Arjun Kumar Gaire , Yogendra Bahadur Gurung

© Arjun Kumar Gaire, Yogendra Bahadur Gurung. Article available under the CC BY-SA 4.0 licence


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This paper introduces a novel three-parameter skew-log-logistic distribution. The research involves the development of a new random variable based on Azzalini and Capitanio’s (2013) proposition. Additionally, various statistical properties of this distribution are explored. The paper presents a maximum likelihood method for estimating the distribution’s parameters. The density function exhibits unimodality with heavy right tails, while the hazard function exhibits rapid increase, unimodality, and slow decrease, resulting in a right-skewed curve. Furthermore, four real datasets are utilized to assess the applicability of this new distribution. The AIC and BIC criteria are employed to assess the goodness of fit, revealing that the new distribution offers greater flexibility compared to the baseline distribution.


Log-Logistic, skew, marriage, menarche, age-specific fertility rate.


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

Alexander, C., Cordeiro, G. M., Ortega, E. M. and Sarabia, J. M. (2012). Generalized Betagenerated distributions. Computational Statistics and Data Analysis, 56(6), pp. 1880–1897.

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

Ashkar, F., Mahdi, S., (2006). Fitting the log-logistic distribution by generalized moments. Journal of Hydrology, 328(3-4), pp. 694–703.

Asili, S., Rezaei, S. and Najjar, L., (2014). Using skew-logistic probability density function as a model for age-specific fertility rate pattern. BioMed Research International, 10, pp. 1–5.

Azzalini A., (1985). A class of distributions that includes the normal ones. Scandinavian Journal of Statistics, pp. 171–178.

Azzalini A., (2005). The skew-normal distribution and related multivariate families. Scandinavian Journal of Statistics, 32(2), pp. 159–188.

Azzalini, A., Capitanio, A., (2013). The skew-normal and related families (Vol. 3), London: Cambridge University Press.

Barlow, R. E., Marshall, A. W. and Proschan, F., (1963). Properties of probability distributions with monotone hazard rate. The Annals of Mathematical Statistics, 34(2), pp. 375–389.

Collett, D., (2015). Modeling survival data in medical research. Boca Raton, Florida USA: CRC press.

Cordeiro, G. M., De-Castro M., (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81, pp. 883–898.

Cordeiro, G. M., Ortega, E. M. and Da-Cunha, D. C., (2013). The exponentiated generalized class of distributions. Journal of Data Science, 11(1), pp. 1–27.

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.

Fisk, P. R., (1961). The graduation of income distribution. Econometrica, 29(2), pp. 171– 185.

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., 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.

Gaire, A. K., (2022). Skew Lomax distribution, parameter estimation, its properties, and applications. Journal of Science and Engineering, 10, pp. 1–11.

Ghitany, M.E., Atieh, B. and Nadarajah, S., (2008). Lindley distribution and its application. Mathematics and Computers in Simulation, 78(4), pp. 493–506.

Gradshteyn, I. S., Ryzhik, I. M., (2000). Table of integrals, series, and products, San Diego, CA: Academic Press.

Gupta, A. K., Chang, F. C. and Huang, W. J., (2002). Some skew-symmetric models. Random Operators and Stochastic Equations, 10(2), pp. 133–140.

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

Harvda, J., Charvat, F., (1967). Quantification method of classification processes. Concept of structural a-entropy. Kybernetika, 3(1), pp. 30–35.

Hassan, A. S., Hemeda, S. E., (2016). The additive Weibull-G family of probability distributions. International Journals of Mathematics and Its Applications, 4(2), pp. 151- 164.

Hemeda, S., (2018). Additive Weibull 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.

Kleiber, C., Kotz, S., (2003). Statistical size distributions in economics and actuarial sciences (Vol. 470). New York: John Wiley and Sons.

Lawless, J. F., (2003). Statistical models and methods for lifetime data. Vol. 362. New York: John Wiley and Sons.

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

Lima, S. R., Cordeiro, G. M., (2017). The extended Log-Logistic distribution: Properties and application. Anais da Academia Brasileira de Ciencias, 89(1), pp. 3–17.

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

Mazzuco, S., Scarpa, B., (2011). Fitting an age-specific fertility rate by Skew-symmetrical probability density function, the University of Padova, Working paper Series, Italy, 10, pp. 1–18.

Mishra, R., Singh, K. K. and Singh, A., (2017). A model for age-specific fertility rate pattern of India using skew-logistic distribution function. American Journal of Theoretical and Applied Statistics, 6(1), pp. 32–37.

Nadarajah, S., (2009). The skew logistic distribution. Advances in Statistical Analysis, 93(2), pp. 187–203.

NDHS, (2022). Nepal Demographic and Health Survey 2022: Key Indicators Report.

Kathmandu, Nepal: Ministry of Health and Population; New ERA; and ICF., Nepal. Peristera, P., Kostaki, A., (2007). Modeling fertility in modern populations. Demographic Research, 16, pp. 141–194.

Renyi, A., (1961). On measures of entropy and information, Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, 1, pp. 547–561, Barkeley: The University of California Press.

Rodriguez, G., (2010). Parametric survival models. New Jersey: Rapport technique, Princeton University.

Rosaiah, K., Nagarjuna, K. M., Kumar, D. C. U. S. and Rao, B. S., (2014). Exponentiallog- logistic additive failure rate model. Int J Sci Res Publ., 4(3), pp. 1–5.

Shaw, W. T., Buckley, I. R., (2007). The alchemy of probability distributions: Beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map. arXiv preprint arXiv:0901.0434.

Surendran, S., Tota-Maharaj, K., (2015). Log logistic distribution to model water demand data. Procedia Engineering, 119, pp. 798–802.

Tadikamalla, P. R., (1980). A look at the Burr and related distributions. International Statistical Review/Revue Internationale de Statistique, pp. 337–344.

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.

Tsallis, C., (1988). Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics, 52, pp. 479–487.

Yilmaz, V., Erisoglu, M. and Çelik, H. E., (2011). Probabilistic prediction of the next earthquake in the NAFZ (North Anatolian Fault Zone), Turkey: Do˘gu¸s Üniversitesi Dergisi, 5(2), pp. 243–250.

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

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