropy measures. Parameters are estimated using maximum likelihood estimation, and the model's robustness is assessed through Monte Carlo simulations. The OBP-Gompertz model is applied to three real-world COVID-19 mortality datasets from China, the Netherlands, and Nepal. The results demonstrate that the OBP-Gompertz model provides superior fits compared to the traditional Gompertz and other models. This work highlights the OBP-Gompertz distribution as a valuable tool for survival analysis, reliability studies, and epidemiological research.

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References

  1. I.E. Ragab, H. Daud, A.A. Suleiman, N. Alsadat, V.B. Nagarjuna, M. Elgarhy, Type Ii Topp-Leone Exponentiated Gamma Distribution with Application to Breaking Stress Data, J. Radiat. Res. Appl. Sci. 17 (2024), 101045. https://doi.org/10.1016/j.jrras.2024.101045.
  2. A.A. Alahmadi, R.A. ZeinEldin, O. Albalawi, M.M. Badr, T.A.A. Abdelfadel, A.W. Shawki, Modified Kies Power Lomax Model with Applications in Different Sciences, J. Radiat. Res. Appl. Sci. 18 (2025), 101239. https://doi.org/10.1016/j.jrras.2024.101239.
  3. Y. Yu, Y. Jia, M.A. Alshahrani, O.A. Alamri, H. Daud, J.G. Dar, A.A. Suleiman, Adopting a New Sine-Induced Statistical Model and Deep Learning Methods for the Empirical Exploration of the Music and Reliability Data, Alex. Eng. J. 104 (2024), 396-408. https://doi.org/10.1016/j.aej.2024.07.104.
  4. L.A. Al-Essa, M. Muhammad, M.H. Tahir, B. Abba, J. Xiao, F. Jamal, A New Flexible Four Parameter Bathtub Curve Failure Rate Model, and Its Application to Right-Censored Data, IEEE Access 11 (2023), 50130-50144. https://doi.org/10.1109/access.2023.3276904.
  5. M.I.A. Araibi, A.A. Mir, I. Elbatal, E.M. Almetwally, C. Tanış, E. Ozkan, A.M. Gemeay, A New Alternative to the Log-Kumaraswamy Distribution: Properties, Estimation, and Fitting Data, Int. J. Anal. Appl. 23 (2025), 174. https://doi.org/10.28924/2291-8639-23-2025-174.
  6. U. Panitanarak, A.I. Ishaq, A.A. Suleiman, H. Daud, N.S.S. Singh, A.U. Usman, N. Alsadat, M. Elgarhy, A New Beta Distribution with Interdisciplinary Data Analysis, AIMS Math. 10 (2025), 8495-8527. https://doi.org/10.3934/math.2025391.
  7. N. Alsadat, A.S. Hassan, M. Elgarhy, C. Chesneau, R.E. Mohamed, An Efficient Stress–strength Reliability Estimate of the Unit Gompertz Distribution Using Ranked Set Sampling, Symmetry 15 (2023), 1121. https://doi.org/10.3390/sym15051121.
  8. T.N. Sindhu, A. Shafiq, M.B. Riaz, T.A. Abushal, H. Ahmad, E.M. Almetwally, S. Askar, Introducing the New Arcsine-Generator Distribution Family: An In-Depth Exploration with an Illustrative Example of the Inverse Weibull Distribution for Analyzing Healthcare Industry Data, J. Radiat. Res. Appl. Sci. 17 (2024), 100879. https://doi.org/10.1016/j.jrras.2024.100879.
  9. A.A. Osi, G.S.S. Abdalla, N.S. Sawaran Singh, A.A. Suleiman, Advancing Lifetime Data Modeling via the Marshall-Olkin Cosine Topp-Leone Distribution Family, Comput. J. Math. Stat. Sci. (2025). https://doi.org/10.21608/cjmss.2025.374786.1155.
