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


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.


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

Mathematics Subject Classification

60E05 62F10 



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.


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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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