Skip to main content
SpringerLink
Log in

The exponentiated exponential distribution: a survey

  • Book Review
  • Published:
AStA Advances in Statistical Analysis Aims and scope Submit manuscript

Abstract

The exponentiated exponential distribution, a most attractive generalization of the exponential distribution, introduced by Gupta and Kundu (Aust. N. Z. J. Stat. 41:173–188, 1999) has received widespread attention. It appears, however, that many mathematical properties of this distribution have not been known or have not been known in simpler/general forms. In this paper, we provide a comprehensive survey of the mathematical properties. We derive expressions for the moment generating function, characteristic function, cumulant generating function, the nth moment, the first four moments, variance, skewness, kurtosis, the nth conditional moment, the first four cumulants, mean deviation about the mean, mean deviation about the median, Bonferroni curve, Lorenz curve, Bonferroni concentration index, Gini concentration index, Rényi entropy, Shannon entropy, cumulative residual entropy, Song’s measure, moments of order statistics, L moments, asymptotic distribution of the extreme order statistics, reliability, distribution of the sum of exponentiated exponential random variables, distribution of the product of exponentiated exponential random variables and the distribution of the ratio of exponentiated exponential random variables. We also discuss estimation by the method of maximum likelihood, including the case of censoring, and provide simpler expressions for the Fisher information matrix than those given by Gupta and Kundu. It is expected that this paper could serve as a source of reference for the exponentiated exponential distribution and encourage further research.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abdel-Hamid, A.H., Al-Hussaini, E.K.: Estimation in step-stress accelerated life tests for the exponentiated exponential distribution with type-I censoring. Comput. Stat. Data Anal. 53, 1328–1338 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Asgharzadeh, A.: Approximate MLE for the scaled generalized exponential distribution under progressive type-II censoring. J. Korean Stat. Soc. 38, 223–229 (2009)

    Article  MathSciNet  Google Scholar 

  • Aslam, M., Kundu, D., Ahmad, M.: Time truncated acceptance sampling plans for generalized exponential distribution. J. Appl. Stat. 37, 555–566 (2010)

    Article  Google Scholar 

  • Baklizi, A.: Likelihood and Bayesian estimation of Pr (X<Y) using lower record values from the generalized exponential distribution. Comput. Stat. Data Anal. 52, 3468–3473 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Biondi, F., Kozubowski, T.J., Panorska, A.K., Saito, L.: A new stochastic model of episode peak and duration for eco-hydro-climatic applications. Ecol. Model. 211, 383–395 (2008)

    Article  Google Scholar 

  • Chen, D.G., Lio, Y.L.: Parameter estimations for generalized exponential distribution under progressive type-I interval censoring. Comput. Stat. Data Anal. 54, 1581–1591 (2010)

    Article  Google Scholar 

  • Cleveland, W.S.: Robust locally weighted regression and smoothing scatterplots. J. Am. Stat. Assoc. 74, 829–836 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  • Cleveland, W.S.: LOWESS: a program for smoothing scatterplots by robust locally weighted regression. Am. Stat. 35, 54 (1981)

    Article  Google Scholar 

  • Cota-Felix, J.E., Rivas-Davalos, F., Maximov, S.: An alternative method for estimating mean life of power system equipment with limited end-of-life failure data. In: Toma, L., Otomega, B. (eds.) IEEE Bucharest Powertech, vols. 1–5, pp. 2342–2345 (2009)

    Google Scholar 

  • Dagum, C.: Lorenz curve. In: Kotz, S., Johnson, N.L., Read, C.B. (eds.) Encyclopedia of Statistical Sciences, vol. 5, pp. 156–161. Wiley, New York (1985)

    Google Scholar 

  • Ellah, A.H.A.: Parametric prediction limits for generalized exponential distribution using record observations. Appl. Math. Inf. Sci. 3, 135–149 (2009)

    MathSciNet  MATH  Google Scholar 

  • Escalante-Sandoval, C.: Design rainfall estimation using exponentiated and mixed exponentiated distributions in the Coast of Chiapas. Ing. Hidrául. Méx. 22, 103–113 (2007)

    Google Scholar 

  • Gail, M.H., Gastwirth, J.L.: A scale-free goodness-of-fit test for the exponential distribution based on the Lorenz curve. J. Am. Stat. Assoc. 73, 787–793 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  • Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 6th edn. Academic Press, San Diego (2000)

    MATH  Google Scholar 

  • Gupta, R.D., Kundu, D.: Generalized exponential distributions. Aust. N. Z. J. Stat. 41, 173–188 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Gupta, R.D., Kundu, D.: Generalized exponential distributions: different methods of estimation. J. Stat. Comput. Simul. 69, 315–338 (2001b)

    Article  MathSciNet  MATH  Google Scholar 

  • Gupta, R.D., Kundu, D.: Discriminating between the Weibull and the GE distributions. Comput. Stat. Data Anal. 43, 179–196 (2003a)

    MathSciNet  MATH  Google Scholar 

  • Gupta, R.D., Kundu, D.: Closeness of gamma and generalized exponential distribution. Commun. Stat., Theory Methods 32, 705–721 (2003b)

    Article  MathSciNet  MATH  Google Scholar 

  • Gupta, R.D., Kundu, D.: Discriminating between gamma and generalized exponential distributions. J. Stat. Comput. Simul. 74, 107–121 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Gupta, R.D., Kundu, D.: On the comparison of Fisher information of the Weibull and GE distributions. J. Stat. Plan. Inference 136, 3130–3144 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Hosking, J.R.M.: L-moments: analysis and estimation of distributions using linear combinations of order statistics. J. R. Stat. Soc. B 52, 105–124 (1990)