  10. C.M. Dalah, V. Singh, I. Abdullahi, A. Suleiman, The Study of HIV/AIDS Trend in Yobe State for the Prescribed Period (1999–2019), Int. J. Stat. Appl. 10 (2020), 10-16.
  11. H. Daud, A.S. Mohammed, A.I. Ishaq, B. Abba, Y. Zakari, J. Abdullahi, D.A. Shobanke, A.A. Suleiman, Modeling and Prediction of Exchange Rates Using Topp-Leone Burr Type X, Machine Learning and Deep Learning Models, Eur. J. Stat. 4 (2024), 11. https://doi.org/10.28924/ada/stat.4.11.
  12. L.P. Sapkota, V. Kumar, G. Tekle, H. Alrweili, M.S. Mustafa, M. Yusuf, Fitting Real Data Sets by a New Version of Gompertz Distribution, Mod. J. Stat. 1 (2025), 25-48. https://doi.org/10.64389/mjs.2025.01109.
  13. U. Danjuma Maiwada, R. Yusuf Zakari, A.A. Janisar, Distribution Function-Driven Handover Solutions for 5G Mobile Networks, J. Stat. Sci. Comput. Intell. 1 (2025), 46-60. https://doi.org/10.64497/jssci.1.
  14. A.A. Suleiman, H. Daud, A.I. Ishaq, R. Sokkalingam, S.A. Suleiman, et al. Extension of the Log-Logistic Distribution for Groundwater Analysis and Potability Prediction Using Machine Learning Models, in: Proceedings of the 5th International Electronic Conference on Applied Sciences, 2024, MDPI, Basel, Switzerland, 2024.
  15. U. Panitanarak, A. Ismail Ishaq, A. Adewole Abiodun, H. Daud, A. Abubakar Suleiman, A New Maxwell-Log Logistic Distribution and Its Applications for Mortality Rate Data, J. Niger. Soc. Phys. Sci. 7 (2025), 1976. https://doi.org/10.46481/jnsps.2025.1976.
  16. Q.N. Husain, A.S. Qaddoori, N.A. Noori, K.N. Abdullah, A.A. Suleiman, O.S. Balogun, New Expansion of Chen Distribution According to the Nitrosophic Logic Using the Gompertz Family, Innov. Stat. Probab. 1 (2025), 60-75. https://doi.org/10.64389/isp.2025.01105.
  17. U. Panitanarak, A.I. Ishaq, A. Usman, I.A. Sadiq, A.S. Mohammed, The Modified Sine Distribution and Machine Learning Models for Enhancing Crude Oil Production Prediction, J. Stat. Sci. Comput. Intell. 1 (2025), 29-45. https://doi.org/10.64497/jssci.3.
  18. A. Ibrahim, A.A. Suleiman, U.A. Abdullahi, S.A. Suleiman, Monitoring Groundwater Quality Using Probability Distribution in Gwale, Kano State, Nigeria, J. Stat. Model. Anal. 3 (2021), 95-108. https://doi.org/10.22452/josma.vol3no2.6.
  19. A.I. Ishaq, A.A. Abiodun, A.A. Suleiman, A. Usman, A.S. Mohammed, M. Tasiu, Modelling Nigerian Inflation Rates from January 2003 to June 2023 Using Newly Developed Inverse Power Chi-Square Distribution, in: 2023 4th International Conference on Data Analytics for Business and Industry (ICDABI), IEEE, 2023, pp. 644-651. https://doi.org/10.1109/icdabi60145.2023.10629442.
  20. A.I. Ishaq, A.U. Usman, H.N. Alqifari, A. Almohaimeed, H. Daud, S.I. Abba, A.A. Suleiman, A New Log-Lomax Distribution, Properties, Stock Price, and Heart Attack Predictions Using Machine Learning Techniques, AIMS Math. 10 (2025), 12761-12807. https://doi.org/10.3934/math.2025575.
  21. A. Suleiman, A. Usman, H. Daud, F.A. Idris, R. Sokkalingam, A.I. Ishaq, A Voting Regressor Ensemble Model for Crude Oil Price Prediction, J. Stat. Sci. Comput. Intell. 1 (2025), 61-72. https://doi.org/10.64497/jssci.4.
