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Annals of Data Science

, Volume 4, Issue 1, pp 31–61 | Cite as

A New Extension of Weibull Distribution with Application to Lifetime Data

  • Sanku Dey
  • Vikas Kumar Sharma
  • Mhamed Mesfioui
Article

Abstract

The Weibull distribution has been generalized by many authors in recent years. Here, we introduce a new generalization, called alpha-power transformed Weibull distribution that provides better fits than the Weibull distribution and some of its known generalizations. The distribution contains alpha-power transformed exponential and alpha-power transformed Rayleigh distributions as special cases. Various properties of the proposed distribution, including explicit expressions for the quantiles, mode, moments, conditional moments, mean residual lifetime, stochastic ordering, Bonferroni and Lorenz curve, stress–strength reliability and order statistics are derived. The distribution is capable of modeling monotonically increasing, decreasing, constant, bathtub, upside-down bathtub and increasing–decreasing–increasing hazard rates. The maximum likelihood estimators of unknown parameters cannot be obtained in explicit forms, and they have to be obtained by solving non-linear equations only. Two data sets have been analyzed to show how the proposed models work in practice. Further, a bivariate extension based on Marshall–Olkin and copula concept of the proposed model are developed but the properties of the distribution not considered in detail in this paper that can be addressed in future research.

Keywords

Weibull distribution Weighted distribution Moments Quantile function Stochastic ordering Entropy Stress–strength reliability Maximum likelihood estimation 

Mathematics Subject Classification

60E05 62F10 

Notes

Acknowledgements

The authors are thankful to the Editorial Board and to the reviewers for their valuable comments and suggestions which led to this improved version. Mhamed Mesfioui acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada No 261968-2013.

