A weighted Gompertz-G family of distributions for reliability and lifetime data analysis

Abstract

This article is set to push new boundaries with leading-edge innovations in statistical distribution for generating up-to-the-minute contemporary distributions by a mixture of the second record value of the Gompertz distribution and the classical Gompertz model (weighted Gompertz model) using T-X characterization, especially used for two-sided schemes that provide an accurate model. The quantile, ordinary, and complete moments, order statistics, probability, and moments generating functions, entropies, probability weighted moments, Lin’s condition random variable, reliability in multicomponent stress strength system, reversed, and moments of residuals life and other reliability characteristics in engineering, actuarial, economics, and environmental technology were derived in their closed form. To investigate and test the flexibility, viability, tractability, and performance of the proposed Weighted Gompertz-G (WGG) generated model, the shapes of some sub-models of the WGG model were examined. The shapes of the sub-models indicated J-shapes, increasing, decreasing, and bathtub hazard rate functions. The maximum likelihood estimation of the WGG-generated model parameters was examined. An illustration with simulation and real-life data analysis indicated that the WGG-generated model provides consistently better goodness-of-fit statistics than some competitive models in the literature.

Keywords

• Gompertz distribution
• Poisson processes
• record values
• weighted Gompertz
• weighted model

References

1. Abd-AL-Motalib, R. S., Abed AL-Kadim, K., The odd truncated inverse exponential Weibull exponential distribution, Journal of Positive School Psychology , 6(2) (2022), 5361-5375.
2. Aljarrah, M. A., Lee, C., Famoye, F., On generating T-X family of distributions using quantile functions, Journal of Statistical Distributions and Applications, 1(1) (2014), 1-17. https://doi.org/10.1186/2195-5832-1-2
3. Alizadeh, M., Altun, E., Ozel, G., Afshari, M., Eftekharian, A., A new odd log-logistic Lindley distribution with properties and applications, Sankhya A, 81 (2019), 323-346. https://doi.org/10.1007/s13171-018-0142-x
4. Alzaatreh, A., Famoye, F., Lee, C., A new method for generating families of continuous distributions, Metron, 71 (2013), 63-79. https://doi.org/10.1007/s40300-013-0007-y
5. Alzaghal, A., Famoye, F., Lee, C., Exponentiated T-X family of distributions with some applications, International Journal of Statistics and Probability, 2(3) (2013), 31–49. https://doi.org/10.5539/ijsp.v2n3p31
6. Arshad, M. Z., Iqbal, M. Z., Mutairi, A. A., A comprehensive review of datasets for statistical research in probability and quality control, Journal of Mathematical and Computational Science, 11(3) (2021), 3663-3728. https://doi.org/10.28919/jmcs/5692
7. Bakouch, S. H., Ahmed, M. T., El-Bar, A., A new weighted Gompertz distribution with applications to reliability data, Applications of Mathematics, 3 (2017), 269–296. https://doi.org/10.21136/AM.2017.0277-16
8. Bemmaor, A. C., Modelling the Diffusion of New Durable Goods: Word-of-mouth Effect Versus Consumer Heterogeneity: In G. Laurent, G.L. Lilien, B. Pras eds., Research 98 Traditions in Marketing. Boston, Kluwer Academic Publishers, 1, 201-229, 1992. https://doi.org/10.1007/978-94-011-1402-8-6