  22. A. Usman, A.I. Ishaq, A.A. Suleiman, M. Othman, H. Daud, Y. Aliyu, Univariate and Bivariate Log-Topp-Leone Distribution Using Censored and Uncensored Datasets, Comput. Sci. Math. Forum 7 (2023), 32. https://doi.org/10.3390/iocma2023-14421.
  23. A. A. Suleiman, H. Daud, M. Othman, A. Husin, A.I. Ishaq et al. Forecasting the Southeast Asian Currencies against the British Pound Sterling Using Probability Distributions, Data Sci. Insights 1 (2023), 31-51.
  24. F. Jamal, S. Kanwal, S. Shafiq, M. Hashim, M. Kayid, M. Muhammad, S. Dutta, A.W. Shawki, The New Extended Exponentiated Burr Xii Distribution: Properties and Applications, J. Radiat. Res. Appl. Sci. 18 (2025), 101200. https://doi.org/10.1016/j.jrras.2024.101200.
  25. A.I. Ishaq, A. Usman, A.A. Suleiman, M. Othman, H. Daud, et al. Perspective Chapter: A New Bivariate Inverted Nakagami Distribution – Properties and Applications, in: New Trends and Challenges in Open Data , IntechOpen, 2023. https://doi.org/10.5772/intechopen.1001446.
  26. U. Panitanarak, A.I. Ishaq, N.S.S. Singh, A. Usman, A.U. Usman, et al. Machine Learning Models in Predicting Failure Times Data Using a Novel Version of the Maxwell Model, Eur. J. Stat. 5 (2025), 1. https://doi.org/10.28924/ada/stat.5.1.
  27. R.A.R. Bantan, F. Jamal, C. Chesneau, M. Elgarhy, Theory and Applications of the Unit Gamma/gompertz Distribution, Mathematics 9 (2021), 1850. https://doi.org/10.3390/math9161850.
  28. I. Elbatal, F. Jamal, C. Chesneau, M. Elgarhy, S. Alrajhi, The Modified Beta Gompertz Distribution: Theory and Applications, Mathematics 7 (2018), 3. https://doi.org/10.3390/math7010003.
  29. A.I. Ishaq, A.A. Suleiman, H. Daud, N.S.S. Singh, M. Othman, R. Sokkalingam, P. Wiratchotisatian, A.G. Usman, S.I. Abba, Log-kumaraswamy Distribution: Its Features and Applications, Front. Appl. Math. Stat. 9 (2023), 1258961. https://doi.org/10.3389/fams.2023.1258961.
  30. H. Semary, A.A. Suleiman, A.I. Ishaq, J.Y. Falgore, U.K. Abdullahi, H. Daud, M.A. Abd Elgawad, M. Elgarhy, A New Modified Sine-Weibull Distribution for Modeling Medical Data with Dynamic Structures, J. Radiat. Res. Appl. Sci. 18 (2025), 101427. https://doi.org/10.1016/j.jrras.2025.101427.
  31. S.F. Salleh, A.A. Suleiman, H. Daud, M. Othman, R. Sokkalingam, K. Wagner, Tropically Adapted Passive Building: a Descriptive-Analytical Approach Using Multiple Linear Regression and Probability Models to Predict Indoor Temperature, Sustainability 15 (2023), 13647. https://doi.org/10.3390/su151813647.
  32. C.K. Onyekwere, O.C. Aguwa, O.J. Obulezi, An Updated Lindley Distribution: Properties, Estimation, Acceptance Sampling, Actuarial Risk Assessment and Applications, Innov. Stat. Probab. 1 (2025), 1-27. https://doi.org/10.64389/isp.2025.01103.
  33. A.M. Gemeay, T. Moakofi, O.S. Balogun, E. Ozkan, M.M. Hossain, Analyzing Real Data by a New Heavy-Tailed Statistical Model, Mod. J. Stat. 1 (2025), 1-24. https://doi.org/10.64389/mjs.2025.01108.
