Journal of Statistical Theory and Practice

, Volume 11, Issue 4, pp 573–593 | Cite as

On a new class of skewed Birnbaum-Saunders models

  • N. Balakrishnan
  • Helton SauloEmail author
  • Jeremias Leão


Scale mixtures of Birnbaum–Saunders (SBS) distributions are attractive models in lifetime analysis. These models are based on scale mixture of normal (SMN) distributions and provide flexible heavy-tailed distributions. In this article, we propose a skewed version of SBS distributions and we establish some of its probabilistic and inferential properties. We then discuss the maximum likelihood estimation of the model parameters. An illustration of the methodology is provided, using real data.


Birnbaum–Saunders distribution maximum likelihood estimation hazard rate scale-mixture distribution 

AMS Subject Classification

Primary 62F10 secondary 60E05 


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  1. Andrews, D. R., and C. L. Mallows. 1974. Scale mixtures of normal distributions. Journal of the Royal Statistical Society Series B 36:99–102.MathSciNetzbMATHGoogle Scholar
  2. Azzalini, A. 1985. A class of distributions which includes the normal ones. Scandinavian Journal of Statistics 12:171–178.MathSciNetzbMATHGoogle Scholar
  3. Azevedo, C., V. Leiva, E. Athayde, and N. Balakrishnan. 2012. Shape and change point analyses of the Birnbaum–Saunders-t hazard rate. Computational Statistics & Data Analysis 56:3887–3897.MathSciNetCrossRefGoogle Scholar
  4. Balakrishnan, N., V. Leiva, and J. López. 2007. Acceptance sampling plans from truncated life tests based on the generalized Birnbaum–Saunders distribution. Communications in Statistics—Simulation and Computation 36:643–656.MathSciNetCrossRefGoogle Scholar
  5. Balakrishnan, N., V. Leiva, A. Sanhueza, and F. Vilca. 2009. Estimation in the Birnbaum–Saunders distribution based on scale-mixture of normals. Statistics and Operations Research Transactions 33:171–192.MathSciNetzbMATHGoogle Scholar
  6. Balakrishnan, N., R. C. Gupta, D. Kundu, V. Leiva, and A. Sanhueza. 2011. On some mixture models based on the Birnbaum–Saunders distribution and associated inference. Journal of Statistical Planning and Inference 141:2175–2190.MathSciNetCrossRefGoogle Scholar
  7. Bhatti, C. R. 2010. The Birnbaum–Saunders autoregressive conditional duration model. Mathematics and Computers in Simulation 80:2062–2078.MathSciNetCrossRefGoogle Scholar
  8. Birnbaum, Z. W., and S. C. Saunders. 1969. A new family of life distributions. Journal of Applied Probability 6:319–327.MathSciNetCrossRefGoogle Scholar
  9. Cox, D., and D. Hinkley. 1974. Theoretical statistics. London, UK: Chapman and Hall.CrossRefGoogle Scholar
  10. Cysneiros, A., F. Cribari-Neto, and C. A. Araujo, Jr. 2008. On Birnbaum–Saunders inference. Computational Statistics & Data Analysis 52:4939–4950.MathSciNetCrossRefGoogle Scholar
  11. Dempster, A. P., N. M. Laird, and D. B. Rubin. 1977. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal statistical Society Series B 39:1–38.MathSciNetzbMATHGoogle Scholar
  12. Díaz-García, J. A., and V. Leiva. 2005. A new family of life distributions based on elliptically contoured distributions. Journal of Statistical Planning and Inference 128:445–457.MathSciNetCrossRefGoogle Scholar
  13. Efron, B., and D. Hinkley. 1978. Assessing the accuracy of the maximum likelihood estimator: observed vs. expected Fisher information. Biometrika 65:457–487.MathSciNetCrossRefGoogle Scholar
  14. Ferreira, C. S., H. Bolfarine, and V. H. Lachos. 2011. Skew scale mixtures of normal distributions: properties and estimation. Statistical Methods 8:154–171.MathSciNetCrossRefGoogle Scholar
  15. Ferreira, M., M. I. Gomes, and V. Leiva. 2012. On an extreme value version of the Birnbaum–Saunders distribution. Revstat Statistical Journal 10:181–210.MathSciNetzbMATHGoogle Scholar
  16. Gómez, H. W., O. Venegas, and H. Bolfarine. 2007. Skew-symmetric distributions generated by the normal distribution function. Environmetrics 18:395–407.MathSciNetCrossRefGoogle Scholar
