, Volume 72, Issue 1, pp 65–76 | Cite as

Generalized exponential records: existence of maximum likelihood estimates and its comparison with transforming based estimates

  • Mohammad Z. RaqabEmail author
  • Khalaf S. Sultan


In this paper, and based on records of a sequence of iid random variables from the generalized exponential distribution, we consider the problem of the existence of the maximum likelihood estimates of the shape and scale parameters. Existence and uniqueness of the MLE’s are proved. Different transforming based estimates and confidence intervals of these parameters are then derived. The performances of the so obtained estimates and confidence intervals are compared through an extensive numerical simulation study. Analysis of a real data set has also been presented for illustrative purposes.


Record statistics Generalized exponential distribution  Maximum likelihood estimate Pivotal quantity  Confidence interval Uniform distribution Exponential distribution 

Mathematics Subject Classification

62G30 62E15 62F10 



The authors are grateful to the associate editor and referees for their helpful comments and suggestions.


  1. 1.
    Aarset, M.V.: How to identify a bathtub hazard rate. IEEE Trans. Reliab. 36, 106–108 (1987)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ahmadi, J.: Record values, theory and applications, Ph. D. Dissertation, Ferdowsi University of Mashhad, Iran (2000)Google Scholar
  3. 3.
    Ahsanullah, M.: Record Values-Theory and Applications. University Press of America Inc., New York (2004)Google Scholar
  4. 4.
    Al-Hussaini, E., Ahmad, A.: On Bayesian interval prediction of future records. Test 12(1), 79–99 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Arnold, B.C., Balakrishnan, N., Nagaraja, H.N.: Records. Wiley, New York (1998)CrossRefzbMATHGoogle Scholar
  6. 6.
    Carlin, P.B., Gelfand, A.E.: Parametric likelihood inference for record breaking problems. Biometrika 80(3), 507–515 (1993)CrossRefzbMATHGoogle Scholar
  7. 7.
    Ghitani, M.E., Al-Jarallah, R.A., Balakrishnan, N.: On the existence and uniqueness of the MLEs of the parameters of a general class of exponentiated distributions. Statistics 47(3), 605–612 (2013)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Gulati, S., Padgett, W.J.: Smooth nonparametric estimation of the distribution and density functions from record-breaking data. Commun. Stat. Theor. Meth. 23, 1256–1274 (1994)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Gulati, S., Padgett, W.J.: Nonparametric function estimation from inversely sampled record breaking data. Can. J. Stat. 23, 359–368 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Gupta, R.D., Kundu, D.: Generalized exponential distribution. Aust. N. Z. Stat. 41(2), 173–188 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Gupta, R.D., Kundu, D.: Exponentiated exponential distribution, an aternative to gamma and Weibull distributions. Biometr. J. 43(1), 117–130 (2001a)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Gupta, R.D., Kundu, D.: Exponentiated exponential distributions, different methods of estimations. J. Stat. Comput. Simul. 69(4), 315–338 (2001b)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Madi, M.T., Raqab, M.Z.: Bayesian prediction of rainfall records using the generalized exponential distribution. Environmetrics 18, 541–549 (2007)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Raqab, M.Z.: Inferences for generalized exponential distribution based on record statistics. J. Stat. Plann. Inference 104(2), 339–350 (2002)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Sapienza Università di Roma 2013

Authors and Affiliations

  1. 1.Department of Statistics and Operations ResearchKuwait UniversitySafatKuwait
  2. 2.Department of Statistics and Operation ResearchKing Saud UniversityRiyadhKingdom of Saudi Arabia

Personalised recommendations