© V. Ranjbar, A. Eftekharian, O. Kharazmi, M. Alizadeh. Article available under the CC BY-SA 4.0 licence

In this paper, a new three-parameter lifetime model, called the odd log-logistic generalised Lindley distribution, is introduced. Some structural properties of the new distribution including ordinary and incomplete moments, quantile and generating functions and order statistics are obtained. The new density function can be expressed as a linear mixture of exponentiated Lindley densities. Different methods are discussed to estimate the model parameters and a simulation study is carried out to show the performance of the new distribution. The importance and flexibility of the new model are also illustrated empirically by means of two real data sets. Finally, Bayesian analysis and Gibbs sampling are performed based on the two real data sets.

Lindley distribution, odd log-logistic generalised family, moments, Bayesian analysis, simulation study

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 Ci^encias, 91(3).

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, 35, pp. 281–308.

Alizadeh, M., Afshari, M., Hosseini, B., , and Ramires, T. G., (2020). Extended exp-g family of distributions: Properties and applications. Communication in Statistics-Simulation and Computation, 49 (7), pp. 1730–1745.

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

Alizadeh, M., K MirMostafaee, S., Altun, E., Ozel, G., and Khan Ah- madi, M., (2017). 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., (2017). The odd log-logistic marshall-olkin lindley model for lifetime data. Journal of Statistical Theory and Applications, 16(3), pp. 382–400.

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

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

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

Congdon, P., (2001). Bayesian statistical analysis. Wiley, New York.

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.

De Haan, L., Ferreira, A., and Ferreira, A., (2006). Extreme value theory: an introduction, Springer, volume 21.

Doomik, J., (2007). Object-Oriented Matrix Programming Using OX. International Thomson Business Press and Oxford, London.

Geman, S. and Geman, D., (1984). Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Transactions on pattern analysis and machine intelligence,6, pp. 721– 741.

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. and 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., Ven^ancio, R., and Cordeiro, G. M., (2017). The odd lindley-g family of distributions. Austrian Journal of Statistics, 46(1), pp. 65–87.

Gradshteyn, I. and Ryzhik, I., (2007). Table of Integrals, Series, and Products. Elsevier/Academic Press.

Hastings,W. K., (1970). Monte Carlo sampling methods using Markov chains and their applications. Oxford University Press.

Jodr´ a, 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.

Leadbetter, M. R., Lindgren, G., and Rootz´en, H., (2012). Extremes and related properties of random sequences and processes. Springer Science and Business Media.

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

Lo, G. S., Ngom, M., Kpanzou, T. A., and Diallo, M., (2018). Weak convergence (iia)-functional and random aspects of the univariate extreme value theory. arXiv preprint arXiv:1810.01625.

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

Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E., (1953). Equation of state calculations by fast computing machines. The journal of chemical physics, 21(6), pp. 1087–1092.

Murthy, D. P., Xie, M., and Jiang, R., (2004). Weibull models, volume 505. John Wiley and 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, pp. 1–34.

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., (2018). Extended generalized lindley distribution: Properties and applications. Journal of Mathematical Extension, 13(1), pp. 117–142.

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