Skip to main content
SpringerLink
Log in

Bayesian inference for the Birnbaum–Saunders nonlinear regression model

  • Published:
Statistical Methods & Applications Aims and scope Submit manuscript

Abstract

We develop a Bayesian analysis for the class of Birnbaum–Saunders nonlinear regression models introduced by Lemonte and Cordeiro (Comput Stat Data Anal 53:4441–4452, 2009). This regression model, which is based on the Birnbaum–Saunders distribution (Birnbaum and Saunders in J Appl Probab 6:319–327, 1969a), has been used successfully to model fatigue failure times. We have considered a Bayesian analysis under a normal-gamma prior. Due to the complexity of the model, Markov chain Monte Carlo methods are used to develop a Bayesian procedure for the considered model. We describe tools for model determination, which include the conditional predictive ordinate, the logarithm of the pseudo-marginal likelihood and the pseudo-Bayes factor. Additionally, case deletion influence diagnostics is developed for the joint posterior distribution based on the Kullback–Leibler divergence. Two empirical applications are considered in order to illustrate the developed procedures.

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

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 

  • Aslam M, Jun CH, Ahmad M (2011) New acceptance sampling plans based on life tests for Birnbaum–Saunders distributions. J Stat Comput Simul 81: 461–470

    Article  MathSciNet  MATH  Google Scholar 

  • Balakrishnan N, Leiva V, López J (2007) Acceptance sampling plans from truncated life tests from generalized Birnbaum–Saunders distribution. Commun Stat Simul Comput 36: 643–656

    Article  MATH  Google Scholar 

  • Berger JO, Pericchi LR (2001) Objective Bayesian methods for model selection: introduction and comparison. IMS Lect Notes Monogr Ser 38: 135–207

    Article  MathSciNet  Google Scholar 

  • Bhatti CR (2010) The Birnbaum–Saunders autoregressive conditional duration model. Math Comput Simul 80: 2062–2078

    Article  MathSciNet  MATH  Google Scholar 

  • 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: 328–377

    Article  MathSciNet  MATH  Google Scholar 

  • Cancho VG, Ortega EEM, Paula GA (2010) On estimation and influence diagnostics for log-Birnbaum–Saunders Student-t regression models: full Bayesian analysis. J Stat Plan Inference 140: 2486–2496

    Article  MathSciNet  MATH  Google Scholar 

  • Cho H, Ibrahim JG, Sinha D, Zhu H (2009) Bayesian case influence diagnostics for survival models. Biometrics 65: 116–124

    Article  MathSciNet  MATH  Google Scholar 

  • Cook RD, Weisberg S (1982) Residuals and influence in regression. Chapman and Hall, London

    MATH  Google Scholar 

  • Cordeiro GM, Lemonte AJ (2011) The β-Birnbaum–Saunders distribution: an improved distribution for fatigue life modeling. Comput Stat Data Anal 55: 1445–1461

    Article  Google Scholar 

  • Cox DR, Hinkley DV (1974) Theoretical statistics. Chapman and Hall, London

    MATH  Google Scholar 

  • Desmond AF (1985) Stochastic models of failure in random environments. Can J Stat 13: 171–183

    Article  MathSciNet  MATH  Google Scholar 

  • Desmond AF (1986) On the relationship between two fatigue-life models. IEEE Trans Reliab 35: 167–169

    Article  MATH  Google Scholar 

  • Díaz-García JA, Leiva V (2005) A new family of life distributions based on the elliptically contoured distributions. J Stat Plan Inference 128: 445–457

    Article  MATH  Google Scholar 

  • Farias RBA, Lemonte AJ (2010) Birnbaum–Saunders nonlinear regression model: a full Bayesian analysis. Article presented in the 19◦ SINAPE (Simpósio Nacional de Probabilidade e Estatística), Estãncia de São Pedro, São Paulo, Brazil. (Available at http://www.ime.unicamp.br/sinape)

  • Galea M, Leiva V, Paula GA (2004) Influence diagnostics in log-Birnbaum–Saunders regression models. J Appl Stat 31: 1049–1064

    Article  MathSciNet  MATH  Google Scholar 

  • Geisser S, Eddy W (1979) A predictive approach to model selection. J Am Stat Assoc 74: 153–160

    Article  MathSciNet  MATH  Google Scholar 

  • Gelfand AE, Dey DK, Chang H (1992) Model determination using predictive distributions with implementation via sampling-based methods (with discussion). In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics. 4. Oxford University Press, Oxford, pp 147–169

    Google Scholar 

  • Gómes HW, Olivares-Pacheco JF, Bolfarine H (2009) An extension of the generalized Birnbaum–Saunders distribution. Stat Probab Lett 79: 331–338

