Skip to main content
SpringerLink
Log in

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

  • Original Article
  • Published:
International Journal of System Assurance Engineering and Management Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  • Berger JO, Sun D (1992) Bayesian analysis for poly-Weibull distribution. Technical report no 92–05C Purdue University

  • Block HW, Basu A (1974) A continuous, bivariate exponential extension. J Am Stat Assoc 69(348):1031–1037

    MathSciNet  MATH  Google Scholar 

  • 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 175–197

    Google Scholar 

  • Freund JE (1961) A bivariate extension of the exponential distribution. J Am Stat Assoc 56(296):971–977

    Article  MathSciNet  Google Scholar 

  • Gilks WR, Wild P (1992) Adaptive rejection sampling for Gibbs sampling. Appl Stat 41:337–348

    Article  Google Scholar 

  • Gumbel EJ (1960) Bivariate exponential distributions. J Am Stat Assoc 55(292):698–707

    Article  MathSciNet  Google Scholar 

  • Hanagal DD (2005) A bivariate Weibull regression model. Econ Qual Control 20(1):143–150

    Article  MathSciNet  Google Scholar 

  • Heinrich G, Jensen U (1995) Parameter estimation for a bivariate lifetime distribution in reliability with multivariate extensions. Metrika 42(1):49–65

    Article  MathSciNet  Google Scholar 

  • 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):17–24

    Google Scholar 

  • Jose K, Ristić MM, Joseph A (2011) Marshall–Olkin bivariate Weibull distributions and processes. Stat Pap 52(4):789–798

    Article  MathSciNet  Google Scholar 

  • Kundu D, Dey AK (2009) Estimating the parameters of the Marshall–Olkin bivariate Weibull distribution by em algorithm. Comput Stat Data Anal 53(4):956–965

    Article  MathSciNet  Google Scholar 

  • Kundu D, Gupta AK (2013) Bayes estimation for the Marshall–Olkin bivariate Weibull distribution. Comput Stat Data Anal 57(1):271–281

    Article  MathSciNet  Google Scholar 

  • Lai C-D (2014) Generalized Weibull distributions. In: Generalized Weibull distributions. Springer, pp 23–75

  • 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):156–170

    Article  MathSciNet  Google Scholar 

  • Lin D, Sun W, Ying Z (1999) Nonparametric estimation of the gap time distribution for serial events with censored data. Biometrika 86(1):59–70

    Article  MathSciNet  Google Scholar 

  • Lu J-C (1989) Weibull extensions of the Freund and Marshall–Olkin bivariate exponential models. IEEE Trans Reliab 38(5):615–619

    Article  Google Scholar 

  • 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):608–615

    Article  MathSciNet  Google Scholar 

  • Marshall AW, Olkin I (1967) A multivariate exponential distribution. J Am Stat Assoc 62(317):30–44

    Article  MathSciNet  Google Scholar 

  • McCool J (2012) Using the Weibull distribution: reliability, modeling, and inference. Wiley Series in Probability. Wiley, Hoboken

    Book  Google Scholar 

  • Meintanis SG (2007) Test of fit for Marshall–Olkin distributions with applications. J Stat Plan Inference 137(12):3954–3963

    Article  MathSciNet  Google Scholar 

  • 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 7:68–74

    Google Scholar 

  • Murthy D, Xie M, Jiang R (2004) Weibull models. Wiley Series in Probability and Statistics. Wiley, Hoboken

    MATH  Google Scholar 

  • 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):1559–1569

    Article  MathSciNet  Google Scholar 

  • Patra K, Dey DK (1999) A multivariate mixture of Weibull distributions in reliability modeling. Stat Probab Lett 45(3):225–235

    Article  MathSciNet  Google Scholar 

  • Rachev ST, Wu C, Yakovlev AY (1995) A bivariate limiting distribution of tumor latency time. Math Biosci 127(2):127–147

    Article  MathSciNet  Google Scholar 

  • 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 144:35–44

    Article  Google Scholar 

  • Robert C, Casella G (2013) Monte Carlo statistical methods. Springer Texts in Statistics. Springer, New York

    MATH  Google Scholar 

  • Sarhan AM, Balakrishnan N (2007) A new class of bivariate distributions and its mixture. J Multivar Anal 98(7):1508–1527

    Article  MathSciNet  Google Scholar 

  • Upadhyay SK, Vasishta N, Smith A (2001) Bayes inference in life testing and reliability via Markov chain Monte Carlo simulation. Sankhyā 63(1):15–40

    MATH  Google Scholar 

  • 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. https://journal.r-project.org/archive/2013/RJ-2013-002/index.html

Download references

Author information

Authors and Affiliations

  1. DST-Centre for Interdisciplinary Mathematical Sciences, Banaras Hindu University, Varanasi, 221 005, India

    Rakesh Ranjan

  2. Department of Statistics, Government Arts & Science College, Ratlam, India

    Vastoshpati Shastri

Authors
  1. Rakesh Ranjan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vastoshpati Shastri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rakesh Ranjan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ranjan, R., Shastri, V. Posterior and predictive inferences for Marshall Olkin bivariate Weibull distribution via Markov chain Monte Carlo methods. Int J Syst Assur Eng Manag 10, 1535–1543 (2019). https://doi.org/10.1007/s13198-019-00903-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13198-019-00903-9

Keywords

Search

Navigation