Skip to main content
SpringerLink
Log in

Bayesian analysis of Birnbaum–Saunders distribution via the generalized ratio-of-uniforms method

  • Original Paper
  • Published:
Computational Statistics Aims and scope Submit manuscript

Abstract

This paper deals with the Bayesian inference for the parameters of the Birnbaum–Saunders distribution. We adopt the inverse-gamma priors for the shape and scale parameters because the continuous conjugate joint prior distribution does not exist and the reference prior (or independent Jeffreys’ prior) results in an improper posterior distribution. We propose an efficient sampling algorithm via the generalized ratio-of-uniforms method to compute the Bayesian estimates and the credible intervals. One appealing advantage of the proposed procedure over other sampling techniques is that it efficiently generates independent samples from the required posterior distribution. Simulation studies are conducted to investigate the behavior of the proposed method, and two real-data applications are analyzed for illustrative purposes.

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
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Achcar JA (1993) Inferences for the Birnbaum–Saunders fatigue life model using Bayesian methods. Comput Stat Data Anal 15:367–380

    Article  MathSciNet  MATH  Google Scholar 

  • Albert J (2012) LearnBayes: functions for learning Bayesian inference. R package version 2.12

  • Balakrishnan N, Zhu X (2014) On the existence and uniqueness of the maximum likelihood estimates of the parameters of Birnbaum-Saunders distribution based on Type-I, Type-II and hybrid censored samples. Statistics 48:1013–1032

    Article  MathSciNet  MATH  Google Scholar 

  • Barros M, Leiva V, Paula GA (2012) gbs: package for generalized Birnbaum–Saunders Distributions. R package version 1.0

  • Birnbaum ZW, Saunders SC (1969a) A new family of life distributions. J Appl Probab 6:319–327

    Article  MathSciNet  MATH  Google Scholar 

  • Birnbaum ZW, Saunders SC (1969b) Estimation for a family of life distributions with applications to fatigue. J Appl Probab 6:328–347

    Article  MathSciNet  MATH  Google Scholar 

  • Broyden CG (1972) Quasi-Newton methods in numerical methods for unconstrained optimization. Academic Press, New York (Murray, W. Ed.)

    Google Scholar 

  • Chen M, Shao Q (1999) Monte Carlo estimation of Bayesian credible and HPD intervals. J Comput Graph Stat 8:69–92

    MathSciNet  Google Scholar 

  • Congdon P (2001) Bayesian statistical modeling. Wiley, Chichester

    MATH  Google Scholar 

  • Dupuis D, Mills J (1998) Robust estimation of the Birnbaum–Saunders distribution. IEEE Trans Reliab 47:88–95

    Article  Google Scholar 

  • Engelhardt M, Bain LJ, Wright FT (1981) Inferences on the parameters of the Birnbaum–Saunders fatigue life distribution based on maximum likelihood estimation. Technometrics 23:251–256

    Article  MathSciNet  MATH  Google Scholar 

  • Farias R, Lemonte A (2011) Bayesian inference for the Birnbaum–Saunders nonlinear regression model. Stat Methods Appl 20:423–438

    Article  MathSciNet  MATH  Google Scholar 

  • Leiva V, Barros M, Paula GA (2009) Generalized Birnbaum–Saunders models using R. Brazilian Statistical Association, São Paulo

    MATH  Google Scholar 

  • Lemonte AJ, Cribari-Neto F, Vasconcellos KLP (2007) Improved statistical inference for the two-parameter Birnbaum–Saunders distribution. Comput Stat Data Anal 51:4656–4681

    Article  MathSciNet  MATH  Google Scholar 

  • Lemonte AJ, Simas AB, Cribari-Neto F (2008) Bootstrap-based improved estimators for the two-parameter Birnbaum–Saunders distribution. J Stat Comput Simul 78:37–49

    Article  MathSciNet  MATH  Google Scholar 

  • Lindley DV (1980) Approximate Bayesian methods. Trabajos Estadist 31:223–237

    Article  MathSciNet  MATH  Google Scholar 

  • McCool J (1974) Inferential techniques for Weibull populations. Aerospace Research Laboratories Report ARL TR74-0180. Wright-Patterson Air Force Base, Dayton

  • Neal P, Roberts G (2008) Optimal scaling for random walk Metropolis on spherically constrained target densities. Methodol Comput Appl Probab 10:277–297

    Article  MathSciNet  MATH  Google Scholar 

  • Ng HKT, Kundu D, Balakrishnan N (2003) Modified moment estimation for the two-parameter Birnbaum–Saunders distribution. Comput Statist Data Anal 43:283–298

    Article  MathSciNet  MATH  Google Scholar 

  • R Core Team (2012) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0

  • Shanno DF (1970) Conditioning of quasi-newton methods for function minimization. Math Comput 24:647–656

    Article  MathSciNet  MATH  Google Scholar 

  • Snow G (2013) TeachingDemos: demonstrations for teaching and learning. R package version 2:9

  • Sun D, Berger JO (1998) Reference priors with partial information. Biometrika 85:55–71

    Article  MathSciNet  MATH  Google Scholar 

  • Vilca F, Balakrishnan N, Zeller CB (2014) A robust extension of the bivariate Birnbaum–Saunders distribution and associated inference. J Multivar Anal 124:418–435

    Article  MathSciNet  MATH  Google Scholar 

  • Wakefield JC, Gelfand AE, Smith AFM (1991) Efficient generation of random variates via the ratio-of-uniforms method. Stat Comput 1:129–133

    Article  Google Scholar 

  • Wang M, Zhao J, Sun X, Park C (2013) Robust explicit estimation of the two-parameter Birnbaum–Saunders distribution. J Appl Statist 40:2259–2274

    Article  MathSciNet  Google Scholar 

  • Xu A, Tang Y (2010) Reference analysis for Birnbaum–Saunders distribution. Comput Stat Data Anal 54:185–192

    Article  MathSciNet  MATH  Google Scholar 

  • Xu A, Tang Y (2011) Bayesian analysis of Birnbaum–Saunders distribution with partial information. Comput Stat Data Anal 55:2324–2333

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We thank the Editor, an associate editor and three referees for their suggestions and comments which have significantly improved the paper. The first author was partially supported by the New Faculty Start-Up Fund at Michigan Technological University.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA

    Min Wang

  2. Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634, USA

    Xiaoqian Sun

  3. Department of Industrial Engineering, Pusan National University, Geumjeong-gu, Pusan, Republic of Korea

    Chanseok Park

Authors
  1. Min Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoqian Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chanseok Park
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Min Wang.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (R 2 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, M., Sun, X. & Park, C. Bayesian analysis of Birnbaum–Saunders distribution via the generalized ratio-of-uniforms method. Comput Stat 31, 207–225 (2016). https://doi.org/10.1007/s00180-015-0629-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00180-015-0629-z

Keywords

Search

Navigation