Reliability Models Using the Composite Generalizers of Weibull Distribution

  • Gokarna R. AryalEmail author
  • Keshav P. Pokhrel
  • Netra Khanal
  • Chris P. Tsokos


In this article, we study the composite generalizers of Weibull distribution using exponentiated, Kumaraswamy, transmuted and beta distributions. The composite generalizers are constructed using both forward and reverse order of each of these distributions. The usefulness and effectiveness of the composite generalizers and their order of composition is investigated by studying the reliability behavior of the resulting distributions. Two sets of real-world data are analyzed using the proposed generalized Weibull distributions.


Weibull distribution Exponentiated distribution Kumaraswamy distribution Beta distribution Transmutation map Composition map 



The authors are grateful to the editor and anonymous reviewers for their valuable suggestions. This work was completed while GA was in sabbatical leave for which he would like to thank the Purdue University Northwest.


  1. 1.
    Achcar J, Coelho-Barros E, Cordeiro G (2013) Beta generalized distributions and related exponentiated models: a Bayesian approach. Braz J Probab Stat 27(1):1–19Google Scholar
  2. 2.
    Adam AM, Farouk RM, Abd El-aziz ME (2016) Blind image separation based on exponentiated transmuted Weibull distribution. Int J Comput Sci Inf Secur (IJCSIS) 14(3):434–447Google Scholar
  3. 3.
    Akinsete AA, Famoye F, Lee C (2014) The Kumaraswamy geometric distribution. J Stat Distrib Appl 1:1–21Google Scholar
  4. 4.
    Alexander C, Cordeiro GM, Ortega EMM, Sarabia JM (2012) Generalized beta generated distributions. Comput Stat Data Anal 56:1880–1897Google Scholar
  5. 5.
    Andrews D, Herzberg A (1985) Data: a collection of problems from many fields for the student and research worker. Springer series in statistics. Springer, New YorkGoogle Scholar
  6. 6.
    Aryal GR, Tsokos CP (2011) Transmuted Weibull distribution: a generalization of the Weibull probability distribution. Eur J Pure Appl Math 4:89–102Google Scholar
  7. 7.
    Aryal G, Tsokos CP (2009) On the transmuted extreme value distribution with application. Nonlinear Anal Theory Methods Appl 71:1401–1407Google Scholar
  8. 8.
    Bebbington M, Lai CD, Zitikis R (2007) A flexible Weibull extension. Reliab Eng Syst Saf 92:719–726Google Scholar
  9. 9.
    Bourguignon M, Silva RB, Cordeiro GM (2014) The Weibull-G family of probability distributions. J Data Sci 12(1):53–68Google Scholar
  10. 10.
    Castillo E, Hadi A, Balakrishnan N, Sarabia J (2004) Extreme value and related models with applications in engineering and science. Wiley series in probability and statistics. Wiley, HobokenGoogle Scholar
  11. 11.
    Cordeiro GM, Afify AZ, Yousof HM, Pescim RR, Aryal GR (2017) The exponentiated Weibull-H family of distributions: theory and applications. Mediterr J Math 14, 155Google Scholar
  12. 12.
    Cordeiro GM, Ortega EMM, Popovic BV, Pescim RR (2014) The Lomax generator of distributions: properties, minification process and regression model. Appl Math Comput 247:465–486Google Scholar
  13. 13.
    Cordeiro GM, Gomes AE, de Silva CQ, Ortega EMM (2013) The beta exponentiated Weibull distribution. J Stat Comput Simul 83(1):114–138Google Scholar
  14. 14.
    Cordeiro GM, Ortega EM, da Cunha DC (2013) The exponentiated generalized class of distributions. J Data Sci 11:1–27Google Scholar
  15. 15.
    Cordeiro GM, Simas AB, Stosic B (2011) Closed form expressions for moments of the beta Weibull distribution. Anais da Academia Brasileira de Ci\({\hat{e}}\)ncias 83: 357–373Google Scholar
