Home Mathematics The Lindley negative-binomial distribution: Properties, estimation and applications to lifetime data
Article
Licensed
Unlicensed Requires Authentication

The Lindley negative-binomial distribution: Properties, estimation and applications to lifetime data

  • Muhammad Mansoor EMAIL logo , Muhammad Hussain Tahir , Gauss M. Cordeiro , Sajid Ali and Ayman Alzaatreh
Published/Copyright: July 24, 2020
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 70 Issue 4

Abstract

A generalization of the Lindley distribution namely, Lindley negative-binomial distribution, is introduced. The Lindley and the exponentiated Lindley distributions are considered as sub-models of the proposed distribution. The proposed model has flexible density and hazard rate functions. The density function can be decreasing, right-skewed, left-skewed and approximately symmetric. The hazard rate function possesses various shapes including increasing, decreasing and bathtub. Furthermore, the survival and hazard rate functions have closed form representations which make this model tractable for censored data analysis. Some general properties of the proposed model are studied such as ordinary and incomplete moments, moment generating function, mean deviations, Lorenz and Bonferroni curve. The maximum likelihood and the Bayesian estimation methods are utilized to estimate the model parameters. In addition, a small simulation study is conducted in order to evaluate the performance of the estimation methods. Two real data sets are used to illustrate the applicability of the proposed model.

Keywords: Bathtub failure rate; Bayesian estimation; Lindley distribution; maximum likelihood estimation; Markov Chain Monte Carlo simulation; negative-binomial distribution
MSC 2010: 60E05; 62E15; 62E20

  1. (Communicated by Gejza Wimmer)

Acknowledgement

We would like to express our sincere thanks to the referees, an Associate Editor and the Editor-in-Chief for valuable comments and suggestions that significantly improved the presentation of the results. The financial support of the CNPq (Brazil) is gratefully acknowledged by the the third author.

References

[1] Aarset, M. V.: How to identify bathtub hazard rate, IEEE Transactional on Reliability 36 (1987), 106–108.10.1109/TR.1987.5222310Search in Google Scholar

[2] Ali, S.: On the Bayesian estimation of the weighted Lindley Distribution, J. Stat. Comput. Simul. 85 (2015), 955–880.10.1080/00949655.2013.847442Search in Google Scholar

[3] Ashour, S. K.—Eltehiwy, M. A.: Exponentiated power Lindley distribution, J. Adv. Res. 6 (2015), 895–905.10.1016/j.jare.2014.08.005Search in Google Scholar PubMed PubMed Central

[4] 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 (1996), 329–359.10.1007/BF02124750Search in Google Scholar

[5] Ghitany, M. E.—Atieh, B.—Nadarajah, S.: Lindley distribution and its application, Math. Comput. Simulation 78 (2008), 493–506.10.1016/j.matcom.2007.06.007Search in Google Scholar

[6] Ghitany, M. E.—Al-Mutairi, D. K.—Al-Awadhi, F. A.—Al-Burais, M. M.: Marshall-Olkin extended Lindley distribution and its application, Int. J. Appl. Math. Comput. Sci. 25 (2012), 709–721.Search in Google Scholar

[7] Ghitany, M. E.—Al-Mutairi, D. K.—Balakrishnan, N.—Al-Enezi, L. J.: Power Lindley distribution and associated inference, Comput. Statist. Data Anal. 64 (2013), 20–33.10.1016/j.csda.2013.02.026Search in Google Scholar

[8] Ghitany, M. E.—Al-Mutairi, D. K.—Aboukhamseen, S. M.: Estimation of the reliability of a stress-strength system from power Lindley distributions, Comm. Statist. Theory Methods 44 (2015), 118–136.10.1080/03610918.2013.767910Search in Google Scholar

[9] Jodrá, J.: Computer generation of random variables with Lindley or Poisson–Lindley distribution via the Lambert W function, Math. Comput. Simulation 81 (2010), 851–859.10.1016/j.matcom.2010.09.006Search in Google Scholar

[10] Kenney, J.—Keeping, E.: Mathematics of Statistics, Van Nostrand, New Jersy, 1962.Search in Google Scholar

[11] Lawless, J. F.: Statistical Models and Methods for Lifetime Data, Wiley, New York, 2003.10.1002/9781118033005Search in Google Scholar

[12] Lemonte, A. J.: A new exponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtub-shaped failure rate function, Comput. Statist. Data Anal. 62 (2013), 149–170.10.1016/j.csda.2013.01.011Search in Google Scholar

[13] Lindley, D .V.: Fiducial distributions and Bayes’ theorem, J. R. Stat. Soc. Ser. B 20 (1958), 102–107.10.1111/j.2517-6161.1958.tb00278.xSearch in Google Scholar

