Marshall-Olkin Extended Power Lindley Distribution with Application
Main Article Content
Abstract
Lindley distribution was introduced by Lindley (1958) in the context of Bayes inference. Its density function is obtained by mixing the exponential distribution, with scale parameter β, and the gamma distribution, with shape parameter 2 and scale parameter β. Recently, a new generalization of the Lindley distribution was proposed by Ghitany et al. (2013), called power Lindley distribution. This paper will introduce an extension of the power Lindley distribution using the Marshall-Olkin method, resulting in Marshall-Olkin Extended Power Lindley (MOEPL) distribution. The MOEPL distribution offers a flexibility in representing data with various shapes. This flexibility is due to the addition of a parameter to the power Lindley distribution. Some properties of the MOEPL were explored, such as probability density function (pdf), cumulative distribution function (cdf), hazard rate, survival function, quantiles, and moments. Estimation of the MOEPL parameters was conducted using maximum likelihood method. The proposed distribution was applied to data. The results were given which illustrate the MOEPL distribution and were compared to Lindley, power Lindley, gamma, and Weibull. Model comparison using the log likelihood, AIC, and BIC showed that MOEPL fit the data better than the other distributions.
Article Details
Section
Actuarial Science and Financial Mathematics
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References
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[7]M.E. Ghitany, D.K. Al-Mutairi, N. Balakrishnan, & L.J. Al-Enezi, Power Lindley distribution and associated inference. Computational Statistics & Data Analysis, Vol.64, pp.20-33, 2013.
[8] M.E. Ghitany, B. Atieh & S. Nadarajah, Lindley distribution and its application. Mathematics and computers in simulation, Vol.78[4], pp.493-506, 2008.
[9] M. Javed, T. Nawaz & M. Irfan, The Marshall-Olkin Kappa Distribution: Properties and Applications. Journal of King Saud University-Science, 2018.
[10] P. Jodra, Computer generation of random variables with Lindley or PoissonLindley distribution via the Lambert W function, Mathematics and Computers in Simulation, Vol.81[4], pp.851-859, 2010.
[11] K.K. Jose, Marshall-Olkin family of distributions and their applications in reliability theory, time series modeling and stress-strength analysis. Proc. ISI 58th World Statist. Congr Int Stat Inst, 21st-26th August, Vol.201, pp.3918-392, 2011.
[12] E. Krishna, E., & K.K. Jose, Marshall-Olkin generalized asymmetric Laplace distributions and processes. Statistica, Vol.71[4], pp.453-467, 2011.
[13] H. Krishna & K. Kumar, Reliability estimation in Lindley distribution with progressively type II right censored sample. Mathematics and Computers in Simulation, Vol.82[2], pp.281-294 , 2011.
[14] D.V. Lindley, Fiducial distributions and Bayes' theorem, Journal of the Royal Statistical Society, Series B (Methodological), pp.102-107, 1958.
[15] A.W. Marshall, & I. Olkin, A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families, Biometrika, Vol.84[3], pp.641-652 ,1997.
[16] J. Mazucheli & J.A. Achcar, The Lindley distribution applied to competing risks lifetime data. Computer methods and programs in biomedicine, Vol.104[2], pp.188-1, 2011.
[17] M. Pérez-Casany & A. Casellas, Marshall-Olkin Extended Zipf Distribution. arXiv preprint arXiv:pp.1304.4540, 2013.
[18] R. Rocha, S. Nadarajah, V. Tomazella, & Louzada, F, A new class of defective models based on the MarshallOlkin family of distributions for cure rate modeling. Computational Statistics & Data Analysis, Vol.107, pp.48-63, 2017.
[2] Alizadeh, M., Cordeiro, G. M., De Brito, E., & Demétrio, C. G. B. (2015). The beta Marshall-Olkin family of distributions. Journal of Statistical Distributions and Applications, Vol.2[1], p.4, 2015.
[3] Alizadeh, M., Tahir, M. H., Cordeiro, G. M., Mansoor, M., Zubair, M., & Hamedani, G. G, The Kumaraswamy Marshal-Olkin family of distributions. Journal of the Egyptian Mathematical Society, Vol.23[3], pp.546-557, 2015.
[4] Y.A. Al-Saiari, L.A. Baharith & S.A Mousa, S. A, Marshall-Olkin extended Burr type XII distribution. International Journal of Statistics and probability, Vol.3[1], p.78, 2014.
[5] I. Elbatal, F. Merovci, & M. Elgarhy, A new generalized Lindley distribution. Mathematical theory and Modeling, Vol.3[13], pp.30-47, 2013.
[6] M.E. Ghitany, M. E., Al-Mutairi, D. K., Al-Awadhi, F. A., & M. Al-Burais, Marshall-Olkin extended Lindley distribution and its application. International Journal of Applied Mathematics, Vol.25[5], pp.709-721, 2012.
[7]M.E. Ghitany, D.K. Al-Mutairi, N. Balakrishnan, & L.J. Al-Enezi, Power Lindley distribution and associated inference. Computational Statistics & Data Analysis, Vol.64, pp.20-33, 2013.
[8] M.E. Ghitany, B. Atieh & S. Nadarajah, Lindley distribution and its application. Mathematics and computers in simulation, Vol.78[4], pp.493-506, 2008.
[9] M. Javed, T. Nawaz & M. Irfan, The Marshall-Olkin Kappa Distribution: Properties and Applications. Journal of King Saud University-Science, 2018.
[10] P. Jodra, Computer generation of random variables with Lindley or PoissonLindley distribution via the Lambert W function, Mathematics and Computers in Simulation, Vol.81[4], pp.851-859, 2010.
[11] K.K. Jose, Marshall-Olkin family of distributions and their applications in reliability theory, time series modeling and stress-strength analysis. Proc. ISI 58th World Statist. Congr Int Stat Inst, 21st-26th August, Vol.201, pp.3918-392, 2011.
[12] E. Krishna, E., & K.K. Jose, Marshall-Olkin generalized asymmetric Laplace distributions and processes. Statistica, Vol.71[4], pp.453-467, 2011.
[13] H. Krishna & K. Kumar, Reliability estimation in Lindley distribution with progressively type II right censored sample. Mathematics and Computers in Simulation, Vol.82[2], pp.281-294 , 2011.
[14] D.V. Lindley, Fiducial distributions and Bayes' theorem, Journal of the Royal Statistical Society, Series B (Methodological), pp.102-107, 1958.
[15] A.W. Marshall, & I. Olkin, A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families, Biometrika, Vol.84[3], pp.641-652 ,1997.
[16] J. Mazucheli & J.A. Achcar, The Lindley distribution applied to competing risks lifetime data. Computer methods and programs in biomedicine, Vol.104[2], pp.188-1, 2011.
[17] M. Pérez-Casany & A. Casellas, Marshall-Olkin Extended Zipf Distribution. arXiv preprint arXiv:pp.1304.4540, 2013.
[18] R. Rocha, S. Nadarajah, V. Tomazella, & Louzada, F, A new class of defective models based on the MarshallOlkin family of distributions for cure rate modeling. Computational Statistics & Data Analysis, Vol.107, pp.48-63, 2017.