New version of Log-Logistic distribution: properties and applications to survival time and demographic data

Arjun Kumar Gaire

doi:https://doi.org/10.59139/stattrans-2026-019

New version of Log-Logistic distribution: properties and applications to survival time and demographic data

Arjun Kumar Gaire Department of Science and Humanities, Khwopa Engineering College, Purbanchal University, Bhaktapur, Nepal ORCID:https://orcid.org/0000-0002-1958-9797 Statistics in Transition new series, vol. 27, 2026, 2, pages: 111-127 Published online: 9 June 2026 https://doi.org/10.59139/stattrans-2026-019 Citation: Gaire A. K., 2026. New version of Log-Logistic distribution: properties and applications to survival time and demographic data. Statistics in Transition new series, 27(2), pp. 111-127. https://doi.org/10.59139/stattrans-2026-019

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ABSTRACT

This paper proposes a Deflation-Inflation Log-Logistic (DILLog) distribution as a sub-model of the Deflation-Inflation Distributed (DID) family, introduced by Alodat and Al-Rawwash (2021). The proposed model offers greater flexibility than the original model in fitting data from real-world problems, especially for survival times and demographic data. The DILLog model is characterized by unimodal right-tailed density and hazard rate functions, and its key statistical properties, including the cumulative distribution function and a closed-form quantile function,are derived. To test the performance of the distribution, a simulation study has been used as well as an application to two real data sets: the age at menarche of Nepalese girls and the survival times of patients suffering from melanoma disease. To illustrate the usefulness and application of the proposed distribution, its parameters were estimated by using the maximum likelihood estimation method. The analysis and plots of the fitted results attest to the the DILLog model being flexible enough to fit the right-skewed real data. The actual application of demographic and survival datasets demonstrates that the DILLog distribution outperforms comparative models.

KEYWORDS

deflation-inflation, Log-Logistic, melanoma, menarche, survival

REFERENCES

Adeyinka, F. S., Olapade, A. K., (2019). On transmuted four parameters generalized Log- Logistic distribution. International Journal of Statistical Distributions and Applications, 5(2), pp. 32–37.

Alodat, M. D. T., Al-Rawwash, M., (2021). A proposed mechanism for skewing symmetric distributions. Communications in Statistics-Theory and Methods, 50(11), pp. 2674–2695.

Aryal, G. R. (2013). Transmuted log-logistic distribution. Journal of Statistics Applications and Probability, 2(1), pp. 11–20.

De-Santana, T. V. F., Ortega, E. M., Cordeiro, G. M. and Silva, G. O., (2012). The Kumaraswamy-log-logistic distribution. Journal of Statistical Theory and Applications, 11(3), pp. 265–291.

Eugene, N., Lee, C. and Famoye, F., (2002). Beta-normal distribution and its applications. Communications in Statistics-Theory and Methods, 31(4), pp. 497–512.

Gaire, A. K., (2023). Skew Lomax distribution, parameter estimation, its properties, and applications. Journal of Science and Engineering, 10, pp. 1–11.

Gaire, A. K., Aryal, R., (2015). Inverse Gaussian model to describe the distribution of age-specific fertility rates of Nepal. Journal of Institute of Science and Technology, 20(2), pp. 80–83.

Gaire, A. K., Gurung, Y. B., (2024a). Rayleigh generated Log-Logistic distribution: Properties and performance analysis. Istatistik Journal of Turkish Statistical Association, 15(1), pp. 13–28.

Gaire, A. K., Gurung, Y. B., (2024b). Skew Log-logistic distribution: Properties and application. Statistics in Transition New Series, 25(1), pp. 43–62.

Gaire, A. K., Gurung, Y. B., and Bhusal, T. P., (2024a). Age at first marriage of Nepalese women: a statistical analysis (status, differential, determinants, and distributional pattern). Journal of Population and Social Studies, 32, pp. 308—328.

Gaire, A. K., Gurung, Y. B., and Bhusal, T. P., (2023). Stochastic modeling of age at menopause for Nepalese women and development of menopausal life table. Global Health Economics and Sustainability. 1(2), pp. 1–11.

Gaire, A. K., Gurung, Y. B., and Bhusal, T. P., (2024b). Fertility model evolution: a survey on mathematical models of age-specific fertility with application to Nepalese and Malaysian data. Global Health Economics and Sustainability. 3(1), pp. 222–234.

Gaire, A. K., Gurung, Y. B., Bhusal, T. P., (2024c). Stochastic modeling of age at menarche for Nepalese girls and developing menarchial life table. Population Review, 63(2), pp. 146-161.

Gaire, A. K., Thapa G. B. and KC, S., (2019). Preliminary results of Skew Log-logistic distribution, properties, and application. Proceeding of the 2nd International Conference on Earthquake Engineering and Post Disaster Reconstruction Planning, 25–27 April 2019, Bhaktapur, Nepal, pp. 37–43.

Gaire, A. K., Thapa, G. B. and KC, S., (2022). Mathematical modeling of age-specific fertility rates of Nepali mothers. Pakistan Journal of Statistics and Operation Research, 18(2), pp. 417–426. https://doi.org/10.18187/pjsor.v18i2.3319.

Gui, W., (2013). Marshall-Olkin extended log-logistic distribution and its application in minification processes. Applied Mathematical Science, 7(80), pp. 3947–3961.

Hemeda, S., (2018). AdditiveWeibull Log Logistic distribution: Properties and application. Journal of Advanced Research in Applied Mathematics and Statistics, 3(4), pp. 8–15.

Hamedani, G., (2013). The Zografos-Balakrishnan log-logistic distribution: Properties and applications. Journal of Statistical Theory and Applications, 12(3), pp. 225–244.

Jones, M., (2004). Families of distributions arising from distributions of order statistics. Test, 13(1), pp. 1–43.

Lemonte, A. J., (2014). The Beta log-logistic distribution. Brazilian Journal of Probability and Statistics, 28(3), pp. 313–332.

Marinho, P. R. D., Silva, R. B., Bourguignon, M., Cordeiro, G. M., and Nadarajah, S., (2019). AdequacyModel: An R package for probability distributions and generalpurpose optimization. PloS One, 14(8), e0221487.

Marshall, A. W., Olkin, I., (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84(3), pp. 641–652.

Ministry of Health and Population, New ERA, and ICF International Inc. (2023). Nepal demographic and health survey 2022. Kathmandu, Nepal: Ministry of Health and Population.

Mohammad, S., Cooray, K., (2025). Modeling actuarial data using iterated trigonometric distributions. Communications in Statistics-Theory and Methods, 54(16), pp. 5112–5128.

Mohammad, S., Gaire, A.K. (2025). Exponential Arctan-G Family of Distribution with Properties and Applications, Journal of Probability and Statistics, 25(1), pp.–13.

R Core Team, (2025). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.

Susarla, V., Van Ryzin, J., (1978). Empirical Bayes estimation survival distribution function from right censored data. Anal. Statist, 6, pp. 740–755.

Tahir, M. H., Mansoor, M., Zubair, M. and Hamedani, G., (2014). McDonald log-logistic distribution with an application to breast cancer data. Journal of Statistical Theory and Applications, 13(1), pp. 65–82.

Zografos, K., Balakrishnan, N., (2009). On families of Beta-and generalized Gammagenerated distributions and associated inference. Statistical Methodology, 6(4), pp. 344–362.