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Generalized mixtures of Weibull components

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Abstract

Weibull mixtures have been used extensively in reliability and survival analysis, and they have also been generalized by allowing negative mixing weights, which arise naturally under the formation of some structures of reliability systems. These models provide flexible distributions for modeling dependent lifetimes from heterogeneous populations. In this paper, we study conditions on the mixing weights and the parameters of the Weibull components under which the considered generalized mixture is a well-defined distribution. Specially, we characterize the generalized mixture of two Weibull components. In addition, some reliability properties are established for these generalized two-component Weibull mixture models. One real data set is also analyzed for illustrating the usefulness of the studied model.

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References

  • Attardi L, Guida M, Pulcini C (2005) A mixed-Weibull regression model for the analysis of automotive warranty data. Reliab Eng Syst Saf 87:265–273

    Article  Google Scholar 

  • Badar MG, Priest AM (1982) Statistical aspects of fiber and bundle strength in hybrid composites. In: Hayashi T, Kawata K, Umekawa S (eds) Progress in science and engineering composites, ICCM-IV, Tokyo, pp 1129–1136

  • Baggs GE, Nagaraja HN (1996) Reliability properties of order statistics from bivariate exponential distributions. Commun Stat Stoch Models 12:611–631

    Article  MATH  MathSciNet  Google Scholar 

  • Bagnoli M, Bergstrom T (2005) Log-concave probability and its applications. Econ Theory 26:445–469

    Article  MATH  MathSciNet  Google Scholar 

  • Barlow RE, Proschan F (1981) Statistical theory of reliability and life testing: probability models. To begin with. Silver Spring, Maryland

    Google Scholar 

  • Bartholomew DJ (1969) Sufficient conditions for a mixture of exponentials to be a probability density function. Ann Math Stat 40:2183–2188

    Article  MATH  Google Scholar 

  • Block HW, Langberg NA, Savits TH (2012) A mixture of exponential and IFR gamma distributions having an upsidedown bathtub-shaped failure rate. Probab Eng Inf Sci 26:573–580

    Article  MATH  MathSciNet  Google Scholar 

  • Block HW, Langberg NA, Savits TH, Wang J (2010) Continuous mixtures of exponentials and IFR gammas having bathtub-shaped failure rates. J Appl Probab 47:899–907

    Article  MATH  MathSciNet  Google Scholar 

  • Bučar T, Nagode M, Fajdiga M (2004) Reliability approximation using finite Weibull mixture distributions. Reliab Eng Syst Saf 84:241–251

    Article  Google Scholar 

  • Carta JA, Ramírez P (2007) Analysis of two-component mixture Weibull statistics for estimation of wind speed distributions. Renew Energy 32:518–531

    Article  Google Scholar 

  • Ebden M, Stranjak A, Roberts S (2010) Visualizing uncertainty in reliability functions with application to aero engine overhaul. J R Stat Soc Ser C 59:163–173

    Article  MathSciNet  Google Scholar 

  • Elmahdy EE, Aboutahoun AW (2013) A new approach for parameter estimation of finite Weibull mixture distributions for reliability modeling. Appl Math Model 37:1800–1810

    Article  MathSciNet  Google Scholar 

  • Farcomeni A, Nardi A (2010) A two-component Weibull mixture to model early and late mortality in a Bayesian framework. Comput Stat Data Anal 54:416–428

    Article  MATH  MathSciNet  Google Scholar 

  • Franco M, Vivo JM (2002) Reliability properties of series and parallel systems from bivariate exponential models. Commun Stat Theory Methods 31:2349–2360

    Article  MATH  MathSciNet  Google Scholar 

  • Franco M, Vivo JM (2006) On log-concavity of the extremes from Gumbel bivariate exponential distributions. Statistics 40:415–433

    Article  MATH  MathSciNet  Google Scholar 

  • Franco M, Vivo JM (2007) Generalized mixtures of gamma and exponentials and reliability properties of the maximum from Friday and Patil bivariate exponential model. Commun Stat Theory Methods 36:2011–2025

    Article  MATH  MathSciNet  Google Scholar 

  • Franco M, Vivo JM (2009a) Constraints for generalized mixtures of Weibull distributions with a common shape parameter. Stat Probab Lett 79:1724–1730

    Article  MATH  MathSciNet  Google Scholar 

  • Franco M, Vivo JM (2009b) Stochastic aging classes for the maximum statistic from Friday and Patil bivariate exponential distribution family. Commun Stat Theory Methods 38:902–911