  17. Henze, N. 1986. A probabilistic representation of the skew-normal distribution. Scandinavian Journal of Statistics 13:271–275.MathSciNetzbMATHGoogle Scholar
  18. Hubert, M., and E. Vandervieren. 2008. An adjusted boxplot for skewed distributions. Computational Statistics & Data Analysis 52:5186–5201.MathSciNetCrossRefGoogle Scholar
  19. Johnson, N., S. Kotz, and N. Balakrishnan. 1994. Continuous univariate distributions, vol. 1. New York, NY: Wiley.zbMATHGoogle Scholar
  20. Johnson, N., S. Kotz, and N. Balakrishnan. 1995. Continuous univariate distributions, vol. 2. New York, NY: Wiley.zbMATHGoogle Scholar
  21. Kotz, S., V. Leiva, and A. Sanhueza. 2010. Two new mixture models related to the inverse Gaussian distribution. Methods and Computing in Applied Probability 12:199–212.MathSciNetCrossRefGoogle Scholar
  22. Kundu, D., N. Kannan, and N. Balakrishnan. 2008. On the hazard function of Birnbaum–Saunders distribution and associated inference. Computational Statistics & Data Analysis 52:2692–2702.MathSciNetCrossRefGoogle Scholar
  23. Laslett, G. M. 1994. Kriging and splines: an empirical comparison of their predictive performance in some applications. Journal of the American Statistical Association 89:391–400.MathSciNetCrossRefGoogle Scholar
  24. Leiva, V., C. Marchant, H. Saulo, M. Aslam, and F. Rojas. 2014a. Capability indices for Birnbaum–Saunders processes applied to electronic and food industries. Journal of Applied Statistics 41:1881–1902.MathSciNetCrossRefGoogle Scholar
  25. Leiva, V., E. Rojas, M. Galea, and A. Sanhueza. 2014b. Diagnostics in Birnbaum–Saunders accelerated life models with an application to fatigue data. Applied Stochastic Models in Business and Industry 30:15–131.MathSciNetCrossRefGoogle Scholar
  26. Leiva, V., M. Santos-Neto, F. J. A. Cysneiros, and M. Barros. 2014c. Birnbaum–Saunders statistical modelling: a new approach. Statistical Modelling 14:21–48.MathSciNetCrossRefGoogle Scholar
  27. Paula, G. A., V. Leiva, M. Barros, and S. Liu. 2012 Robust statistical modeling using the Birnbaum–Saunders-t distribution applied to insurance. Applied Stochastic Models in Business and Industry 28:16–34.MathSciNetCrossRefGoogle Scholar
  28. Rieck, J. R., and J. R. Nedelman. 1991. A log-linear model for the Birnbaum–Saunders distribution. Technometrics 33:51–60.zbMATHGoogle Scholar
  29. Saulo, H., V. Leiva, F. A. Ziegelmann, and C. Marchant. 2013. A nonparametric method for estimating asymmetric densities based on skewed Birnbaum–Saunders distributions applied to environmental data. Stochastic Environmental Research and Risk Assessment 27:1479–1491.CrossRefGoogle Scholar
  30. Vilca, F., A. Sanhueza, V. Leiva, and G. Christakos. 2010. An extended Birnbaum–Saunders model and its application in environmental quality in Santiago. Stochastic Environmental Research and Risk Assessment 24:771–782.CrossRefGoogle Scholar
  31. Vilca, F., L. Santana, V. Leiva, and N. Balakrishnan. 2011. Estimation of extreme percentiles in Birnbaum–Saunders distributions. Computational Statistics 55:1665–1678.MathSciNetzbMATHGoogle Scholar
  32. Villegas, C., G. A. Paula, and V. Leiva. 2011. Birnbaum–Saunders mixed models for censored reliability data analysis. IEEE Transactions on Reliability 60:748–758.CrossRefGoogle Scholar
  33. West, M. 1987. On scale mixtures of normal distributions. Biometrika 74:646–648.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing, 20 Middlefield Ct, Greensboro, NC 27455 2017

Authors and Affiliations

  • N. Balakrishnan
    • 1
  • Helton Saulo
    • 1
    • 2
    Email author
  • Jeremias Leão
    • 3
  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada
  2. 2.Instituto de Matemática e EstatísticaUniversidade Federal de GoiásGoiâniaBrazil
  3. 3.Departamento de EstatísticaUniversidade Federal do AmazonasManausBrazil

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