An extended odd log-logistic-Lindley distribution with properties, applications and Bayesian estimation

Abbas Eftekharian; Morad Alizadeh; Vahid Ranjbar; Omid Kharazmi; Gholamhossein Hamedani

doi:https://doi.org/10.59139/stattrans-2026-005

An extended odd log-logistic-Lindley distribution with properties, applications and Bayesian estimation

Abbas Eftekharian 1University of Hormozgan, Iran ORCID:https://orcid.org/0000-0002-5343-8597 , Morad Alizadeh Persian Gulf University, Iran ORCID:https://orcid.org/0000-0001-6638-2185 , Vahid Ranjbar Corresponding author. Golestan University, Iran ORCID:https://orcid.org/0000-0003-3743-0330 , Omid Kharazmi Vali-e-Asr University of Rafsanjan, Iran ORCID:https://orcid.org/0000-0003-4176-9708 , Gholamhossein Hamedani 5Marquette University, United States ORCID:https://orcid.org/0000-0003-3216-0511 Statistics in Transition new series, vol. 27, 2026, 1, pages: 85-103 Published online: 11 March 2026 https://doi.org/10.59139/stattrans-2026-005 Citation: Eftekharian A., Alizadeh M., Ranjbar, Kharazmi O., Hamedani G., 2026. An extended odd log-logistic - Llindley distribution with properties, applications and Bayesian estimation. Statistics in Transition new series, 27(1), pp. 85-103 https://doi.org/10.59139/stattrans-2026-005

271 Views 2 Downloads

ARTICLE

(English) PDF

ABSTRACT

This paper introduces a four-parameter extended odd log-logistic-Lindley distribution from which moments, hazard, and quantile functions are then obtained. The statistical properties of this distribution show the high flexibility of the proposed distribution. The maximum likelihood and least-squares estimators of the extended odd log-logistic-Lindley parameters are studied. Moreover, a simulation study is carried out for evaluating the performance of the estimation methods, and the usefulness of the new distribution is illustrated using two real data sets. Finally, Bayesian analysis and efficiency of Gibbs sampling are provided on the basis of two real data sets.

KEYWORDS

Bayesian estimation, Gibbs sampling, Lindley distribution, moment, odd loglogistic, simulation

REFERENCES

Abouammoh, A., Alshangiti, A. M. and Ragab, I., (2015). A new generalized Lindley distribution. Journal of Statistical Computation and Simulation, 85(18), pp. 3662–3678.

Afify, A. Z., Cordeiro, G. M., Maed, M. E., Alizadeh, M., Al-Mofleh, H. and Nofal, Z. M., (2019). The generalized odd lindley-g family: properties and applications. Anais da Academia Brasileira de Ciencias, 91(3).

Akinsete, A., Famoye, F. and Lee, C., (2008). The beta-pareto distribution. Statistics, 42(6), pp. 547–563.

Alizadeh, M., Afshari, M., Hosseini, B. and Ramires, T. G., (2018a). Extended expg family of distributions: Properties and applications. Communication in Statistics- Simulation and Computation, accepted.

Alizadeh, M., Afify, A. Z., Eliwa, M. and Ali, S., (2019). The odd log-logistic lindley-g family of distributions:properties, bayesian and non-bayesian estimation with applications. Computational Statistics, pp. 1–28.

Alizadeh, M., Altun, E., Ozel, G., Afshari, M., and Eftekharian, A., (2018b). A new odd log-logistic lindley distribution with properties and applications. Sankhya A, 81(2), pp. 323–346.

Alizadeh, M., K MirMostafaee, S., Altun, E., Ozel, G. and Khan Ahmadi, M., (2017a). The odd log-logistic marshall-olkin power lindley distribution: Properties and applications. Journal of Statistics and Management Systems, 20(6), pp. 1065–1093.

Alizadeh, M., Ozel, G., Altun, E., Abdi, M. and Hamedani, G. (2017b). The odd loglogistic marshall-olkin lindley model for lifetime data. Journal of Statistical Theory and Applications, 16(3), pp. 382–400.

Alizadeh, M., Afshari, M., Cordeiro, G. M., Ramaki, Z., Contreras-Reyes, J. E., Dirnik, F. and Yousof, H. M., (2025). A Weighted Lindley Claims Model with Applications to Extreme Historical Insurance Claims. Stats, 8(1), p. 8.