9. CDC, Obesity among children and adolescelent between the aged 2- 19 years by selected characteristics, https://www.cdc.gov/nchs/hus/contents.htm-Table-027, (2020).
10. Cordeiro, G. M., Alizadeh, M., Diniz Marinho, P. R., The type I half-logistic family of distributions, Journal of Statistical Computation and Simulation, 86(4) (2016), 707-728. https://doi.org/10.1080/00949655.2015.1031233
11. Dijoux, Y., Construction of the tetration distribution based on the continuous iteration of the exponential-minus-one function, Applied Stochastic Models in Business and Industry, (2020), 1-26. https://doi.org/10.1002/asmb.2538
12. Eghwerido, J. T., The alpha power Teissier distribution and its applications, Afr. Stat., 16(2) (2021), 2731-2745. http://dx.doi.org/10.16929/as/2021.2731.181
13. Eghwerido, J. T., Agu, F. I., The shifted Gompertz-G family of distributions: properties, and applications, Mathematica Slovaca, 71(5) (2021a), 1291-1308. http://dx.doi.org/10.1515/ms-2021-0053
14. Eghwerido, J. T., Nzei, L. C, Agu, F. I., Alpha power Gompertz distribution: properties and applications, Sankhya A - The Indian Journal of Statistics, 83(1) (2021b), 449-475. http://dx.doi.org/10.1007/s13171-020-00198-0
15. Eghwerido, J. T., Oguntunde, P. E., Agu, F. I., The alpha power Marshall-Olkin-G family of distributions: properties, and applications, Sankhya A, 85 (2023), 172-197. http://dx.doi.org/10.1007/s13171-020.00235-y
16. Eghwerido, J. T., The Marshall-Olkin Teissier generated model for lifetime data, Journal of the Belarusian State University: Mathematics and Informatics, 1 (2022), 46-65. https://doi.org/10.33581/2520-6508-2022-1-46-65
17. Eghwerido, J. T., Agu, F. I., The statistical properties and applications of the alpha power Topp-Leone-G distribution, Heliyon, 8(6) (2020a), 1-10. https://doi.org/10.1016/j.heliyon.2022.e09775
18. Eghwerido, J. T., Agu, F. I., The alpha power Muth-G distribution and its applications in survival and reliability analyses, Mathematica Slovaca, 73(6) (2023), 1-18.
19. Hassana, A. S., Shawkia, A. W., Muhammeda, A. H., Weighted Weibull-G family of distributions: theory and application in the analysis of renewable energy sources,Journal of Positive School Psychology, 6(3) (2022), 9201-9216.
20. Jamal, F., Chesneau, C., Saboor, A., Aslam, M., Tahir, M. H., Mashwani, W. K., The U family of distributions: properties and applications, Mathematica Slovaca, 72(1) (2022), 217-240. https://doi.org/10.1515/ms-2022-0015
21. Lin, G. D., Dou, X., Kuriki, S., The bivariate lack-of-memory distributions, Sankhya A, 81 (2019), 273-297. https://doi.org/10.1007/s13171-017-0119-1
22. Marshall, A., Olkin, I., Life Distributions. Structure of Nonparametric, Semiparametric, and Parametric Families. Springer Series in Statistics, Springer, New York, 2007.
23. Nzei, L. C., Eghwerido, J. T., Ekhosuehi, N., Topp-Leone Gompertz distribution: properties and application, Journal of Data Science, 18(4) (2020), 782-794. https://doi.org/10.6339/JDS.202010-18(4)-0012
24. Osatohanmwen, P., Oyegue, F. O., Ogbonmwan, S. M., A new member from the T-X family of distributions: the Gumbel-Burr XII distribution and its properties, Sankhya A, 81 (2019), 298-322. https://doi.org/10.1007/s13171-017-0110-x
25. Osatohanmwen, P., Efe-Eyefia, E., Oyegue, F. O., Osemwenkhae, J. E., Ogbonmwan, S. M., Afere, B. A., The exponentiated Gumbel-Weibull(logistic) distributions with application to Nigeria’s COVID-19 infections data, Annal of Data Science, 9 (2022), 909-943. https://doi.org/10.1007/s40745-022-00373-0
26. Peng, X., Yan, Z., Estimation and application for a new extendedWeibull distribution, Reliab. Eng. Syst. Saf., 121 (2014), 34-42. https://doi.org/10.1016/j.ress.2013.07.007
27. Pollard, J. H., Valkovics, E. J., The Gompertz distribution and its applications, Genus, 48 (1992), 15-28. https://doi.org/10.2307/29789100
28. Rastogi, M. K., Merovci, F., Bayesian estimation for parameters and reliability characteristic of the Weibull Rayleigh distribution, Journal of King Saud University-Science, 30 (2018), 472-478. https://doi.org/10.1016/j.jksus.2017.05.008
29. Shama,M. S., Dey, S., Altun, E., Afify, A. Z., The gamma–Gompertz distribution: theory and applications, Mathematics and Computers in Simulation, 193 (2022), 689-712. https://doi.org/10.1016/j.matcom.2021.10.024
30. Zografos, K., Balakrishnan, N., On families of beta and generalized gamma-generated distributions and associated inference, Statistical Methodology, 6(4) (2009), 344–362. https://doi.org/10.1016/j.stamet.2008.12.003