# zbMATH — the first resource for mathematics

The exponentiated generalized power Lindley distribution: properties and applications. (English) Zbl 1438.60011
Summary: In this paper, we introduce a new extension of the power Lindley distribution, called the exponentiated generalized power Lindley distribution. Several mathematical properties of the new model such as the shapes of the density and hazard rate functions, the quantile function, moments, mean deviations, Bonferroni and Lorenz curves and order statistics are derived. Moreover, we discuss the parameter estimation of the new distribution using the maximum likelihood and diagonally weighted least squares methods. A simulation study is performed to evaluate the estimators. We use two real data sets to illustrate the applicability of the new model. Empirical findings show that the proposed model provides better fits than some other well-known extensions of Lindley distributions.

##### MSC:
 60E05 Probability distributions: general theory 62F10 Point estimation
##### Software:
EISPACK; nleqslv; R
Full Text:
##### References:
 [1] Alizadeh, M.; MirMostafaee, S. M T. K.; Altun, E.; Ozel, G.; Ahmadi, M. K., The odd log-logistic Marshall-Olkin power Lindley distribution: Properties and applications, J Stat Manag Syst, 20, 1065-1093, (2017) [2] Alizadeh, M.; MirMostafaee, S. M T. K.; Ghosh, I., A new extension of power Lindley distribution for analyzing bimodal data, Chilean J ^Stat, 8, 67-86, (2017) [3] Andrade, T.; Rodrigues, H.; Bourguignon, M.; Cordeiro, G. M., The exponentiated generalized Gumbel distribution, Rev Colombiana Estadíst, 38, 123-143, (2015) [4] Aryal, G.; Elbatal, I., On the exponentiated generalized modified Weibull distribution, Commun Stat Appl Methods, 22, 333-348, (2015) [5] Ashour, S. K.; Eltehiwy, M. A., Exponentiated power Lindley distribution, J Adv Res, 6, 895-905, (2015) [6] Bourguignon, M.; Silva, R. B.; Cordeiro, G. M., The Weibull-G family of probability distributions, J Data Sci, 12, 53-68, (2014) [7] Chen, G.; Balakrishnan, N., A general purpose approximate goodness-of-fit test, J Qual Technol, 27, 154-161, (1995) [8] Chhikara, R. S.; Folks, J. L., The inverse Gaussian distribution as a lifetime model, Technometrics, 19, 461-468, (1977) · Zbl 0372.62076 [9] Cordeiro, G. M.; Lemonte, A. J., The exponentiated generalized Birnbaum-Saunders distribution, Appl Math Comput, 247, 762-779, (2014) · Zbl 1338.62054 [10] Cordeiro, G. M.; Ortega, E. M M.; Cunha, D. C C., The exponentiated generalized class of distributions, J Data Sci, 11, 1-27, (2013) [11] Corless, R. M.; Gonnet, G. H.; Hare, D. E G.; Jeffrey, D. J.; Knuth, D. E., On the Lambert W function, Adv Comput Math, 5, 329-359, (1996) · Zbl 0863.65008 [12] Ghitany, M. E.; Atieh, B.; Nadarajah, S., Lindley distribution and its application, Math Comput Simulation, 78, 493-506, (2008) · Zbl 1140.62012 [13] Ghitany, M. E.; Al-Mutairi, D. K.; Balakrishnan, N.; Al-Enezi, L. J., Power Lindley distribution and associated inference, Comput Statist Data Anal, 64, 20-33, (2013) · Zbl 06958940 [14] I S Gradshteyn, I M Ryzhik. Table of Integrals, Series, and Products, 6th ed., Corrected by A Jeffrey, D Zwillinger, Academic Press, 2000, San Diego. · Zbl 0981.65001 [15] B Hasselman. Solve systems of nonlinear equations, 2018, R package version 3.3.2. https://cran.r-project.org/package=nleqslv. [16] Jodrá, P., Computer generation of random variables with Lindley or Poisson-Lindley distribution via the Lambert W function, Math Comput Simulation, 81, 851-859, (2010) · Zbl 1207.65012 [17] E L Lehmann, G Casella. Theory of Point Estimation, 2nd ed., Springer-Verlag, 1998, New York. · Zbl 0916.62017 [18] Lindley, D. V., Fiducial distributions and Bayes’ theorem, J R Stat Soc Ser B Stat Methodol, 20, 102-107, (1958) · Zbl 0085.35503 [19] Nadarajah, S.; Bakouch, H. S.; Tahmasbi, R., A generalized Lindley distribution, Sankhyā B, 73, 331-359, (2011) · Zbl 1268.62018 [20] Oluyede, B. O.; Mutiso, F.; Huang, S., The log generalized Lindley-Weibull distribution with application, J Data Sci, 13, 281-310, (2015) [21] P E Oguntunde, O A Odetunmibi, A O Adejumo. On the exponentiated generalized Weibull distribution: a generalization of the Weibull distribution, Indian J Sci Technol, 2015, 8(35), DOI: 10.17485/ijst/2015/v8i35/67611. [22] Pinheiro, J.; Bates, D.; DebRoy, S.; Sarkar, D.; authors, EISPACK; Heisterkamp, S.; Willigen, B. V., R Core Team, (2018) [23] R Core Team. R: A language and environment for statistical computing, 2018, R Foundation for Statistical Computing, Vienna, Austria, URL: http://www.R-project.org/. [24] Silva, A. O.; Silva, L. C M.; Cordeiro, G. M., The extended Dagum distribution: Properties and application, J Data Sci, 13, 53-72, (2015) [25] W H Von Alven. (Ed.) Reliability Engineering, ARINC Research Corporation, 1964, Prentice-Hall. [26] G Warahena-Liyanage, M Pararai. A generalized power Lindley distribution with applications, Asian J Math Appl, 2014, vol. 2014, Article ID ama0169, 23 pages. · Zbl 1307.62028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.