Bayesian Estimation of 3-Component Mixture of the Inverse Weibull Distributions

  • Tabasam SultanaEmail author
  • Muhammad Aslam
  • Javid Shabbir
Research Paper


This article focuses on the study of a 3-component mixture of the inverse Weibull distributions under Bayesian perspective. The censored sampling scheme is used because it is popular in reliability theory and survival analysis. To achieve this objective, the Bayes estimates of the parameter of the mixture model along with their posterior risks using informative and non-informative priors are attained. These estimates have been acquired under two cases: (a) when the shape parameter is known and (b) when all parameters are unknown. For the case (a), Bayes estimates are gained under three loss functions while for the case (b) only the squared error loss function is used. To study numerically, the performance of the Bayes estimators under different loss functions, their statistical properties have been simulated for different sample sizes and test termination times.


Bayes estimators Censoring Informative prior Loss functions Posterior risks 


Author Contributions

Design the experiments: TS. Performed the experiments and analyzed the data: TS. Wrote the paper: TS. Professor DR. MA is my PhD advisor, while Dr. JS is also my PhD advisor and Head of the Department.

Supplementary material

40995_2017_443_MOESM1_ESM.docx (42 kb)
Supplementary material 1 (DOCX 42 kb)
40995_2017_443_MOESM2_ESM.docx (332 kb)
Supplementary material 2 (DOCX 331 kb)


  1. Ali S (2015) Mixture of the inverse Rayleigh distribution: properties and estimation in a Bayesian framework. Appl Math Model 39:515–530MathSciNetCrossRefGoogle Scholar
  2. Aslam M, Tahir M, Hussain Z, Al-Zahrani B (2015) A 3-component mixture of rayleigh distributions: properties and estimation in Bayesian framework. PLoS ONE 10:e0126183CrossRefGoogle Scholar
  3. Berger JO (1985) Statistical decision theory and Bayesian analysis. Springer, New YorkCrossRefzbMATHGoogle Scholar
  4. DeGroot MH (2005) Optimal statistical decision. Wiley, New YorkGoogle Scholar
  5. Gijbels I (2010) Censored data. Wiley Interdiscip Rev 2:178–188CrossRefGoogle Scholar
  6. Jamal F, Nasir MA, Nasir JA (2014) A mixture of modified inverse Weibull distribution. J Stat Appl Probab Lett 2:31–46Google Scholar
  7. Jeffreys H (1946) An invariant form for the prior probability in estimation problems. Proc R Soc Lond A 186:453–461MathSciNetCrossRefzbMATHGoogle Scholar
  8. Jeffreys H (1961) Theory of probability. Clarendon Press, OxfordzbMATHGoogle Scholar
  9. Jiang R, Murthy DNP, Ji P (2001) Models involving two inverse Weibull distributions. Reliab Eng Syst Saf 73:73–81CrossRefGoogle Scholar
  10. Kalbfleisch JD, Prentice RL (2011) The statistical analysis of failure time data. Wiley, New YorkzbMATHGoogle Scholar
  11. Khan MS, Pasha GR, Pasha AH (2008) Theoretical analysis of inverse Weibull distribution. WSEAS Trans Math 7:30–38zbMATHGoogle Scholar
  12. Kundu D, Howlader H (2010) Bayesian inference and prediction of the inverse Weibull distribution for type-II censored data. Comput Stat Data Anal 54:1547–1558MathSciNetCrossRefzbMATHGoogle Scholar
  13. Legendre AM (1805) Nouvelles méthodes pour la détermination des orbites des comètes. F. Didot., ParisGoogle Scholar
  14. Mendenhall W, Hader RJ (1958) Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data. Biometrika 45:504–520MathSciNetCrossRefzbMATHGoogle Scholar
  15. Noor F, Aslam M (2013) Bayesian inference of the inverse weibull mixture distribution using type-i censoring. J Appl Stat 40:1076–1089MathSciNetCrossRefGoogle Scholar
  16. Norstrom JG (1996) The use of precautionary loss functions in risk analysis. IEEE Trans Reliab 45:400–403CrossRefGoogle Scholar
  17. Panaitescu E, Popescu PG, Cozma P (2010) Bayesian and non-Bayesian estimators using record statistics of the modified-inverse Weibull distribution. Proc Romanian Acad Ser A 11:224–231zbMATHGoogle Scholar
  18. Pawlas P, Szynal D (2000) Characterizations of the inverse Weibull distribution and generalized extreme value distributions by moments of kth record values. Appl Math 27:197–202MathSciNetzbMATHGoogle Scholar
  19. Saleem M, Aslam M, Economu P (2010) On the Bayesian analysis of the mixture of power function distribution using the complete and the censored sample. J Appl Stat 37(1):25–40MathSciNetCrossRefGoogle Scholar
  20. Shi Y, Yan W (2010) The EB estimation of scale-parameter for two parameter exponential distribution under the type-i censoring life testGoogle Scholar
  21. Soland RM (1968) Bayesian analysis of the Weibull process with unknown scale parameter and its application to acceptance sampling. IEEE Trans Reliab 17:84–90CrossRefGoogle Scholar
  22. Sultan KS, Ismail MA, Al-Moisheer AS (2007) Mixture of two inverse Weibull distributions: properties and estimation. Comput Stat Data Anal 51:5377–5387MathSciNetCrossRefzbMATHGoogle Scholar
  23. Tahir M, Aslam M, Hussain Z (2016) On the Bayesian analysis of 3-component mixture of exponential distributions under different loss functions. Hacet J Math Stat 45:609–628MathSciNetzbMATHGoogle Scholar

Copyright information

© Shiraz University 2017

Authors and Affiliations

  • Tabasam Sultana
    • 1
    Email author
  • Muhammad Aslam
    • 2
  • Javid Shabbir
    • 1
  1. 1.Department of StatisticsQuaid-i-Azam UniversityIslamabadPakistan
  2. 2.Department of Basic SciencesRiphah International UniversityIslamabadPakistan

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