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Lifetime Data Analysis

, Volume 16, Issue 3, pp 409–430 | Cite as

The beta modified Weibull distribution

  • Giovana O. Silva
  • Edwin M. M. OrtegaEmail author
  • Gauss M. Cordeiro
Article

Abstract

A five-parameter distribution so-called the beta modified Weibull distribution is defined and studied. The new distribution contains, as special submodels, several important distributions discussed in the literature, such as the generalized modified Weibull, beta Weibull, exponentiated Weibull, beta exponential, modified Weibull and Weibull distributions, among others. The new distribution can be used effectively in the analysis of survival data since it accommodates monotone, unimodal and bathtub-shaped hazard functions. We derive the moments and examine the order statistics and their moments. We propose the method of maximum likelihood for estimating the model parameters and obtain the observed information matrix. A real data set is used to illustrate the importance and flexibility of the new distribution.

Keywords

Beta distribution Exponentiated exponential Exponentiated Weibull Generalized modified Weibull Maximum likelihood Modified Weibull Observed information matrix Weibull distribution 

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References

  1. Aarset MV (1987) How to identify bathtub hazard rate. IEEE Trans Reliab 36: 106–108zbMATHCrossRefGoogle Scholar
  2. Barakat HM, Abdelkader YH (2004) Computing the moments of order statistics from nonidentical random variables. Stat Methods Appl 13: 15–26zbMATHCrossRefMathSciNetGoogle Scholar
  3. Bebbington M, Lai CD, Zitikis R (2007) A flexible Weibull extension. Reliab Eng Syst Saf 92: 719–726CrossRefGoogle Scholar
  4. Brown BW, Floyd MS, Levy LB (2002) The log F: a distribution for all seasons. Comput Stat 17: 47–58zbMATHCrossRefGoogle Scholar
  5. Carrasco JMF, Ortega EMM, Cordeiro GM (2008) A generalized modified Weibull distribution for lifetime modeling. Comput Stat Data Anal 53: 450–462zbMATHCrossRefGoogle Scholar
  6. Cox C (2008) The generalized F distribution: an umbrella for parametric survival analysis. Stat Med 27: 4301–4312CrossRefMathSciNetGoogle Scholar
  7. Cox C, Chu H, Schneider MF, Mũoz A (2007) Tutorial in biostatistics: parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution. Stat Med 26: 4352–4374CrossRefMathSciNetGoogle Scholar
  8. Doornik J (2007) Ox 5: object-oriented matrix programming language, 5th ed. Timberlake Consultants, LondonGoogle Scholar
  9. Eugene N, Lee C, Famoye F (2002) Beta-normal distribution and its applications. Commun Stat Theory Methods 31: 497–512zbMATHCrossRefMathSciNetGoogle Scholar
  10. Gradshteyn IS, Ryzhik IM (2000) Table of integrals, series, and products. Academic Press, New YorkzbMATHGoogle Scholar
  11. Gupta RD, Kundu D (1999) Generalized exponential distributions. Aust NZ J Stat 41: 173–188zbMATHCrossRefMathSciNetGoogle Scholar
  12. Gupta RD, Kundu D (2001) Exponentiated exponential distribution: an alternative to gamma and Weibull distributions. Biomet J 43: 117–130zbMATHCrossRefMathSciNetGoogle Scholar
  13. Gupta AK, Nadarajah S (2004) Handbook of beta distribution and its applications. Marcel Dekker, New YorkzbMATHGoogle Scholar
  14. Haupt E, Schabe H (1992) A new model for a lifetime distribution with bathtub shaped failure rate. Microelectron Reliab 32: 633–639CrossRefGoogle Scholar
  15. Hosking JRM (1986) The theory of probability weighted moments. Research Report RC12210, IBM Thomas J. Watson Research Center, New York.Google Scholar
  16. Hosking JRM (1990) L-moments: analysis and estimation of distributions using linear combinations of order statistics. J R Stat Soc Ser B 52: 105–124zbMATHMathSciNetGoogle Scholar
  17. Jones MC (2004) Family of distributions arising from distribution of order statistics. Test 13: 1–43zbMATHCrossRefMathSciNetGoogle Scholar
  18. Kundu D, Rakab MZ (2005) Generalized Rayleigh distribution: different methods of estimation. Comput Stat Data Anal 49: 187–200zbMATHCrossRefGoogle Scholar
  19. Lai CD, Xie M, Murthy DNP (2003) A modified Weibull distribution. Trans Reliab 52: 33–37CrossRefGoogle Scholar
  20. Lee C, Famoye F, Olumolade O (2007) Beta-Weibull distribution: some properties and applications to censored data. J Mod Appl Stat Methods 6: 173–186Google Scholar
  21. Mudholkar GS, Srivastava DK (1993) Exponentiated Weibull family for analyzing bathtub failure-real data. IEEE Trans Reliab 42: 299–302zbMATHCrossRefGoogle Scholar
  22. Mudholkar GS, Srivastava DK, Friemer M (1995) The exponentiated Weibull family: a reanalysis of the bus-motor-failure data. Technometrics 37: 436–445zbMATHCrossRefGoogle Scholar
  23. Mudholkar GS, Srivastava DK, Kollia GD (1996) A generalization of the Weibull distribution with application to the analysis of survival data. J Am Stat Assoc 91: 1575–1583zbMATHCrossRefMathSciNetGoogle Scholar
  24. Nadarajah S, Gupta AK (2004) The beta Fréchet distribution. Far East J Theor Stat 14: 15–24zbMATHMathSciNetGoogle Scholar
  25. Nadarajah S, Kotz S (2004) The beta Gumbel distribution. Math Prob Eng 10: 323–332CrossRefMathSciNetGoogle Scholar
  26. Nadarajah S, Kotz S (2006) The beta exponential distribution. Reliab Eng Syst Saf 91: 689–697CrossRefMathSciNetGoogle Scholar
  27. Nelson W (1990) Accelerated life testing: statistical models, data analysis and test plans. Wiley, New YorkGoogle Scholar
  28. Pham H, Lai CD (2007) On recent generalizations of the Weibull distribution. IEEE Trans Reliab 56: 454–458CrossRefGoogle Scholar
  29. Rajarshi S, Rajarshi MB (1988) Bathtub distributions: a review. Commun Stat Theory Methods 17: 2521–2597MathSciNetGoogle Scholar
  30. Wang FK (2000) A new model with bathtub-shaped failure rate using an additive Burr XII distribution. Reliab Eng Syst Saf 70: 305–312CrossRefGoogle Scholar
  31. Xie M, Lai CD (1995) Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function. Reliab Eng Syst Saf 52: 87–93CrossRefGoogle Scholar
  32. Xie M, Tang Y, Goh TN (2002) A modified Weibull extension with bathtub failure rate function. Reliab Eng Syst Saf 76: 279–285CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Giovana O. Silva
    • 1
  • Edwin M. M. Ortega
    • 1
    Email author
  • Gauss M. Cordeiro
    • 2
  1. 1.ESALQUniversidade de São PauloPiracicabaBrazil
  2. 2.DEINFOUniversidade Federal Rural de PernambucoRecifeBrazil

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