  34. A.A. Suleiman, A. Suleiman, U.A. Abdullahi, S.A. Suleiman, Estimation of the Case Fatality Rate of COVID-19 Epidemiological Data in Nigeria Using Statistical Regression Analysis, Biosaf. Health 3 (2021), 4-7. https://doi.org/10.1016/j.bsheal.2020.09.003.
  35. A.A. Suleiman, H. Daud, A.G. Usman, S.I. Abba, M. Othman, M. Elgarhy, A New Two-Parameter Half-Logistic Distribution with Numerical Analysis and Applications, J. Stat. Sci. Comput. Intell. 1 (2025), 1-28. https://doi.org/10.64497/jssci.2.
  36. A.A. Suleiman, H. Daud, A.I. Ishaq, A.U. Farouk, A.S. Mohammed, M. Kayid, V.B. Nagarjuna, S. Mohammad, M. Elgarhy, A New Statistical Model for Advanced Modeling of Cancer Disease Data, Kuwait J. Sci. 52 (2025), 100429. https://doi.org/10.1016/j.kjs.2025.100429.
  37. A.I. Ishaq, A.A. Suleiman, A. Usman, H. Daud, R. Sokkalingam, Transformed Log-Burr III Distribution: Structural Features and Application to Milk Production, in: The 4th International Electronic Conference on Applied Sciences, MDPI, Basel Switzerland, 2023, pp. 322. https://doi.org/10.3390/asec2023-15289.
  38. S.O. Bashiru, M. Kayid, R. Sayed, O.S. Balogun, A. Hammad, M.A. El-Raouf, Transmuted Inverse Unit Teissier Distribution: Properties, Estimations and Applications to Medical and Radiation Sciences, J. Radiat. Res. Appl. Sci. 18 (2025), 101208. https://doi.org/10.1016/j.jrras.2024.101208.
  39. A. Usman, A.I. Ishaq, M. Tasi’U, A.A. Suleiman, U.A. Abdullahi, Inverse Power Log-Toppleone distribution and its competitors for estimating the exchange rate datasets, in: 2023 4th International Conference on Data Analytics for Business and Industry (ICDABI), IEEE, 2023, pp. 639-643. https://doi.org/10.1109/icdabi60145.2023.10629436.
  40. R.B. Yunus, N. Zainuddin, K. Kamfa, B.D. Garba, M.A. Lawan, S.I. Mohammed, An Efficient Dai-Yuan Cg Method Based on Structured Secant Conditions for Nls Problems and Its Application, J. Stat. Sci. Comput. Intell. 1 (2025), 73-84. https://doi.org/10.64497/jssci.5.
  41. B. Gompertz, On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies. In a Letter to Francis Baily, Esq. FRS &c. By Benjamin Gompertz, Esq. F. R. S., Phil. Trans. R. Soc.115 (1825), 513–583. http://doi.org/10.1098/rstl.1825.0026.
  42. N.L. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions, John Wiley & Sons, (1995).
  43. K. Ohishi, H. Okamura, T. Dohi, Gompertz Software Reliability Model: Estimation Algorithm and Empirical Validation, J. Syst. Softw. 82 (2009), 535-543. https://doi.org/10.1016/j.jss.2008.11.840.
  44. A.C. Economos, Rate of Aging, Rate of Dying and the Mechanism of Mortality, Arch. Gerontol. Geriatr. 1 (1982), 3-27. https://doi.org/10.1016/0167-4943(82)90003-6.
  45. A.C. Bemmaor, N. Glady, Modeling Purchasing Behavior with Sudden “death”: A Flexible Customer Lifetime Model, Manag. Sci. 58 (2012), 1012-1021. https://doi.org/10.1287/mnsc.1110.1461.