    Article  MATH  MathSciNet  Google Scholar 

  • Franco M, Vivo JM, Balakrishnan N (2011) Reliability properties of generalized mixtures of Weibull distributions with a common shape parameter. J Stat Plan Inference 141:2600–2613

    Article  MATH  MathSciNet  Google Scholar 

  • Jiang R, Zuo MJ, Li HX (1999) Weibull and inverse Weibull mixture models allowing negative weights. Reliab Eng Syst Saf 66:227–234

    Article  Google Scholar 

  • Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1, 2nd edn. Wiley, New York

  • Kokonendji C, Mizere D, Balakrishnan N (2008) Connections of the Poisson weight function to overdispersion and underdispersion. J Stat Plan Inference 138:1287–1296

    Article  MATH  MathSciNet  Google Scholar 

  • Kundu D, Basu S (2000) Analysis of incomplete data in presence of competing risks. J Stat Plan Inference 87:231–239

    MathSciNet  Google Scholar 

  • Marín JM, Rodríguez MT, Wiper MP (2005) Using Weibull mixture distributions to model heterogeneous survival data. Commun Stat Simul Comput 34:673–684

    Article  MATH  Google Scholar 

  • Mosler K, Scheicher C (2008) Homogeneity testing in a Weibull mixture model. Stat Pap 49:315–332

    Article  MATH  MathSciNet  Google Scholar 

  • Navarro J, Guillamon A, Ruiz MC (2009) Generalized mixtures in reliability modelling: applications to the construction of bathtub shaped hazard models and the study of systems. Appl Stoch Models Bus Ind 25:323–337

    Article  MATH  MathSciNet  Google Scholar 

  • Patra K, Dey DK (1999) A multivariate mixture of Weibull distributions in reliability modeling. Stat Probab Lett 45:225–235

    Article  MATH  MathSciNet  Google Scholar 

  • Qin X, Zhang JS, Yan XD (2012) Two improved mixture Weibull models for the analysis of wind speed data. J Appl Meteorol Climatol 51:1321–1332

    Article  Google Scholar 

  • Shaked M, Shanthikumar JG (2006) Stochastic orders, 2nd edn. Springer, New York

    Google Scholar 

  • Sinha SK (1987) Bayesian estimation of the parameters and reliability function of mixture of Weibull life distributions. J Stat Plan Inference 16:377–387

    Article  MATH  Google Scholar 

  • Steutel FW (1967) Note on the infinite divisibility of exponential mixtures. Ann Math Stat 38:1303–1305

    Article  MATH  MathSciNet  Google Scholar 

  • Tsionas EG (2002) Bayesian analysis of finite mixtures of Weibull distributions. Commun Stat Theory Methods 31:37–48

    Article  MATH  MathSciNet  Google Scholar 

  • Wondmagegnehu ET (2004) On the behavior and shape of mixture failure rates from a family of IFR Weibull distributions. Nav Res Logist 51:491–500

    Article  MATH  MathSciNet  Google Scholar 

  • Wondmagegnehu ET, Navarro J, Hernandez PJ (2005) Bathtub shaped failure rates from mixtures: a practical point of view. IEEE Trans Reliab 54:270–275

    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 

Download references

Acknowledgments

The authors would like to thank two referees, the Associated Editor and the Editor-in-Chief for many constructive suggestions of an earlier version of this manuscript, which have improved the presentation of this paper. The authors are also thankful to Prof. J.G. Surles for providing the real data set. This work was partially supported by Fundación Séneca of the Regional Government of Murcia (Spain) under Grant 11886/PHCS/09.

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Authors and Affiliations

  1. Department of Statistics and Operations Research, University of Murcia, 30100 , Murcia, Spain

    Manuel Franco

  2. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada

    Narayanaswamy Balakrishnan

  3. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, India

    Debasis Kundu

  4. Regional Campus of International Excellence “Campus Mare Nostrum”, University of Murcia, 30100 , Murcia, Spain

    Juana-María Vivo

Authors
  1. Manuel Franco
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  2. Narayanaswamy Balakrishnan
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  3. Debasis Kundu
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  4. Juana-María Vivo
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Correspondence to Manuel Franco.

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Franco, M., Balakrishnan, N., Kundu, D. et al. Generalized mixtures of Weibull components. TEST 23, 515–535 (2014). https://doi.org/10.1007/s11749-014-0362-x

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