Skip to main content
SpringerLink
Log in

The beta modified Weibull distribution

Lifetime Data Analysis volume 16, pages 409–430 (2010)Cite this 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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  • Aarset MV (1987) How to identify bathtub hazard rate. IEEE Trans Reliab 36: 106–108

    Article  MATH  Google Scholar 

  • Barakat HM, Abdelkader YH (2004) Computing the moments of order statistics from nonidentical random variables. Stat Methods Appl 13: 15–26

    Article  MATH  MathSciNet  Google Scholar 

  • Bebbington M, Lai CD, Zitikis R (2007) A flexible Weibull extension. Reliab Eng Syst Saf 92: 719–726

    Article  Google Scholar 

  • Brown BW, Floyd MS, Levy LB (2002) The log F: a distribution for all seasons. Comput Stat 17: 47–58

    Article  MATH  Google Scholar 

  • Carrasco JMF, Ortega EMM, Cordeiro GM (2008) A generalized modified Weibull distribution for lifetime modeling. Comput Stat Data Anal 53: 450–462

    Article  MATH  Google Scholar 

  • Cox C (2008) The generalized F distribution: an umbrella for parametric survival analysis. Stat Med 27: 4301–4312

    Article  MathSciNet  Google Scholar 

  • 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: 4352–4374

    Article  MathSciNet  Google Scholar 

  • Doornik J (2007) Ox 5: object-oriented matrix programming language, 5th ed. Timberlake Consultants, London

    Google Scholar 

  • Eugene N, Lee C, Famoye F (2002) Beta-normal distribution and its applications. Commun Stat Theory Methods 31: 497–512

    Article  MATH  MathSciNet  Google Scholar 

  • Gradshteyn IS, Ryzhik IM (2000) Table of integrals, series, and products. Academic Press, New York

    MATH  Google Scholar 

  • Gupta RD, Kundu D (1999) Generalized exponential distributions. Aust NZ J Stat 41: 173–188

    Article  MATH  MathSciNet  Google Scholar 

  • Gupta RD, Kundu D (2001) Exponentiated exponential distribution: an alternative to gamma and Weibull distributions. Biomet J 43: 117–130

    Article  MATH  MathSciNet  Google Scholar 

  • Gupta AK, Nadarajah S (2004) Handbook of beta distribution and its applications. Marcel Dekker, New York

    MATH  Google Scholar 

  • Haupt E, Schabe H (1992) A new model for a lifetime distribution with bathtub shaped failure rate. Microelectron Reliab 32: 633–639

    Article  Google Scholar 

  • Hosking JRM (1986) The theory of probability weighted moments. Research Report RC12210, IBM Thomas J. Watson Research Center, New York.

  • Hosking JRM (1990) L-moments: analysis and estimation of distributions using linear combinations of order statistics. J R Stat Soc Ser B 52: 105–124

    MATH  MathSciNet  Google Scholar 

  • Jones MC (2004) Family of distributions arising from distribution of order statistics. Test 13: 1–43

    Article  MATH  MathSciNet  Google Scholar 

  • Kundu D, Rakab MZ (2005) Generalized Rayleigh distribution: different methods of estimation. Comput Stat Data Anal 49: 187–200

    Article  MATH  Google Scholar 

  • Lai CD, Xie M, Murthy DNP (2003) A modified Weibull distribution. Trans Reliab 52: 33–37

    Article  Google Scholar 

  • Lee C, Famoye F, Olumolade O (2007) Beta-Weibull distribution: some properties and applications to censored data. J Mod Appl Stat Methods 6: 173–186

    Google Scholar 

  • Mudholkar GS, Srivastava DK (1993) Exponentiated Weibull family for analyzing bathtub failure-real data. IEEE Trans Reliab 42: 299–302

    Article  MATH  Google Scholar 

  • Mudholkar GS, Srivastava DK, Friemer M (1995) The exponentiated Weibull family: a reanalysis of the bus-motor-failure data. Technometrics 37: 436–445

    Article  MATH  Google Scholar 

  • 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: 1575–1583

    Article  MATH  MathSciNet  Google Scholar 

  • Nadarajah S, Gupta AK (2004) The beta Fréchet distribution. Far East J Theor Stat 14: 15–24

    MATH  MathSciNet  Google Scholar 

  • Nadarajah S, Kotz S (2004) The beta Gumbel distribution. Math Prob Eng 10: 323–332

    Article  MathSciNet  Google Scholar 

  • Nadarajah S, Kotz S (2006) The beta exponential distribution. Reliab Eng Syst Saf 91: 689–697

    Article  MathSciNet  Google Scholar 

  • Nelson W (1990) Accelerated life testing: statistical models, data analysis and test plans. Wiley, New York

    Google Scholar 

  • Pham H, Lai CD (2007) On recent generalizations of the Weibull distribution. IEEE Trans Reliab 56: 454–458

    Article  Google Scholar 

  • Rajarshi S, Rajarshi MB (1988) Bathtub distributions: a review. Commun Stat Theory Methods 17: 2521–2597

    MathSciNet  Google Scholar 

  • Wang FK (2000) A new model with bathtub-shaped failure rate using an additive Burr XII distribution. Reliab Eng Syst Saf 70: 305–312

    Article  Google Scholar 

  • Xie M, Lai CD (1995) Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function. Reliab Eng Syst Saf 52: 87–93

    Article  Google Scholar 

  • Xie M, Tang Y, Goh TN (2002) A modified Weibull extension with bathtub failure rate function. Reliab Eng Syst Saf 76: 279–285

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ESALQ, Universidade de São Paulo, Piracicaba, Brazil

    Giovana O. Silva & Edwin M. M. Ortega

  2. DEINFO, Universidade Federal Rural de Pernambuco, Recife, Brazil

    Gauss M. Cordeiro

Authors
  1. Giovana O. Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Edwin M. M. Ortega
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gauss M. Cordeiro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Edwin M. M. Ortega.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Silva, G.O., Ortega, E.M.M. & Cordeiro, G.M. The beta modified Weibull distribution. Lifetime Data Anal 16, 409–430 (2010). https://doi.org/10.1007/s10985-010-9161-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10985-010-9161-1

Keywords

search

Navigation