  46. S. Dey, T. Kayal, Y.M. Tripathi, Evaluation and Comparison of Estimators in the Gompertz Distribution, Ann. Data Sci. 5 (2017), 235-258. https://doi.org/10.1007/s40745-017-0126-z.
  47. S. Dey, F.A. Moala, D. Kumar, Statistical Properties and Different Methods of Estimation of Gompertz Distribution with Application, J. Stat. Manag. Syst. 21 (2018), 839-876. https://doi.org/10.1080/09720510.2018.1450197.
  48. T. Missov, A. Lenart, Linking Period and Cohort Life-Expectancy Linear Increases in Gompertz Proportional Hazards Models, Demogr. Res. 24 (2011), 455-468. https://doi.org/10.4054/demres.2011.24.19.
  49. F.A. MOALA, S. DEY, Objective and Subjective Prior Distributions for the Gompertz Distribution, An. Acad. Bras. Ciências 90 (2018), 2643-2661. https://doi.org/10.1590/0001-3765201820171040.
  50. J.H. Pollard, E. J. Valkovics, The Gompertz Distribution and Its Applications, Genus, 48 (1992), 15-28.
  51. C. Lai, M. Xie, D. Murthy, Ch. 3. Bathtub-shaped failure rate life distributions, in: Handbook of Statistics, Elsevier, 2001, pp. 69-104. https://doi.org/10.1016/s0169-7161(01)20005-4.
  52. A. El-Gohary, A. Alshamrani, A.N. Al-Otaibi, The Generalized Gompertz Distribution, Appl. Math. Model. 37 (2013), 13-24. https://doi.org/10.1016/j.apm.2011.05.017.
  53. A. Al-Khedhairi, A. El-Gohary, A New Class of Bivariate Gompertz Distributions and Its Mixture, Int. J. Math. Anal. 2 (2008), 235-253.
  54. L.J. Bain, Analysis for the Linear Failure-Rate Life-Testing Distribution, Technometrics 16 (1974), 551-559. https://doi.org/10.2307/1267607.
  55. A.M. Sarhan, D. Kundu, Generalized Linear Failure Rate Distribution, Commun. Stat. - Theory Methods 38 (2009), 642-660. https://doi.org/10.1080/03610920802272414.
  56. M. Muhammad, B. Abba, J. Xiao, N. Alsadat, F. Jamal, M. Elgarhy, A New Three-Parameter Flexible Unit Distribution and Its Quantile Regression Model, IEEE Access 12 (2024), 156235-156251. https://doi.org/10.1109/access.2024.3485219.
  57. A.A. Jafari, S. Tahmasebi, Gompertz-power Series Distributions, Commun. Stat. - Theory Methods 45 (2015), 3761-3781. https://doi.org/10.1080/03610926.2014.911904.
  58. A.A. Jafari, S. Tahmasebi, M. Alizadeh, The Beta-Gompertz Distribution, arXiv:1407.0743 (2014). http://arxiv.org/abs/1407.0743v1.
  59. A. El-Gohary, A. Alshamrani, A.N. Al-Otaibi, The Generalized Gompertz Distribution, Appl. Math. Model. 37 (2013), 13-24.
  60. S.R. Haile, J. Jeong, X. Chen, Y. Cheng, A 3-Parameter Gompertz Distribution for Survival Data with Competing Risks, with an Application to Breast Cancer Data, J. Appl. Stat. 43 (2016), 2239-2253. https://doi.org/10.1080/02664763.2015.1134450.
  61. 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.
  62. M.E. Ghitany, S.M. Aboukhamseen, A.A. Baqer, R.C. Gupta, Gompertz-lindley Distribution and Associated Inference, Commun. Stat. - Simul. Comput. 51 (2022), 2599-2618. https://doi.org/10.1080/03610918.2019.1699113.
  63. M.A.A. Boshi, S.H. Abid, N.H. Al-Noor, Generalized Gamma – Generalized Gompertz Distribution, J. Phys.: Conf. Ser. 1591 (2020), 012043. https://doi.org/10.1088/1742-6596/1591/1/012043.