References

  1. 1.
    Aarset MV (1987) How to identify bathtub hazard rate. IEEE Trans Reliab 36:106–108CrossRefGoogle Scholar
  2. 2.
    Alexander C, Cordeiro GM, Ortega EMM, Sarabia JM (2012) Generalized beta-generated distributions. Comput Stat Data Anal 56:1880–1897CrossRefGoogle Scholar
  3. 3.
    Alzaatreh A, Famoye F, Lee C (2013) A new method for generating families of continuous distributions. Metron 71(1):63–79Google Scholar
  4. 4.
    Andrews DF, Herzberg AM (2012) Data: a collection of problems from many fields for the student and research worker. Springer, BerlinGoogle Scholar
  5. 5.
    Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stat 12:171–178Google Scholar
  6. 6.
    Azzalini A (1986) Further results on a class of distributions which includes the normal ones. Statistica 46:199–208Google Scholar
  7. 7.
    Azzalini A, Capitanio A (2003) Distribution generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. J R Stat Soc Ser B 65:367–389CrossRefGoogle Scholar
  8. 8.
    Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing probability models. Holt, Rinehart and Winston, New YorkGoogle Scholar
  9. 9.
    Barlow RE, Toland RH, Freeman T (1984) A Bayesian analysis of stress rupture life of kevlar 49/epoxy spherical pressure vessels. In: Proceedings of the conference on applications of statistics. Marcel Dekker, New YorkGoogle Scholar
  10. 10.
    Barreto-Souza W, de Morais AL, Cordeiro GM (2011) The Weibull geometric distribution. J Stat Comput Simul 81:645–657CrossRefGoogle Scholar
  11. 11.
    Barreto-Souza W, Santos AHS, Cordeiro GM (2010) The beta generalized exponential distribution. J Stat Comput Simul 80:159–172CrossRefGoogle Scholar
  12. 12.
    Bebbington M, Lai CD, Zitikis R (2007) A flexible Weibull extension. Reliab Eng Syst Saf 92:719–726CrossRefGoogle Scholar
  13. 13.
    Burr IW (1942) Cumulative frequency functions. Ann Math Stat 13:215–232CrossRefGoogle Scholar
  14. 14.
    Cooray K, Ananda MMA (2005) Modeling actuarial data with a composite lognormal-pareto model. Scand Actuar J 5:321–334CrossRefGoogle Scholar
  15. 15.
    Cordeiro GM, Ortega EM, Nadarajah S (2010) The Kumaraswamy Weibull distribution with application to failure data. J Frankl Inst 347(8):1399–1429CrossRefGoogle Scholar
  16. 16.
    Cordeiro GM, Silva RB (2014) The complementary extended Weibull power series class of distributions. Cincia Nat 36:1–13Google Scholar
  17. 17.
    Eugene N, Lee C, Famoye F (2002) The beta-normal distribution and its applications. Commun Stat Theory Methods 31:497–512CrossRefGoogle Scholar
  18. 18.
    Famoye F, Lee C, Olumolade O (2005) The beta-Weibull distribution. J Stat Theory Appl 4:121–136Google Scholar
  19. 19.
    Ferreira JTAS, Steel MFJ (2006) A constructive representation of univariate skewed distributions. J Am Stat Assoc 101:823–829CrossRefGoogle Scholar
  20. 20.
    Genest C, Remillard B, Beaudoin D (2009) Goodness-of-fit tests for copulas: a review and a power study. Insur Math Econ 44:199–213CrossRefGoogle Scholar
  21. 21.
    Gupta R, Kirmani S (1990) The role of weighted distribution in stochastic modeling. Commun Stat Theory Methods 19:3147–3162CrossRefGoogle Scholar
  22. 22.
    Gupta RC, Keating JP (1985) Relations for reliability measures under length biased sampling. Scan J Stat 13:49–56Google Scholar
  23. 23.
    Gupta RD, Kundu D (1999) Generalized exponential distribution. Aust NZ J Stat 41:173–188CrossRefGoogle Scholar
  24. 24.
    Hansen BE (1994) Autoregressive conditional density estimation. Int Econ Rev 35:705–730CrossRefGoogle Scholar
  25. 25.
    Hastings JC, Mosteller F, Tukey JW, Windsor C (1947) Low moments for small samples: a comparative study of order statistics. Ann Stat 18:413–426CrossRefGoogle Scholar
  26. 26.
    Johnson NL (1949) Systems of frequency curves generated by methods of translation. Biometrika 36:149–176CrossRefGoogle Scholar
  27. 27.
    Kotz S, Vicari D (2005) Survey of developments in the theory of continuous skewed distributions. Metron LXIII:225–261Google Scholar
  28. 28.
    Lai CD, Xie M, Murthy DNP (2009) A modified Weibull distribution. IEEE Trans Reliab 52:33–37CrossRefGoogle Scholar
  29. 29.
    Lee ET, Wang JW (2013) Statistical methods for survival data analysis, 3rd edn. Wiley, HobokenGoogle Scholar
  30. 30.
    Mahdavi A, Kundu D (2016) A new method for generating distributions with an application to exponential distribution. Commun Stat Theory Methods. doi: 10.1080/03610926.2015.1130839 Google Scholar
  31. 31.