    Article  Google Scholar 

  • Guiraud P, Leiva V, Fierro R (2009) A non-central version of the Birnbaum–Saunders distribution for reliability analysis. IEEE Trans Reliab 58: 152–160

    Article  Google Scholar 

  • Ibrahim JG, Chen M-H, Sinha D (2001) Bayesian survival analysis. Spring, New York

    MATH  Google Scholar 

  • Kundu D, Kannan N, Balakrishnan N (2008) On the hazard function of Birnbaum–Saunders distribution and associated inference. Comput Stat Data Anal 52: 2692–2702

    Article  MathSciNet  MATH  Google Scholar 

  • Leiva V, Barros M, Paula GA, Sanhueza A (2008) Generalized Birnbaum–Saunders distributions applied to air pollutant concentration. Environmetrics 19: 235–249

    Article  MathSciNet  Google Scholar 

  • Leiva V, Sanhueza A, Angulo JM (2009) A length-biased version of the Birnbaum–Saunders distribution with application in water quality. Stoch Environ Res Risk Assess 23: 299–307

    Article  MathSciNet  Google Scholar 

  • Lemonte AJ, Cordeiro GM (2009) Birnbaum–Saunders nonlinear regression models. Comput Stat Data Anal 53: 4441–4452

    Article  MathSciNet  MATH  Google Scholar 

  • Lemonte AJ, Cordeiro GM (2010) Asymptotic skewness in Birnbaum–Saunders nonlinear regression models. Stat Probab Lett 80: 892–898

    Article  MathSciNet  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, Ferrari SLP (2011) Size and power properties of some tests in the Birnbaum–Saunders regression model. Comput Stat Data Anal 55: 1109–1117

    Article  Google Scholar 

  • Lemonte AJ, Ferrari SLP (2011) Small-sample corrections for score tests in Birnbaum–Saunders regressions. Commun Stat Theory Methods 40: 232–243

    Article  MathSciNet  MATH  Google Scholar 

  • Lemonte AJ, Ferrari SLP (2011) Signed likelihood ratio tests in the Birnbaum–Saunders regression model. J Stat Plan Inference 141: 1031–1040

    Article  MathSciNet  MATH  Google Scholar 

  • Lemonte AJ, Ferrari SLP, Cribari–Neto F (2010) Improved likelihood inference in Birnbaum–Saunders regressions. Comput Stat Data Anal 54: 1307–1316

    Article  MathSciNet  MATH  Google Scholar 

  • Lemonte AJ, Patriota AG (2011) Influence diagnostics in Birnbaum–Saunders nonlinear regression models. J Appl Stat 38: 871–884

    Article  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 

  • Plummer M, Best N, Cowles K, Vines K (2006) CODA: convergence diagnosis and output analysis for MCMC. R News 6: 7–11

    Google Scholar 

  • R Development Core Team: (2009) R: a language and environment for statistical computing. R Development Core Team, Vienna

    Google Scholar 

  • Rieck JR (1999) A moment-generating function with application to the Birnbaum–Saunders distribution. Commun Stat Theory Methods 28: 2213–2222

    Article  MATH  Google Scholar 

  • Rieck JR, Nedelman JR (1991) A log-linear model for the Birnbaum–Saunders distribution. Technometrics 33: 51–60

    Article  MATH  Google Scholar 

  • Saunders SC (1974) A family of random variables closed under reciprocation. J Am Stat Assoc 69: 533–539

    Article  MATH  Google Scholar 

  • Tisionas EG (2001) Bayesian inference in Birnbaum–Saunders regression. Commun Stat Theory Methods 30: 179–193

    Article  Google Scholar 

  • Xi FC, Wei BC (2007) Diagnostics analysis for log-Birnbaum–Saunders regression models. Comput Stat Data Anal 51: 4692–4706

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Williams CR, Lee YL, Rilly JT (2003) A practical method for statistical analysis of strain-life fatigue data. Int J Fatigue 25: 427–436

    Article  Google Scholar 

  • Wu J, Wong ACM (2004) Improved interval estimation for the two-parameter Birnbaum–Saunders distribution. Comput Stat Data Anal 47: 809–821

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, University of São Paulo, São Paulo, Brazil

    Rafael B. A. Farias & Artur J. Lemonte

  2. Departamento de Estatística, Universidade de São Paulo, Rua do Matão, 1010, São Paulo, SP, 05508-090, Brazil

    Rafael B. A. Farias & Artur J. Lemonte

Authors
  1. Rafael B. A. Farias
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Artur J. Lemonte
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Artur J. Lemonte.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Farias, R.B.A., Lemonte, A.J. Bayesian inference for the Birnbaum–Saunders nonlinear regression model. Stat Methods Appl 20, 423–438 (2011). https://doi.org/10.1007/s10260-011-0165-0

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10260-011-0165-0

Keywords

Search

Navigation