Statistical properties and different estimation methods of Inverse Unit Gompertz Distribution with applications on health data sets




IUGD, COVID-19, Gompertz Distribution, Monte Carlo simulation, unit distribution, inversion, reliability measures, probability density function


Continuous probability distributions are always helpful in lifetime data and health-related data sets. Various techniques exist to develop new probability distributions, adding new parameters and applying different transformations. Adding new parameters is not always good; rather, it can also have complex expressions for the function and properties. This research aimed to develop a model without adding new parameters, which will work more efficiently than the existing models. This study proposes a new probability density function by taking the inversion of a random variable whose probability density function is Unit Gompertz Distribution. The newly proposed distribution is called an Inverse Unit Gompertz Distribution (IUGD). Various properties include reliability/survivorship measures, odd function, elasticity, and Mills ratio. Different statistical properties such as moments, quantile function, and Lorenz and Bonferroni curves for IUGD are developed. Five estimation methods are discussed for unknown parameters of the IUGD, and simulations have been conducted. Finally, IUGD is applied to two real-life data sets, i.e., COVID-19 death rates in the Netherlands and the pain relief time of individuals who received analgesics experienced. IUGD is flexible compared to other competing densities. Moreover, the proposed density can be used for health-related data sets to take accurate precautions and treatments.


Abu-Zinadah, H. (2014). Six methods of estimations for the shape parameter of Exponentiated Gompertz Distribution. Applied Mathematical Sciences. 8, 88, 4349–4359. DOI:

Adegoke, T. M., Oladoja, O. M., Bashiru, S. O., Mustapha, A. A., Aderupatan, D. E., & Nzei, L. C. (2023). Topp-Leone Inverse Gompertz Distribution: properties and different estimations techniques and applications. Pakistan Journal of Statistics, 39(4), 433-456.

Alsadat, N., Hassan, A., Elgarhy, M., Chesneau, C., & Mohamed, R. (2023). An efficient stress–strength reliability estimate of the Unit Gompertz Distribution using ranked set sampling. Symmetry, 15, 1121. DOI:

Arshad, M., Azhadc, Q.J., Gupta, N., & Pathake, A.K. (2021). Bayesian inference of Unit Gompertz distribution based on dual generalized order statistics. Communications in Statistics: Simulation and Computation, 52(8), 3657-3675. DOI:

Bantan, R., Jamal, F., Chesneau, C., & Elgarhy, M. (2021). Theory and applications of the Unit Gamma/Gompertz Distribution. Mathematics, 9, 1850. DOI:

Bemmaor, A. C. (1994). Modelling the diffusion of new durable goods: word-of-mouth effect versus consumer heterogeneity. In G. Laurent, G. L. Lilien, & B. Pras (eds.), Research traditions in marketing (pp. 201-229). DOI:

Dey, S., Moala, F., & Kumar, D. (2018). Statistical properties and different methods of estimation of Gompertz distribution with application. Journal of Statistics and Management Systems, 21(5), 839-876. DOI:

Eliwa, M. S., Alhussain, Z. A., & El-Morshedy, M. (2020). Discrete Gompertz-G family of distributions for over- and under-dispersed data with properties, estimation, and applications. Mathematics, 8(3), 358. DOI:

El-Morshedy, M., El-Faheem, A., & El-Dawoody, M. (2020). Kumaraswamy inverse Gompertz distribution: Properties and engineering applications to complete, type-II right censored and upper record data. PLoS ONE 15(12), e0241970. DOI:

Garg, M., Rao, B., & Redmond, C. (1970). Maximum-likelihood estimation of the parameters of the Gompertz survival function. Journal of the Royal Statistical Society, Series C (Applied Statistics), 19(2), 152-159. DOI:

Jaheen, Z. (2003). A Bayesian analysis of record statistic from the Gompertz model. Applied Mathematics and Computation, 145(2), 307-320. DOI:

Jha, M. K., Dey, S., Alotaibi, R. M., Alomani, G., & Tripathi, Y. M. (2020). Reliability estimation of a multicomponent stress-strength model for unit Gompertz distribution under progressive Type II censoring. Qual. Reliab. Eng. Inter. 36, 965–987. DOI:

Jha, M.K., Dey, S., & Tripathi, Y. (2019). Reliability estimation in a multicomponent stress-strength based on unit-Gompertz distribution. Inter. J. Qual. Reliab. Manag., 37, 428–450. DOI:

Khaleel, M., Hashim, N., & Abdal-Hameed, M. (2020). Marshall Olkin exponential Gompertz distribution: Properties and applications. Periodicals of Engineering and Natural Sciences, 8, 1, 298-312.

Kumar, D., Dey, S., Ormoz, E., & MirMostafaee, S. (2020). Inference for the unit-Gompertz model based on record values and inter-record times with an application. Rendiconti del Circolo Matematico di Palermo Series 2, 69, 1295–1319. DOI:

Lee, K., & Seo, J. I. (2020). Different Approaches to Estimation of the Gompertz Distribution under the Progressive Type-II Censoring Scheme. Journal of probability and Statistics, 2020, 3541946. DOI:

Mazucheli, J., Menezes, A. F., & Dey, S. (2019). Unit-Gompertz Distribution with Applications. Statistica, 79(1), 25–43.

Roy, S., & Adnan, M. A. S. (2012). Wrapped Generalized Gompertz distribution: an application to Ornithology. Journal of Biometrics & Biostatistics, 3(6), 1-4. DOI:



How to Cite

Bashir, S., Tayyab, A., Mushtaq, N., Naqvi, I. B., & Vafaeva, K. M. (2023). Statistical properties and different estimation methods of Inverse Unit Gompertz Distribution with applications on health data sets. Natural and Applied Sciences International Journal (NASIJ), 4(2), 41–62.

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