[14] Mervoci, F.—Sharma, V. K.: The Beta Lindley distribution: Properties and Applications, J. Appl. Math. 2014 (2014), Art. ID 198951.10.1155/2014/198951Search in Google Scholar

[15] Moors, J. J. A.: A quantile alternative for kurtosis, The Statistician 37 (1998), 25–32.10.2307/2348376Search in Google Scholar

[16] Nadarajah, S.—Bakouch, H. S.—Tahmasbi, R.: A generalized Lindley distribution, Sankhya B 73 (2011), 331–359.10.1007/s13571-011-0025-9Search in Google Scholar

[17] Nadarajah, S.—Kotz, S.: The beta exponential distribution, Reliability Engineering and System Safety 91 (2006), 689–697.10.1016/j.ress.2005.05.008Search in Google Scholar

[18] Nichols, M. D.—Padgett, W. J.: A bootstrap control chart for Weibull percentiles, Quality and Reliability Engineering International 22 (2006), 141–151.10.1002/qre.691Search in Google Scholar

[19] Pinho, L. G. B.—Cordeiro, G. M.—Nobre, J. S.: The Harris extended exponential distribution, Comm. Statist. Theory Methods 44 (2015), 3486–3502.10.1080/03610926.2013.851221Search in Google Scholar

[20] Shannon, C.E.: A mathematical theory of communication, Bell Telephone System Technical Publications 27 (1948), 379–432.10.1002/j.1538-7305.1948.tb01338.xSearch in Google Scholar

Received: 2019-01-27
Accepted: 2020-01-07
Published Online: 2020-07-24
Published in Print: 2020-08-26

© 2020 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Regular papers
  2. Some relative normality properties in locales
  3. Upper bounds of some special zeros for the Rankin-Selberg L-function
  4. Factorization of polynomials over valued fields based on graded polynomials
  5. Varieties of ∗-regular rings
  6. On reverse Hölder and Minkowski inequalities
  7. Coefficient inequalities related with typically real functions
  8. Existence of wandering and periodic domain in given angular region
  9. The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛*
  10. Uniqueness problem of meromorphic mappings of a complete Kähler manifold into a projective space
  11. Long time decay of 3D-NSE in Lei-Lin-Gevrey spaces
  12. Bn-maximal operator and Bn-singular integral operators on variable exponent Lebesgue spaces
  13. 𝔻-recurrent ∗-Ricci tensor on three-dimensional real hypersurfaces in nonflat complex space forms
  14. More on closed non-vanishing ideals in CB(X)
  15. The Lindley negative-binomial distribution: Properties, estimation and applications to lifetime data
  16. Multi-opponent James functions
  17. An alternative distribution to Lindley and Power Lindley distributions with characterizations, different estimation methods and data applications
  18. A new one-parameter discrete distribution with associated regression and integer-valued autoregressive models
  19. On the bond pricing partial differential equation in a convergence model of interest rates with stochastic correlation
  20. Characterization of linear mappings on (Banach) ⋆-algebras by similar properties to derivations
https://doi.org/10.1515/ms-2017-0404
Keywords for this article
Bathtub failure rate; Bayesian estimation; Lindley distribution; maximum likelihood estimation; Markov Chain Monte Carlo simulation; negative-binomial distribution

Articles in the same Issue

  1. Regular papers
  2. Some relative normality properties in locales
  3. Upper bounds of some special zeros for the Rankin-Selberg L-function
  4. Factorization of polynomials over valued fields based on graded polynomials
  5. Varieties of ∗-regular rings
  6. On reverse Hölder and Minkowski inequalities
  7. Coefficient inequalities related with typically real functions
  8. Existence of wandering and periodic domain in given angular region
  9. The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛*
  10. Uniqueness problem of meromorphic mappings of a complete Kähler manifold into a projective space
  11. Long time decay of 3D-NSE in Lei-Lin-Gevrey spaces
  12. Bn-maximal operator and Bn-singular integral operators on variable exponent Lebesgue spaces
  13. 𝔻-recurrent ∗-Ricci tensor on three-dimensional real hypersurfaces in nonflat complex space forms
  14. More on closed non-vanishing ideals in CB(X)
  15. The Lindley negative-binomial distribution: Properties, estimation and applications to lifetime data
  16. Multi-opponent James functions
  17. An alternative distribution to Lindley and Power Lindley distributions with characterizations, different estimation methods and data applications
  18. A new one-parameter discrete distribution with associated regression and integer-valued autoregressive models
  19. On the bond pricing partial differential equation in a convergence model of interest rates with stochastic correlation
  20. Characterization of linear mappings on (Banach) ⋆-algebras by similar properties to derivations
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 15.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2017-0404/html
Scroll to top button