  64. A.A. Suleiman, H. Daud, M. Othman, A.I. Ishaq, R. Indawati, M.L. Abdullah, A. Husin, The Odd Beta Prime-G Family of Probability Distributions: Properties and Applications to Engineering and Environmental Data, Comput. Sci. Math. Forum 7 (2023), 20. https://doi.org/10.3390/iocma2023-14429.
  65. A.A. Suleiman, H. Daud, A.I. Ishaq, M. Kayid, R. Sokkalingam, Y. Hamed, M. Othman, V.B. Nagarjuna, M. Elgarhy, A New Weibull Distribution for Modeling Complex Biomedical Data, J. Radiat. Res. Appl. Sci. 17 (2024), 101190. https://doi.org/10.1016/j.jrras.2024.101190.
  66. H. Daud, A.A. Suleiman, A.I. Ishaq, N. Alsadat, M. Elgarhy, A. Usman, P. Wiratchotisatian, U.A. Ubale, Y. Liping, A New Extension of the Gumbel Distribution with Biomedical Data Analysis, J. Radiat. Res. Appl. Sci. 17 (2024), 101055. https://doi.org/10.1016/j.jrras.2024.101055.
  67. A.A. Suleiman, H. Daud, A.I. Ishaq, M. Othman, H.M. Alshanbari, S.N. Alaziz, A Novel Extended Kumaraswamy Distribution and Its Application to COVID‐19 Data, Eng. Rep. 6 (2024), e12967. https://doi.org/10.1002/eng2.12967.
  68. A.A. Suleiman, H. Daud, A.I. Ishaq, M. Othman, R. Sokkalingam, A. Usman, A.A. Osi, The Odd Beta Prime Inverted Kumaraswamy Distribution with Application to COVID-19 Mortality Rate in Italy, in: The 4th International Electronic Conference on Applied Sciences, MDPI, Basel Switzerland, 2023, pp. 218. https://doi.org/10.3390/asec2023-16310.
  69. A.A. Suleiman, H. Daud, N.S.S. Singh, A.I. Ishaq, M. Othman, A New Odd Beta Prime-Burr X Distribution with Applications to Petroleum Rock Sample Data and Covid-19 Mortality Rate, Data 8 (2023), 143. https://doi.org/10.3390/data8090143.
  70. A.A. Suleiman, H. Daud, O. Mahmod, N. Singh, A Novel Extension of the Fréchet Distribution: Statistical Properties and Application to Groundwater Pollutant Concentrations, Data Sci. Insights, 1 (2023), 8-24.
  71. A.A. Suleiman, H. Daud, N.S.S. Singh, M. Othman, A.I. Ishaq, R. Sokkalingam, A Novel Odd Beta Prime-Logistic Distribution: Desirable Mathematical Properties and Applications to Engineering and Environmental Data, Sustainability 15 (2023), 10239. https://doi.org/10.3390/su151310239.
  72. A.S. Alghamdi, M.M. Abd El-Raouf, Exploring the Dynamics of Covid-19 with a Novel Family of Models, Mathematics 11 (2023), 1641. https://doi.org/10.3390/math11071641.
  73. A.M.T. Abd El-Bar, I.E. Ragab, On Weighted Exponential-Gompertz Distribution: Properties and Application, J. Taibah Univ. Sci. 13 (2019), 616-627. https://doi.org/10.1080/16583655.2019.1600277.
  74. M. Shama, S. Dey, E. Altun, A.Z. Afify, The Gamma–gompertz Distribution: Theory and Applications, Math. Comput. Simul. 193 (2022), 689-712. https://doi.org/10.1016/j.matcom.2021.10.024.
  75. A.I. Ishaq, U. Panitanarak, A.A. Abiodun, A.A. Suleiman, H. Daud, The Generalized Odd Maxwell-Kumaraswamy Distribution: Its Properties and Applications, Contemp. Math. (2024), 711-742. https://doi.org/10.37256/cm.5120242888.