    Marshall AW, Olkin I (1967) A multivariate exponential distribution. J Am Stat Assoc 62:30–44CrossRefGoogle Scholar
  32. 32.
    Marshall AW, Olkin I (1997) A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84:641–652CrossRefGoogle Scholar
  33. 33.
    Morais AL, Barreto-Souza W (2011) A compound class of Weibull and power series distributions. Comput Stat Data Anal 55:14101425CrossRefGoogle Scholar
  34. 34.
    Mudholkar GS, Srivastava DK (1993) Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Trans Reliab 42:299–302CrossRefGoogle Scholar
  35. 35.
    Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer, New YorkGoogle Scholar
  36. 36.
    Oluyede BO (1999) On inequalities and selection of experiments for length-biased distributions. Probab Eng Inf Sci 13:169–185CrossRefGoogle Scholar
  37. 37.
    Patil GP (2002) Weighted distributions. Encyclopedia of Environmetics, vol 4. pp 2369–2377Google Scholar
  38. 38.
    Patil GP, Rao CR (1978) Weighted distributions and size-biased sampling with application to wildlife populations and human families. Biometrics 34:179–189CrossRefGoogle Scholar
  39. 39.
    Patil GP, Rao CR, Ratnaparkhi MV (1986) On discrete weighted distributions and their use in model choice for observed data. Commun Stat Theory Methods 15:907–918CrossRefGoogle Scholar
  40. 40.
    Pearson K (1895) Contributions to the mathematical theory of evolution. II. Skew variation in homogeneous material. Philos Trans R Soc Lond A 186:343–414CrossRefGoogle Scholar
  41. 41.
    Rao CR (1965) On discrete distributions arising out of methods of ascertainment. In: Patil GP (ed) Classical and contagious discrete distributions. Pergamon Press and Statistical Publishing Society, Calcutta, pp 320–332Google Scholar
  42. 42.
    Renyi A (1961) On measures of entropy and information. In: Proceedings of the 4th Berkeley symposium on mathematical statistics and probability. University of California Press, Berkeley, pp 547–561Google Scholar
  43. 43.
    Sarhan AM, Apaloo J (2013) Exponentiated modified Weibull extension distribution. Reliab Eng Syst Saf 112:137–144CrossRefGoogle Scholar
  44. 44.
    Sarhan AM, Zaindin M (2009) Modified Weibull distribution. Appl Sci 11:2336Google Scholar
  45. 45.
    Shannon CE (1951) Prediction and entropy of printed English. Bell Syst Tech J 30:50–64CrossRefGoogle Scholar
  46. 46.
    Silva GO, Ortega EMM, Cordeiro GM (2010) The beta modified Weibull distribution. Lifetime Data Anal 16:409–430CrossRefGoogle Scholar
  47. 47.
    Singh H, Misra N (1994) On redundancy allocations in systems. J Appl Probab 31:10041014CrossRefGoogle Scholar
  48. 48.
    Singla N, Jain K, Sharma SK (2012) The beta generalized Weibull distribution: properties and applications. Reliab Eng Syst Saf 102:5–15CrossRefGoogle Scholar
  49. 49.
    Sklar A (1959) Fonctions de répartition à \(n\) dimensions et leurs marges. Publ Inst Stat Univ Paris 8:229–233Google Scholar
  50. 50.
    Smith RL, Naylor JC (1987) A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution. Appl Stat 36:358–369CrossRefGoogle Scholar
  51. 51.
    Tukey JW (1960) The Practical Relationship Between the Common Transformations of Percentages of Counts and Amounts, Technical Report, 36. Statistical Techniques Research Group, Princeton University, Princeton, NJGoogle Scholar
  52. 52.
    Wanbo L, Daimin S (2012) A new compounding life distribution: the Weibull–Poisson distribution. J Appl Stat 39:21–38CrossRefGoogle Scholar
  53. 53.
    Weibull WA (1951) Statistical distribution function of wide applicability. J Appl Mech 18:293296Google Scholar
  54. 54.
    Xie M, Lai CD (1996) Reliability analysis using an additive Weibull modelwith bathtub-shaped failure rate function. Reliab Eng Syst Saf 52:87–93CrossRefGoogle Scholar
  55. 55.
    Xie M, Tang Y, Goh TN (2002) A modified Weibull extension with bathtub-shaped failure rate function. Reliab Eng Syst Saf 76:279–285CrossRefGoogle Scholar
  56. 56.
    Zhang T, Xie M (2011) On the upper truncated Weibull distribution and its reliability implications. Reliab Eng Syst Saf 96:194–200CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Sanku Dey
    • 1
  • Vikas Kumar Sharma
    • 2
  • Mhamed Mesfioui
    • 1
    • 3
  1. 1.Department of Statistics, Faculty of SciencesKing Abdulaziz UniversityJeddahSaudi Arabia
  2. 2.Department of MathematicsInstitute of Infrastructure, Technology, Research and Management (IITRAM)AhmedabadIndia
  3. 3.Département de mathématiques et informatiqueUniversité du Québec à Trois-RivièresTrois-RivièresCanada

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