  16. 16.
    Cordeiro GM, de Castro M (2011) A new family of generalized distributions. J Stat Comput Simul 81(7):883–898Google Scholar
  17. 17.
    Cordeiro GM, Ortega EMM, Nadarajah S (2010) The Kumaraswamy Weibull distribution with application to failure data. J Frankl Inst 347:1399–1429Google Scholar
  18. 18.
    de Pascoa MAR, Ortega EM, Cordeiro GM (2011) The Kumaraswamy generalized gamma distribution with application in survival analysis. Stat Methodol 8:411–433Google Scholar
  19. 19.
    Dorey G, Sidey GR, Hutchings J (1978) Impact properties of carbon fibre/Kevlar 49 fibre hybrid composites. Composites 9(1):25–32Google Scholar
  20. 20.
    DuPont Technical Report(2017) Kevlar aramid fiber technical guideGoogle Scholar
  21. 21.
    Ebraheim AE (2014) Exponentiated transmuted Weibull distribution—a generalization of the Weibull distribution. Int J Math Comput Sci 8(6):901–909Google Scholar
  22. 22.
    Eissa FH (2017) The exponentiated Kumaraswamy–Weibull distribution with application to real data. Int J Stat Probab 6(6):167–182Google Scholar
  23. 23.
    Eugene N, Lee C, Famoye F (2002) Beta-normal distribution and its applications. Commun Stat Theory Methods 31:497–512Google Scholar
  24. 24.
    Famoye F, Lee C, Olumolade O (2005) The beta-Weibull distribution. J Stat Theory Appl 4:121–138Google Scholar
  25. 25.
    Gupta RD, Kundu D (1999) Generalized exponential distributions. Aust N Z J Stat 41:173–188Google Scholar
  26. 26.
    Gupta RD, Kundu D (2001) Generalized exponential distribution: an alternative to Gamma and Weibull distributions. Biom J 43:117–130Google Scholar
  27. 27.
    Handique L, Chakraborty S, Hamedani GG (2017) The Marshall–Olkin–Kumaraswamy-G family of distributions. J Stat Theory Appl 16(4):427–447Google Scholar
  28. 28.
    Hanook S, Shahbaz MQ, Mohsin M, Kibria G (2013) A note on beta inverse Weibull distribution. Commun Stat Theory Methods 42(2):320–335Google Scholar
  29. 29.
    Jones MC (2009) Kumaraswamy’s distribution: a beta-type distribution with some tractability advantages. Stat Methodol 6:70–81Google Scholar
  30. 30.
    KalbfleischRoss JD, Prentice L (2011) The statistical analysis of failure time data, 2nd edn. Wiley, Hoboken. Google Scholar
  31. 31.
    Keller AZ, Kamath AR (1982) Reliability analysis of CNC machine tools. Reliab Eng 3:449–473Google Scholar
  32. 32.
    Khan M, King R (2013) Transmuted modified weibull distribution: a generalization of the modified Weibull probability distribution. Eur J Pure Appl Math 6(1):66–88Google Scholar
  33. 33.
    Khan M, King R, Hudson IL (2016) Transmuted new generalized Weibull distribution for lifetime modeling. Commun Stat Appl Methods 23:363–383Google Scholar
  34. 34.
    Khan MS, King R, Hudson IL (2017) Transmuted Kumaraswamy-G family of distributions for modelling reliability data. J Test Eval 45(5):1837–1848Google Scholar
  35. 35.
    Lai CD, Xie M, Murthy DNP (2003) A modified Weibull distribution. IEEE Trans Reliab 52:33–37Google Scholar
  36. 36.
    Makubate B, Oluyede B, Motobetso G, Huang S, Fagbamigbe A (2018) The beta Weibull-G family of distributions: model, properties and application. Int J Stat Probab 7(2):12–32Google Scholar
  37. 37.
    Marshall A, Olkin I (1997) A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84:641–652Google Scholar
  38. 38.
    Mudholkar GS, Srivastava DK (1993) Exponentiated Weibull family for analyzing bathtub failure-real data. IEEE Trans Reliab 42:299–302Google Scholar
  39. 39.
  40. 40.
