Multivariate Generalized Birnbaum-Saunders Models Applied to Case Studies in Bio-Engineering and Industry

  • Víctor LeivaEmail author
  • Carolina Marchant
Part of the Contributions to Statistics book series (CONTRIB.STAT.)


Birnbaum-Saunders models are receiving considerable attention in the literature. Multivariate regression models are a useful tool in the multivariate analysis, which takes into account the correlation between variables. Diagnostic analysis is an important aspect to be considered in the statistical modeling. In this work, we formulate a statistical methodology based on multivariate generalized Birnbaum-Saunders regression models and their diagnostics. We implement the obtained results in the R software, which are illustrated with two real-world multivariate data sets related to case studies in bio-engineering and industry to show their potential applications.



The authors thank the editors and reviewers for their constructive comments on an earlier version of this manuscript. This research work was partially supported by FONDECYT 1160868 grant from the Chilean government.


  1. 1.
    Barros, M., Paula, G.A., Leiva, V.: A new class of survival regression models with heavy-tailed errors: robustness and diagnostics. Lifetime Data Anal. 14, 316–332 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Barros, M., Leiva, V., Ospina, R., Tsuyuguchi, A.: Goodness-of-fit tests for the Birnbaum-Saunders distribution with censored reliability data. IEEE Trans. Reliab. 63, 543–554 (2014)CrossRefGoogle Scholar
  3. 3.
    Díaz-García, J.A., Leiva, V.: A new family of life distributions based on elliptically contoured distributions. J. Stat. Plan. Inference 128, 445–457 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Díaz-García, J.A., Leiva, V., Galea, M.: Singular elliptic distribution: density and applications. Commun. Stat. Theory Methods 31, 665–681 (2002)CrossRefGoogle Scholar
  5. 5.
    Díaz-García, J.A., Galea, M., Leiva, V.: Influence diagnostics for elliptical multivariate linear regression models. Commun. Stat. Theory Methods 32, 625–641 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fang, K.T., Kotz, S., Ng, K.W.: Symmetric Multivariate and Related Distributions. Chapman and Hall, London (1990)CrossRefGoogle Scholar
  7. 7.
    Galea, M., Paula, G.A., Uribe-Opazo, M.A.: On influence diagnostic in univariate elliptical linear regression models. Stat. Pap. 44, 23–45 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Garcia-Papani, F., Uribe-Opazo, M.A., Leiva, V., Aykroyd, R.G.: Birnbaum-Saunders spatial modelling and diagnostics applied to agricultural engineering data. Stoch. Env. Res. Risk A. 31, 105–124 (2017)CrossRefGoogle Scholar
  9. 9.
    Lange, K., Sinsheimer, J.: Normal/independent distributions and their applications in robust regression. J. Comput. Graph. Stat. 2, 175–198 (1993)MathSciNetGoogle Scholar
  10. 10.
    Lange, K., Little, J., Taylor, M.: Robust statistical modeling using the t distribution. J. Am. Stat. Assoc. 84, 881–896 (1989)MathSciNetGoogle Scholar
  11. 11.
    Leiva, V.: The Birnbaum-Saunders Distribution. Academic Press, New York (2016)zbMATHGoogle Scholar
  12. 12.
    Leiva, V., Liu, S., Shi, L., Cysneiros, F.J.A.: Diagnostics in elliptical regression models with stochastic restrictions applied to econometrics. J. Appl. Stat. 43, 627–642 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lepadatu, D., Kobi, A., Hambli, R., Barreau, A.: Lifetime multiple response optimization of metal extrusion die. In: Proceedings of Annual Reliability and Maintainability Symposium, pp. 37–42. IEEE, Piscataway (2005)Google Scholar
  14. 14.
    Li, A., Chen, Z., Xie, F.: Diagnostic analysis for heterogeneous log-Birnbaum-Saunders regression models. Stat. Probab. Lett. 89, 1690–1698 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Marchant, C., Leiva, V., Cysneiros, F.J.A.: A multivariate log-linear model for Birnbaum-Saunders distributions. IEEE Trans. Reliab. 65, 816–827 (2016)CrossRefGoogle Scholar
  16. 16.
    Marchant, C., Leiva, V., Cysneiros, F.J.A., Vivanco, J.F.: Diagnostics in multivariate generalized Birnbaum-Saunders regression models. J. Appl. Stat. 43, 2829–2849 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Marchant, C., Leiva, V., Cysneiros, F.J.A., Liu, S.: A multivariate log-linear model for Birnbaum-Saunders distributions. J. Stat. Comput. Simul. 88, 182–202 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York (2006)zbMATHGoogle Scholar
  19. 19.
    Paula, G.A., Leiva, V., Barros, M., Liu, S.: Robust statistical modeling using the Birnbaum-Saunders-t distribution applied to insurance. Appl. Stoch. Model. Bus. Ind. 28, 16–34 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Vivanco, J.F., Burgers, T.A., García, S., Crookshank, M., Kunz, M., MacIntyre, N.J., Harrison, M.M., Bryant, J.T., Sellens, R.W., Ploeg, H.L.: Estimating the density of femoral head trabecular bone from hip fracture patients using computed tomography scan data. Proc. Inst. Mech. Eng. H J. Eng. Med. 228, 616–626 (2014)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Industrial EngineeringPontificia Universidad Católica de ValparaísoValparaísoChile
  2. 2.Faculty of Basic SciencesUniversidad Católica del MauleTalcaChile

Personalised recommendations