Posterior and predictive inferences for Marshall Olkin bivariate Weibull distribution via Markov chain Monte Carlo methods

  • Rakesh RanjanEmail author
  • Vastoshpati Shastri
Original Article


This paper deals with the well known bi-variate Weibull distribution developed by Marshall and Olkin. In the light of prior information, this paper derives the posterior distribution and performs Markov chain Monte Carlo methods to obtain posterior based inferences. This paper also checks the sensitivity of posterior estimates by changing the prior variances followed by Bayesian prediction using sample-based approaches. Numerical illustrations are provided for real as well as simulated data sets.


Bivariate model ML Estimates MCMC Gibbs sampler Metropolis algorithm Predictive simulation 



  1. Berger JO, Sun D (1992) Bayesian analysis for poly-Weibull distribution. Technical report no 92–05C Purdue UniversityGoogle Scholar
  2. Block HW, Basu A (1974) A continuous, bivariate exponential extension. J Am Stat Assoc 69(348):1031–1037MathSciNetzbMATHGoogle Scholar
  3. Flegal JM, Jones GL (2011) Implementing MCMC: estimating with confidence. In: Brooks et al (eds) Handbook of Markov Chain Monte Carlo. CRC Press, New York, pp 175–197Google Scholar
  4. Freund JE (1961) A bivariate extension of the exponential distribution. J Am Stat Assoc 56(296):971–977MathSciNetCrossRefGoogle Scholar
  5. Gilks WR, Wild P (1992) Adaptive rejection sampling for Gibbs sampling. Appl Stat 41:337–348CrossRefGoogle Scholar
  6. Gumbel EJ (1960) Bivariate exponential distributions. J Am Stat Assoc 55(292):698–707MathSciNetCrossRefGoogle Scholar
  7. Hanagal DD (2005) A bivariate Weibull regression model. Econ Qual Control 20(1):143–150MathSciNetCrossRefGoogle Scholar
  8. Heinrich G, Jensen U (1995) Parameter estimation for a bivariate lifetime distribution in reliability with multivariate extensions. Metrika 42(1):49–65MathSciNetCrossRefGoogle Scholar
  9. Hougaard P, Harvald B, Holm NV (1992) Measuring the similarities between the lifetimes of adult danish twins born between 1881–1930. J Am Stat Assoc 87(417):17–24Google Scholar
  10. Jose K, Ristić MM, Joseph A (2011) Marshall–Olkin bivariate Weibull distributions and processes. Stat Pap 52(4):789–798MathSciNetCrossRefGoogle Scholar
  11. Kundu D, Dey AK (2009) Estimating the parameters of the Marshall–Olkin bivariate Weibull distribution by em algorithm. Comput Stat Data Anal 53(4):956–965MathSciNetCrossRefGoogle Scholar
  12. Kundu D, Gupta AK (2013) Bayes estimation for the Marshall–Olkin bivariate Weibull distribution. Comput Stat Data Anal 57(1):271–281MathSciNetCrossRefGoogle Scholar
  13. Lai C-D (2014) Generalized Weibull distributions. In: Generalized Weibull distributions. Springer, pp 23–75Google Scholar
  14. Lai C-D, Dong Lin G, Govindaraju K, Pirikahu S (2017) A simulation study on the correlation structure of Marshall–Olkin bivariate Weibull distribution. J Stat Comput Simul 87(1):156–170MathSciNetCrossRefGoogle Scholar
  15. Lin D, Sun W, Ying Z (1999) Nonparametric estimation of the gap time distribution for serial events with censored data. Biometrika 86(1):59–70MathSciNetCrossRefGoogle Scholar
  16. Lu J-C (1989) Weibull extensions of the Freund and Marshall–Olkin bivariate exponential models. IEEE Trans Reliab 38(5):615–619CrossRefGoogle Scholar
  17. Lu J-C (1992) Bayes parameter estimation for the bivariate Weibull model of Marshall–Olkin for censored data (reliability theory). IEEE Trans Reliab 41(4):608–615MathSciNetCrossRefGoogle Scholar
  18. Marshall AW, Olkin I (1967) A multivariate exponential distribution. J Am Stat Assoc 62(317):30–44MathSciNetCrossRefGoogle Scholar
  19. McCool J (2012) Using the Weibull distribution: reliability, modeling, and inference. Wiley Series in Probability. Wiley, HobokenCrossRefGoogle Scholar
  20. Meintanis SG (2007) Test of fit for Marshall–Olkin distributions with applications. J Stat Plan Inference 137(12):3954–3963MathSciNetCrossRefGoogle Scholar
  21. Mino T, Yoshikawa N, Suzuki K, Horikawa Y, Abe Y (2003) The mean of lifespan under dependent competing risks with application to mice data. J Health Sports Sci Juntendo Univ 7:68–74Google Scholar
  22. Murthy D, Xie M, Jiang R (2004) Weibull models. Wiley Series in Probability and Statistics. Wiley, HobokenzbMATHGoogle Scholar
  23. Nandi S, Dewan I (2010) An EM algorithm for estimating the parameters of bivariate Weibull distribution under random censoring. Comput Stat Data Anal 54(6):1559–1569MathSciNetCrossRefGoogle Scholar
  24. Patra K, Dey DK (1999) A multivariate mixture of Weibull distributions in reliability modeling. Stat Probab Lett 45(3):225–235MathSciNetCrossRefGoogle Scholar
  25. Rachev ST, Wu C, Yakovlev AY (1995) A bivariate limiting distribution of tumor latency time. Math Biosci 127(2):127–147MathSciNetCrossRefGoogle Scholar
  26. Ranjan R, Singh S, Upadhyay SK (2015) A bayes analysis of a competing risk model based on gamma and exponential failures. Reliab Eng Syst Saf 144:35–44CrossRefGoogle Scholar
  27. Robert C, Casella G (2013) Monte Carlo statistical methods. Springer Texts in Statistics. Springer, New YorkzbMATHGoogle Scholar
  28. Sarhan AM, Balakrishnan N (2007) A new class of bivariate distributions and its mixture. J Multivar Anal 98(7):1508–1527MathSciNetCrossRefGoogle Scholar
  29. Upadhyay SK, Vasishta N, Smith A (2001) Bayes inference in life testing and reliability via Markov chain Monte Carlo simulation. Sankhyā 63(1):15–40zbMATHGoogle Scholar
  30. Xiang Y, Gubian S, Suomela B, Hoeng J (2013) Generalized simulated annealing for efficient global optimization: the GenSA package for R. R J 5/1.

Copyright information

© The Society for Reliability Engineering, Quality and Operations Management (SREQOM), India and The Division of Operation and Maintenance, Lulea University of Technology, Sweden 2019

Authors and Affiliations

  1. 1.DST-Centre for Interdisciplinary Mathematical SciencesBanaras Hindu UniversityVaranasiIndia
  2. 2.Department of StatisticsGovernment Arts & Science CollegeRatlamIndia

Personalised recommendations