    Nadarajah S, Kotz S (2013) The exponentiated Weibull distribution: a survey. Stat Pap 54(3):839–877Google Scholar
  41. 41.
    Nadarajah S (2009) Bathtub-shaped failure rate functions. Qual Quant 43:855–863Google Scholar
  42. 42.
    Nadarajah S, Kotz S (2006) The exponentiated-type distributions. Acta Appl Math 92:97–111Google Scholar
  43. 43.
    Nadarajah S, Kotz S (2004) The beta Gumbel distribution. Math Probl Eng 4:323–332Google Scholar
  44. 44.
    Nadarajah S, Gupta AK (2004) The beta Fréchet distribution. Far East J Theor Stat 14:15–24Google Scholar
  45. 45.
    Nichols MD, Padgett WJ (2006) A Bootstrap control chart for Weibull percentiles. Qual Reliab Eng Int 22:141–151Google Scholar
  46. 46.
    Oseghale OI, Akomolafe AA (2017) Performance rating of the Kumaraswamy transmuted Weibull distribution: an analytical approach. Am J Math Stat 7(3):125–135Google Scholar
  47. 47.
    Pal M, Ali MM, Woo J (2006) Exponentiated Weibull distribution. STATISTICA anno LXVI 2:139–147Google Scholar
  48. 48.
    Pal M, Tiensuwan M (2014) The beta transmuted Weibull distribution. Aust J Stat 43(2):133–149Google Scholar
  49. 49.
    Penn L (1977) Physiochemical properties of Kevlar 49 fiber. J Appl Polym Sci 42:59–73Google Scholar
  50. 50.
    Ramos MWA, Marinho PRD, Cordeiro G, da Silva RV, Hamedani GG (2015) The Kumaraswamy-G Poisson family of distributions. J Stat Theory Appl 14:222–239Google Scholar
  51. 51.
    R Development Core Team (2018) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Viena, Austria. Retrieved from Accessed 5 Sept 2018
  52. 52.
    Shaw W, Buckley I (2009) The alchemy of probability distributions: beyond Gram–Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map. In: Conference on computational finance, IMA, 0901-0434. Research reportGoogle Scholar
  53. 53.
    Shahbaz M, Shahbaz S, Butt N (2012) The Kumaraswamy inverse Weibull distribution. Pak J Stat Oper Res 8(3):479–489Google Scholar
  54. 54.
    Silva G, Ortega EMM, Cordeiro GM (2010) The beta modified Weibull distribution. Lifetime Data Anal 16(3):409–430Google Scholar
  55. 55.
    Tahir M, Nadarajah S (2015) Parameter induction in continuous univariate distributions: well-established G families. Anais da Academia Brasileira de Ci\({\hat{e}}\)ncias 87(2): 539-5-68Google Scholar
  56. 56.
    Tahir M, Cordeiro G (2016) Compounding of distributions: a survey and new generalized classes. J Stat Distrib Appl 3:183–210Google Scholar
  57. 57.
    Xie M, Lai CD (1995) Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function. Reliab Eng Syst Saf 52:87–93Google Scholar
  58. 58.
    Xie M, Tang Y, Goh TN (2002) A modified Weibull extension with bathtub failure rate function. Reliab Eng Syst Saf 76:279–285Google Scholar
  59. 59.
    Yeung KK, Rao KP (2012) Mechanical properties of Kevlar-49 fibre reinforced thermoplastic composites. Polym Polym Compos 20(5):411–424Google Scholar
  60. 60.
    Zografos K, Balakrishnan N (2009) On families of beta and generalized gamma generated distributions and associated inference. Stat Methodol 6(4):344–362Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Gokarna R. Aryal
    • 1
    Email author
  • Keshav P. Pokhrel
    • 2
  • Netra Khanal
    • 3
  • Chris P. Tsokos
    • 4
  1. 1.Department of Mathematics, Statistics and CSPurdue University NorthwestHammondUSA
  2. 2.Department of Mathematics and StatisticsUniversity of Michigan- DearbornDearbornUSA
  3. 3.Department of MathematicsThe University of TampaTampaUSA
  4. 4.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA

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