On the Extended Birnbaum–Saunders Distribution Based on the Skew-t-Normal Distribution

  • Tahereh Poursadeghfard
  • Ahad Jamalizadeh
  • Alireza NematollahiEmail author
Research paper


In this article, a generalized version of the univariate Birnbaum–Saunders distribution based on the skew-t-normal distribution is introduced and its characterizations, properties are studied. Maximum likelihood estimation of the parameters via the ECM algorithm evaluated by Monte Carlo simulations is also discussed. Finally, two real datasets are analyzed for illustrative purposes.


Birnbaum–Saunders distribution ECM algorithm Information matrix Skew-t-normal distribution 


  1. Azzalini A (1985) A class of distribution which includes the normal ones. Scand J Stat 12:171–178MathSciNetzbMATHGoogle Scholar
  2. Azzalini A (1986) Further results on a class of distribution which includes the normal ones. Statistica 46:199–208MathSciNetzbMATHGoogle Scholar
  3. Barros M, Paula GA, Leiva V (2008) A new class of survival regression models with heavy tailed errors: robustness and diagnostics. Lifetime Data Anal 14:316–332MathSciNetCrossRefzbMATHGoogle Scholar
  4. Basford KE, Greenway DR, McLachlan GJ, Peel D (1997) Standard error of fitted means under normal mixture. Comput Stat 12:1–17zbMATHGoogle Scholar
  5. Birnbaum ZW, Saunders SC (1969) A new family of life distribution. J Appl Probab 6:319–327MathSciNetCrossRefzbMATHGoogle Scholar
  6. Cabral CRB, Bolfarine H, Pereira JRG (2008) Bayesian density estimation using skew student-t-normal mixtures. Comput Stat Data Anal 52:5075–5090MathSciNetCrossRefzbMATHGoogle Scholar
  7. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm (with discussion). J R Stat Soc Ser B 39:1–38zbMATHGoogle Scholar
  8. Desmond A (1985) Stochastic models of failure in random environments. Can J Stat 13:171–183MathSciNetCrossRefzbMATHGoogle Scholar
  9. Diaz-Garcia JA, Leiva-Sanchez V (2005) A new family of life distribution based on the elliptically contoured distributions. J Stat Plan Inference 128:445–457MathSciNetCrossRefzbMATHGoogle Scholar
  10. Gomez HW, Venegas O, Bolfarine H (2007) Skew-symmetric distributions generated by the distribution function of the normal distribution. Environmetrics 18:395–407MathSciNetCrossRefGoogle Scholar
  11. Gomez HW, Olivares J, Bolfarine H (2009) An extension of the generalized Birnbaum–Saunders distribution. Stat Probab Lett 79:331–338MathSciNetCrossRefzbMATHGoogle Scholar
  12. Hashmi F, Amirzadeh V, Jamalizadeh A (2015) An extension of the Birnbaum-Saunders distribution based on skew-normal-t distribution. Stat Res Train Cent 12:1–37Google Scholar
  13. Ho HJ, Pyne S, Lin TI (2011) Maximum likelihood inference for mixture of skew Student-t-normal distributions through practical EM-type algorithms. Stat Comput 22:287–299MathSciNetCrossRefzbMATHGoogle Scholar
  14. Kass RE, Raftery AE (1995) Bayes factor. J Am Stat Assoc 90:773–795MathSciNetCrossRefzbMATHGoogle Scholar
  15. Lachos VH, Dey D, Cancho VG, Louzada N (2017) Scale mixtures log-Birnbaum–Saunders regression models with censored data: a Bayesian approach. J Stat Comput Simul 87:2002–2022MathSciNetCrossRefGoogle Scholar
  16. Leiva V, Barros M, Paula GA, Galea M (2007) Influence diagnostics in log-Birnbaum–Saunders regression models with censored data. Comput Stat Data Anal 51:5694–5707MathSciNetCrossRefzbMATHGoogle Scholar
  17. Leiva V, Riquelme M, Balakrishnan N, Sanhueza A (2008) Lifetime analysis based on the generalized Birnbaum–Saunders. Comput Stat Data Anal 52:2079–2097MathSciNetCrossRefzbMATHGoogle Scholar
  18. Leiva V, Vilca F, Balakrishnan N, Sanhueza A (2010) A skewed sinh-normal distribution and its properties and application to air pollution. Commun Stat Theory Methods 39:426–443MathSciNetCrossRefzbMATHGoogle Scholar
  19. McLachlan GJ, Krishnan T (2008) The EM algorithm and extensions, 2nd edn. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  20. Meng X-L, Rubin DB (1993) Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika 80:267–278MathSciNetCrossRefzbMATHGoogle Scholar
  21. Nadarajah S, Kotz S (2003) Skewed distributions generated by the normal kernel. Stat Probab Lett 65(3):269–277MathSciNetCrossRefzbMATHGoogle Scholar
  22. Podaski R (2008) Characterization of diameter data in near-natural forests using the Birnbaum–Saunders distribution. Can J For Res 38:518–527CrossRefGoogle Scholar
  23. Raftery AE (1995) Bayesian model selection in social research. Sociolog Methodol 25:111–163CrossRefGoogle Scholar
  24. R Development Core Team (2016) R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0.
  25. Sanhueza A, Leiva V, Balakrishnan N (2008) The generalized Birnbaum–Saunders distribution and its methodology and application. Commun Stat Theory Methods 37:645–670MathSciNetCrossRefzbMATHGoogle Scholar
  26. Vilca F, Santana L, Leiva V, Balakrishnan N (2011) Estimation of extreme percentiles in Birnbaum–Saunders disribution. Comput Stat Data Anal 55:1665–1678CrossRefzbMATHGoogle Scholar

Copyright information

© Shiraz University 2018

Authors and Affiliations

  • Tahereh Poursadeghfard
    • 1
  • Ahad Jamalizadeh
    • 2
  • Alireza Nematollahi
    • 1
    Email author
  1. 1.Department of Statistics, College of SciencesShiraz UniversityShirazIran
  2. 2.Department of Statistics, Faculty of Mathematics and ComputerShahid Bahonar University of KermanKermanIran

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