A new modified exponential-geometric distribution: properties and applications

Abstract

This paper provides a new three-parameter lifetime distribution with increasing and decreasing hazard function. The various statistical properties of the proposed distribution are also discussed. The maximum likelihood method is used for estimating the unknown parameters, and its performance is assessed using Monte-Carlo simulation. Finally, three real data sets are applied to illustrate the application of the proposed distribution.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

References

  1. 1.

    Aarest, M.V.: How to identify a bathtub hazard rate. IEEE Trans. Reliab. 36(1), 106–108 (1987)

    Article  Google Scholar 

  2. 2.

    Adamidis, K., Dimitrakopoulou, T., Loukas, S.: On an extension of the exponential-geometric distribution. Stat. Probability Latters 73(3), 259–269 (2005)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Adamidis, K., Loukas, S.: A lifetime distribution with decreasing failure rate. Stat. Probability Latters 39, 35–42 (1998)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Ahmad, Z., Elgarhy, M., Abbas, N.: A new extended alpha power transformed family of distributions: properties and applications. J. Stat. Model.: Theory Appl. 1(2), 13–28 (2018)

    Google Scholar 

  5. 5.

    Baily, W.N.: Generalized Hypergeometric Series. University Press, Cambridge (1935)

    Google Scholar 

  6. 6.

    Bjerkedal, T.: Acquisition of resistance in guinea pies infected with different doses of virulent tubercle bacilli. Am. J. Hyg. 72(1), 130–148 (1960)

    Google Scholar 

  7. 7.

    Bordbar, F., Nematollahi, A.R.: The modified exponential-geometric distribution. Commun. Stat.-Theory Methods 45(1), 173–181 (2016)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Brunk, H.D., Barlow, R.E., Bartholomew, D.J., Bremner, J.M.: Stat. Inference under Order Restrict. Wiley, New York (1972)

    Google Scholar 

  9. 9.

    Cordeiro, G.M., Silva, G.O., Ortega, E.M.: An extended-G geometric family. J. Stat. Distrib. Appl. 3(1), 1–16 (2016)

    MATH  Google Scholar 

  10. 10.

    Cox, D.R.: Renew. Theory. Methuen, London (1962)

    Google Scholar 

  11. 11.

    Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood estimation from incomplete data via the EM algorithm (with discussion). J. Royal Stat. Soc.: Series B 39, 1–38 (1977)

    MATH  Google Scholar 

  12. 12.

    Deshpande, JV: and Purohit. Statistical Models and Methods, World Scientific Publishing Company, S. G., Lifetime Data (2006)

  13. 13.

    Dahiya, R.C., Gurland, J.: Goodness of fit tests for the gamma and exponential distributions. Technometrics 14(3), 791–801 (1972)

    Article  Google Scholar 

  14. 14.

    Gleser, L.J.: The gamma distribution as a mixture of exponential distributions. Am. Stat. 43(2), 115–117 (1989)

    MathSciNet  Google Scholar 

  15. 15.

    Gupta, R.D., Kundu, D.: Generalized exponential distribution. Australian New Zealand J. Stat. 41, 173–188 (1999)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Gupta, R.D., Kundu, D.: Exponentiated exponential family: an alternative to gamma and Weibull distributions. Biom. J.: J. Math. Methods Biosci. 43(1), 117–130 (2001)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Gupta, R.D., Kundu, D.: Generalized exponential distribution: different method of estimations. J. Stat. Comput. Simul. 69(4), 315–337 (2001)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Gupta, R.D., Kundu, D.: Generalized exponential distribution: Existing results and some recent developments. J. Stat. Plan. Inference 137(11), 3537–3547 (2007)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Jodrá, P.: On a connection between the polylogarithm function and the Bass diffusion model. Proc. Royal Soc. A: Math., Phys. Eng. Sci. 464(2099), 3081–3088 (2008)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Kus, C.: A new lifetime distribution. Comput. Stat. Data Anal. 51(9), 4497–4509 (2007)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Lawless, J.F.: Statistical models and methods for lifetime data. Wiley, New York (2011)

    Google Scholar 

  22. 22.

    Lewin, L.: Polylogarithms and associated functions. Elsevier, New York (1981)

    Google Scholar 

  23. 23.

    Louzada, F., Marchi, V., Carpenter, J.: The complementary exponentiated exponential geometric lifetime distribution. J. Probability Stat. 2013, 1–12 (2013)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Louzada, F., Ramos, P.L., Perdon, G.S.: Different estimation procedures for the parameters of the extended exponential geometric distribution for medical data. Comput. Math. Methods Med. 2016, 1–12 (2016)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Maguire, B.A., Pearson, E.S., Wynn, A.H.A.: The time intervals between industrial accidents. Biometrika 39, 168–180 (1952)

    Article  Google Scholar 

  26. 26.

    Marshall, A.W., Olkin, I.: A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84(3), 641–652 (1997)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Mudholkar, G.S., Srivastava, D.K.: Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Trans. Reliab. 42(2), 299–302 (1993)

    Article  Google Scholar 

  28. 28.

    Nadarajah, S., Kotz, S.: The exponentiated type distributions. Acta Appl. Math. 9(2), 97–111 (2006)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Preda, V., Panaitescu, E., Ciumara, R.: The modified exponential-Poisson distribution. Proc. Rom. Academy 12(1), 22–29 (2011)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Proschan, F.: Theoretical explanation of observed decreasing failure rate. Technometrics 5(3), 375–383 (1963)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and series. More special functions, vol. 3. Gordon and Breach, New York (1990)

    Google Scholar 

  32. 32.

    Rasekhi, M., Alizadeh, M., Altun, E., Hamedani, G.G., Afify, A.Z., Ahmad, M.: The modified exponential distribution with applications. Pak. J. Stat. 33(5), 383–398 (2017)

    MathSciNet  Google Scholar 

  33. 33.

    Rényi, A., On measures of entropy and information, In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, The Regents of the University of California (1961)

  34. 34.

    Shannon, C.E.: Prediction and entropy of printed English. Bell Labs Tech. J. 30(1), 50–64 (1951)

    Article  Google Scholar 

Download references

Acknowledgements

The authors are thankful the associate editor and two anonymous referees for their useful comments, which led to the improved version of this manuscript.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Karim Zare.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zanboori, A., Zare, K. & Khodadadi, Z. A new modified exponential-geometric distribution: properties and applications. Math Sci (2021). https://doi.org/10.1007/s40096-021-00391-8

Download citation

Keywords

  • Exponential distribution
  • Hazard function
  • Lifetime
  • Monte-Carlo simulation
  • New modified exponential-geometric distribution

Mathematics Subject Classification

  • 60E05
  • 62F12