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A new extension of power Lindley distribution for analyzing bimodal data. (English) Zbl 1449.62025
Summary: In this article, we introduce a new three-parameter odd log-logistic power Lindley distribution and discuss some of its properties. These include the shapes of the density and hazard rate functions, mixture representation, the moments, the quantile function, and order statistics. Maximum likelihood estimation of the parameters and their estimated asymptotic standard errors are derived. Three algorithms are proposed for generating random data from the proposed distribution. A simulation study is carried out to examine the bias and mean square error of the maximum likelihood estimators of the parameters. An application of the model to two real data sets is presented finally and compared with the fit attained by some other well-known two and three-parameter distributions for illustrative purposes. It is observed that the proposed model has some advantages in analyzing lifetime data as compared to other popular models in the sense that it exhibits varying shapes and shows more flexibility than many currently available distributions.

MSC:
62E15 Exact distribution theory in statistics
62N05 Reliability and life testing
60E05 Probability distributions: general theory
Software:
R; AdequacyModel
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References:
[1] Adamidis, K., and Loukas, S. 1998. A lifetime distribution with decreasing failure rate. Statistics and Probability Letters 39, 35-42. · Zbl 0908.62096
[2] Arnold, B.C., Balakrishnan, N., and Nagaraja, H.N. 1992. A First Course in Order Statistics. Wiley, New York. · Zbl 0850.62008
[3] Ashour, S.K., and Eltehiwy, M.A. 2015. Exponentiated power Lindley distribution. Journal of Advanced Research 6, 895-905.
[4] Chen, G., and Balakrishnan, N. 1995. A general purpose approximate goodness-of-fit test. Journal of Quality Technology 27, 154-161.
[5] Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., and Knuth, D.E. 1996. On the LambertWfunction. Advances in Computational Mathematics 5, 329-359. · Zbl 0863.65008
[6] Cooray, K. 2006. Generalization of the Weibull distribution: the odd Weibull family. Statistical Modelling 6, 265-277.
[7] Ghitany, M.E., Atieh, B., and Nadarajah, S. 2008. Lindley distribution and its application. Mathematics and Computers in Simulation 78, 493-506. · Zbl 1140.62012
[8] Ghitany, M.E., Al-Mutairi, D.K., Balakrishnan, N., and Al-Enezi, L.J. 2013. Power Lindley distribution and associated inference. Computational Statistics and Data Analysis 64, 20-33. · Zbl 06958940
[9] Glaser, R.E. 1983. Statistical Analysis of Kevlar 49/Epoxy Composite Stress-Rupture Data. UCID-19849, Lawrence Livermore National Laboratory.
[10] Gleaton, J.U., and Lynch, J.D. 2004. On the distribution of the breaking strain of a bundle of brittle elastic fibers. Advances in Applied Probability 36, 98-115. · Zbl 1041.60075
[11] Gleaton, J.U., and Lynch, J.D. 2006. Properties of generalized log-logistic families of lifetime distributions. Journal of Probability and Statistical Science 4, 51-64.
[12] G´omez, Y.M., Bolfarine, H., and G´omez, H.W. 2014. A new extension of the exponential distribution. Revista Colombiana de Estad´ıstica 37, 25-34.
[13] Gradshteyn, I.S., and Ryzhik, I.M. 2000. Table of Integrals, Series, and Products (6th Ed.), Corrected by A. Jeffrey and D. Zwillinger. Academic Press, San Diego. · Zbl 0981.65001
[14] Jodr´a, P. 2010. Computer generation of random variables with Lindley or Poisson-Lindley Distribution via the LambertWFunction. Mathematics and Computers in Simulation 81, 851-859. · Zbl 1207.65012
[15] Leadbetter, M.R., Lindgren, G., and Rootz´en, H. 1983. Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.
[16] Lee, E.T., and Wang, J.W. 2003. Statistical Methods for Survival Data Analysis (3rd Ed.). Wiley, New Jersey. · Zbl 1026.62103
[17] Lehmann E.L., and Casella, G. 1998. Theory of Point Estimation (2nd Ed.). SpringerVerlag, New York. · Zbl 0916.62017
[18] Lindley, D.V. 1958. Fiducial distributions and Bayes’ theorem. Journal of the Royal Statistical Society, Series B 20, 102-107. · Zbl 0085.35503
[19] Marinho, P.R.D., Bourguignon, M., and Dias, C.R.B. 2013. AdequacyModel: Adequacy of probabilistic models and generation of pseudo-random numbers. R package version 1.0.8. URL:https://CRAN.R-project.org/package=AdequacyModel
[20] Merovci, F., and Sharma, V.K. 2014. The beta-Lindley distribution: properties and applications. Journal of Applied Mathematics, ID 198951,doi:10.1155/2014/198951.
[21] Nadarajah, S., Bakouch, H.S., and Tahmasbi, R. 2011. A generalized Lindley distribution Sankhya B 73, 331-359. · Zbl 1268.62018
[22] Ozel, G., Alizadeh, M., Cakmakyapan, S., Hamedani, G.G., Ortega, E.M.M., and Cancho, V.G. 2016. The odd log-logistic Lindley Poisson model for lifetime data. Communications in Statistics - Simulation and Computation,doi:10.1080/03610918.2016.1206931 · Zbl 1462.62588
[23] 86Alizadeh, MirMostafaee and Ghosh
[24] R Core Team 2016. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL:https://www.R-project.org.
[25] Warahena-Liyanage, G.
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