    MathSciNet  MATH  Google Scholar 

  • Kakade, C.S., Shirke, D.T.: Tolerance interval for exponentiated exponential distribution based on grouped data. Int. J. Agric. Stat. Sci. 3, 625–631 (2007)

    MATH  Google Scholar 

  • Kannan, N., Kundu, D., Nair, P., Tripathi, R.C.: The generalized exponential cure rate model with covariates. J. Appl. Stat. 37, 1625–1636 (2010)

    Article  Google Scholar 

  • Khuong, H.V., Kong, H.-Y.: General expression for pdf of a sum of independent exponential random variables. IEEE Commun. Lett. 10, 159–161 (2006)

    Article  Google Scholar 

  • Kim, C., Song, S.: Bayesian estimation of the parameters of the generalized exponential distribution from doubly censored samples. Stat. Pap. 51, 583–597 (2010)

    Article  MathSciNet  Google Scholar 

  • Kundu, D., Gupta, R.D.: Bivariate generalized exponential distribution. J. Multivar. Anal. 100, 581–593 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Kundu, D., Pradhan, B.: Bayesian inference and life testing plans for generalized exponential distribution. Sci. China Ser. A, Math. 52, 1373–1388 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Kundu, D., Gupta, R.D., Manglick, A.: Discriminating between the log-normal and generalized exponential distribution. J. Stat. Plan. Inference 127, 213–227 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, New York (1987)

    Google Scholar 

  • Madi, M.T., Raqab, M.Z.: Bayesian prediction of rainfall records using the generalized exponential distribution. Environmetrics 18, 541–549 (2007)

    Article  MathSciNet  Google Scholar 

  • Madi, M.T., Raqab, M.Z.: Bayesian inference for the generalized exponential distribution based on progressively censored data. Commun. Stat., Theory Methods 38, 2016–2029 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Meintanis, S.G.: Tests for generalized exponential laws based on the empirical Mellin transform. J. Stat. Comput. Simul. 78, 1077–1085 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Nadarajah, S., Kotz, S.: The exponentiated type distributions. Acta Appl. Math. 92, 97–111 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, vols. 1, 2 and 3. Gordon and Breach Science Publishers, Amsterdam (1986)

    Google Scholar 

  • Rao, M., Chen, Y., Vemuri, B.C., Wang, F.: Cumulative residual entropy: a new measure of information. IEEE Trans. Inf. Theory 50, 1220–1228 (2004)

    Article  MathSciNet  Google Scholar 

  • Raqab, M.Z.: Inferences for generalized exponential distribution based on record statistics. J. Stat. Plan. Inference 104, 339–350 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Raqab, M.Z., Ahsanullah, M.: Estimation of the location and scale parameters of generalized exponential distribution based on order statistics. J. Stat. Comput. Simul. 69, 109–124 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Raqab, M.Z., Madi, M.T., Kundu, D.: Estimation of P(Y<X) for the three-parameter generalized exponential distribution. Commun. Stat., Theory Methods 37, 2854–2864 (2008)

    Article  MathSciNet  Google Scholar 

  • Rényi, A.: On measures of entropy and information. In: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. I, pp. 547–561. University of California Press, Berkeley (1961)

    Google Scholar 

  • Ross, S.M.: Probability Models, 8th edn. Academic Press, Amsterdam (2003)

    MATH  Google Scholar 

  • Sarhan, A.M.: Analysis of incomplete, censored data in competing risks models with generalized exponential distributions. IEEE Trans. Reliab. 56, 132–138 (2007)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  • Shirke, D.T., Kumbhar, R.R., Kundu, D.: Tolerance intervals for exponentiated scale family of distributions. J. Appl. Stat. 32, 1067–1074 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Song, K.-S.: Rényi information, loglikelihood and an intrinsic distribution measure. J. Stat. Plan. Inference 93, 51–69 (2001)

    Article  MATH  Google Scholar 

  • Srivastava, H.M., Nadarajah, S., Kotz, S.: Some generalizations of the Laplace distribution. Appl. Math. Comput. 182, 223–231 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Subburaj, R., Gopal, G., Kapur, P.K.: A software reliability growth model for vital quality metrics. S. Afr. J. Ind. Eng. 18, 93–108 (2007)

    Google Scholar 

  • Yeates, M.P., Tolkamp, B.J., Allcroft, D.J., Kyriazakis, I.: The use of mixed distribution models to determine bout criteria for analysis of animal behaviour. J. Theor. Biol. 213, 413–425 (2001)

    Article  Google Scholar 

  • Zheng, G.: On the fisher information matrix in type II censored data from the exponentiated exponential family. Biom. J. 44, 353–357 (2002)

    Article  MathSciNet  Google Scholar 

  • Zheng, G., Park, S.: A note on time savings in censored life testing. J. Stat. Plan. Inference 124, 289–300 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, University of Manchester, Manchester, M13 9PL, UK

    Saralees Nadarajah

Authors
  1. Saralees Nadarajah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saralees Nadarajah.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nadarajah, S. The exponentiated exponential distribution: a survey. AStA Adv Stat Anal 95, 219–251 (2011). https://doi.org/10.1007/s10182-011-0154-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10182-011-0154-5

Keywords

Search

Navigation