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AStA Advances in Statistical Analysis

, Volume 95, Issue 3, pp 219–251 | Cite as

The exponentiated exponential distribution: a survey

  • Saralees NadarajahEmail author
Book Review

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.

Keywords

Estimation Exponentiated exponential distribution Moments 

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References

  1. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  2. Asgharzadeh, A.: Approximate MLE for the scaled generalized exponential distribution under progressive type-II censoring. J. Korean Stat. Soc. 38, 223–229 (2009) MathSciNetCrossRefGoogle Scholar
  3. Aslam, M., Kundu, D., Ahmad, M.: Time truncated acceptance sampling plans for generalized exponential distribution. J. Appl. Stat. 37, 555–566 (2010) CrossRefGoogle Scholar
  4. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 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) CrossRefGoogle Scholar
  6. 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) CrossRefGoogle Scholar
  7. Cleveland, W.S.: Robust locally weighted regression and smoothing scatterplots. J. Am. Stat. Assoc. 74, 829–836 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  8. Cleveland, W.S.: LOWESS: a program for smoothing scatterplots by robust locally weighted regression. Am. Stat. 35, 54 (1981) CrossRefGoogle Scholar
  9. 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
  10. 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
  11. Ellah, A.H.A.: Parametric prediction limits for generalized exponential distribution using record observations. Appl. Math. Inf. Sci. 3, 135–149 (2009) MathSciNetzbMATHGoogle Scholar
  12. 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
  13. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  14. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 6th edn. Academic Press, San Diego (2000) zbMATHGoogle Scholar
  15. Gupta, R.D., Kundu, D.: Generalized exponential distributions. Aust. N. Z. J. Stat. 41, 173–188 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  16. Gupta, R.D., Kundu, D.: Exponentiated exponential family: an alternative to gamma and Weibull distributions. Biom. J. 43, 117–130 (2001a) MathSciNetzbMATHCrossRefGoogle Scholar
  17. Gupta, R.D., Kundu, D.: Generalized exponential distributions: different methods of estimation. J. Stat. Comput. Simul. 69, 315–338 (2001b) MathSciNetzbMATHCrossRefGoogle Scholar
  18. Gupta, R.D., Kundu, D.: Discriminating between the Weibull and the GE distributions. Comput. Stat. Data Anal. 43, 179–196 (2003a) MathSciNetzbMATHGoogle Scholar
  19. Gupta, R.D., Kundu, D.: Closeness of gamma and generalized exponential distribution. Commun. Stat., Theory Methods 32, 705–721 (2003b) MathSciNetzbMATHCrossRefGoogle Scholar
  20. Gupta, R.D., Kundu, D.: Discriminating between gamma and generalized exponential distributions. J. Stat. Comput. Simul. 74, 107–121 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  22. Gupta, R.D., Kundu, D.: Generalized exponential distribution: existing results and some recent developments. J. Stat. Plan. Inference 137, 3537–3547 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 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) MathSciNetzbMATHGoogle Scholar
  24. 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) zbMATHGoogle Scholar
  25. Kannan, N., Kundu, D., Nair, P., Tripathi, R.C.: The generalized exponential cure rate model with covariates. J. Appl. Stat. 37, 1625–1636 (2010) CrossRefGoogle Scholar
  26. 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) CrossRefGoogle Scholar
  27. Kim, C., Song, S.: Bayesian estimation of the parameters of the generalized exponential distribution from doubly censored samples. Stat. Pap. 51, 583–597 (2010) MathSciNetCrossRefGoogle Scholar
  28. Kundu, D., Gupta, R.D.: Bivariate generalized exponential distribution. J. Multivar. Anal. 100, 581–593 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  29. Kundu, D., Pradhan, B.: Bayesian inference and life testing plans for generalized exponential distribution. Sci. China Ser. A, Math. 52, 1373–1388 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  30. Kundu, D., Gupta, R.D., Manglick, A.: Discriminating between the log-normal and generalized exponential distribution. J. Stat. Plan. Inference 127, 213–227 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  31. Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, New York (1987) Google Scholar
  32. Madi, M.T., Raqab, M.Z.: Bayesian prediction of rainfall records using the generalized exponential distribution. Environmetrics 18, 541–549 (2007) MathSciNetCrossRefGoogle Scholar
  33. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  34. Meintanis, S.G.: Tests for generalized exponential laws based on the empirical Mellin transform. J. Stat. Comput. Simul. 78, 1077–1085 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  35. Nadarajah, S., Kotz, S.: The exponentiated type distributions. Acta Appl. Math. 92, 97–111 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 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
  37. 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) MathSciNetCrossRefGoogle Scholar
  38. Raqab, M.Z.: Inferences for generalized exponential distribution based on record statistics. J. Stat. Plan. Inference 104, 339–350 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  39. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  40. 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) MathSciNetCrossRefGoogle Scholar
  41. 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
  42. Ross, S.M.: Probability Models, 8th edn. Academic Press, Amsterdam (2003) zbMATHGoogle Scholar
  43. Sarhan, A.M.: Analysis of incomplete, censored data in competing risks models with generalized exponential distributions. IEEE Trans. Reliab. 56, 132–138 (2007) CrossRefGoogle Scholar
  44. Shannon, C.E.: Prediction and entropy of printed English. Bell Syst. Tech. J. 30, 50–64 (1951) zbMATHGoogle Scholar
  45. Shirke, D.T., Kumbhar, R.R., Kundu, D.: Tolerance intervals for exponentiated scale family of distributions. J. Appl. Stat. 32, 1067–1074 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  46. Song, K.-S.: Rényi information, loglikelihood and an intrinsic distribution measure. J. Stat. Plan. Inference 93, 51–69 (2001) zbMATHCrossRefGoogle Scholar
  47. Srivastava, H.M., Nadarajah, S., Kotz, S.: Some generalizations of the Laplace distribution. Appl. Math. Comput. 182, 223–231 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  48. 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
  49. 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) CrossRefGoogle Scholar
  50. Zheng, G.: On the fisher information matrix in type II censored data from the exponentiated exponential family. Biom. J. 44, 353–357 (2002) MathSciNetCrossRefGoogle Scholar
  51. Zheng, G., Park, S.: A note on time savings in censored life testing. J. Stat. Plan. Inference 124, 289–300 (2004) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.School of MathematicsUniversity of ManchesterManchesterUK

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