Altun, G., Alizadeh, M., Altun, E. and Ozel, G., (2017). Odd burr lindley distribution with properties and applications. Hacettepe Journal of Mathematics and Statistics, 46(2), pp. 255–276.

Alzaatreh, A., Lee, C. and Famoye, F., (2013). A new method for generating families of continuous distributions. Metron, 71(1), pp. 63–79.

Arellano-Valle, R. B.; Contreras-Reyes, J. E.; Stehlík, M., (2017). Generalized skewnormal negentropy and its application to fish condition factor time series. Entropy, 19, p. 528.

Asgharzadeh, A., Bakouch, H. S., Nadarajah, S., Sharafi, F., et al., (2016). A new weighted lindley distribution with application. Brazilian Journal of Probability and Statistics, 30(1), pp. 1–27.

Asgharzadeh, A., Nadarajah, S. and Sharafi, F., (2018). Weibull lindley distribution. REVSTAT–Statistical Journal, 16(1), pp. 87–113.

Ashour, S. K., Eltehiwy, M. A., (2015). Exponentiated power lindley distribution. Journal of advanced research, 6(6), pp. 895–905.

Calabria, R., Pulcini, G., (1996). Point estimation under asymmetric loss functions for left-truncated exponential samples. Communications in Statistics-Theory and Methods, 25(3), pp. 585–600.

Contreras-Reyes, J. E., Kahrari, F. and Cortés, D. D., (2021). On the modified skewnormal- Cauchy distribution: Properties, inference and applications. Communications in Statistics-Theory and Methods, 50(15), pp. 3615-3631.

Corless, R. M., Gonnet, G. H., Hare, D. E., Jeffrey, D. J. and Knuth, D. E., (1996). On the lambertw function. Advances in Computational mathematics, 5(1), pp. 329–359.

Galambos, J., Kotz, S., (2006). Characterizations of Probability Distributions.: A Unified Approach with an Emphasis on Exponential and Related Models, vol. 675. Springer.

Ghitany, M., Al-Mutairi, D. K., Balakrishnan, N. and Al-Enezi, L., (2013). Power lindley distribution and associated inference. Computational Statistics Data Analysis, 64, pp. 20–33.

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

Gleaton, J. U., Lynch, J. D., (2010). Extended generalized log-logistic families of lifetime distributions with an application. J. Probab. Stat. Sci, 8, pp. 1–17.

Gomes-Silva, F. S., Percontini, A., de Brito, E., Ramos, M.W., Venancio, R., and Cordeiro, G. M., (2017). The odd lindley-g family of distributions. Austrian Journal of Statistics, 46(1), pp. 65–87.

Jodr, P., (2010). Computer generation of random variables with lindley or poisson–lindley distribution via the lambert w function. Mathematics and Computers in Simulation, 81(4), pp. 851–859.

Kim, J. H., Jeon, Y., (2013). Credibility theory based on trimming. Insurance: Mathematics and Economics, 53(1), pp. 36–47.

Kotz, S., Shanbhag, D., (1980). Some new approaches to probability distributions. Advances in Applied Probability, pp. 903–921.

Lindley, D. V., (1958). Fiducial distributions and bayes’ theorem. Journal of the Royal Statistical Society. Series B (Methodological), 20(1), pp. 102–107.

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.

Merovci, F., Sharma, V. K., (2014). The beta-lindley distribution: properties and applications. Journal of Applied Mathematics.

Murthy, D. P., Xie, M. and Jiang, R., (2004). Weibull models, vol. 505, John Wiley Sons.

Nadarajah, S., Bakouch, H. S. and Tahmasbi, R., (2011). A generalized lindley distribution. Sankhya B, 73(2), pp. 331–359.

Oluyede, B. O., Yang, T. and Makubate, B., (2016). A new class of generalized power lindley distribution with applications to lifetime data. Asian Journal of Mathematics and Applications, 2016, p.1.

Ozel, G., Alizadeh, M., Cakmakyapan, S., Hamedani, G., Ortega, E. M. and Cancho, V. G., (2017). The odd log- logistic lindley poisson model for lifetime data. Communications in Statistics-Simulation and Computation, 46(8), pp. 6513–6537.

Ranjbar, V., Alizadeh, M. and Altun, E., (2019). Extended generalized lindley distribution: Properties and applications. Journal of Mathematical Extension, 13(1), pp. 117–142.

Zakerzadeh, H., Dolati, A., (2009). Generalized lindley distribution. Journal of Mathematical Extension, 3(2), pp. 1–17.