Brazilian Journal of Probability and Statistics

A new extension of the Birnbaum–Saunders distribution

Artur J. Lemonte

Full-text: Open access

Abstract

In this paper, a new extension for the Birnbaum–Saunders distribution, which has been applied to the modeling of fatigue failure times and reliability studies, is introduced. The proposed model, called the Marshall–Olkin extended Birnbaum–Saunders distribution, arises based on the scheme introduced by Marshall and Olkin [Biometrika 84 (1997) 641–652]. The maximum likelihood estimators and statistical inference for the new distribution parameters and influence diagnostic for the new distribution are presented. Finally, the proposed new distribution is applied to model three real data sets.

Article information

Source
Braz. J. Probab. Stat. Volume 27, Number 2 (2013), 133-149.

Dates
First available in Project Euclid: 21 February 2013

Permanent link to this document
http://projecteuclid.org/euclid.bjps/1361455031

Digital Object Identifier
doi:10.1214/11-BJPS160

Mathematical Reviews number (MathSciNet)
MR3028800

Zentralblatt MATH identifier
06365955

Keywords
Birnbaum–Saunders distribution fatigue life distribution lifetime data Marshall–Olkin extended distribution maximum likelihood estimation

Citation

Lemonte, Artur J. A new extension of the Birnbaum–Saunders distribution. Braz. J. Probab. Stat. 27 (2013), no. 2, 133--149. doi:10.1214/11-BJPS160. http://projecteuclid.org/euclid.bjps/1361455031.


Export citation

References

  • Balakrishnan, N., Leiva, V. and López, J. (2007). Acceptance sampling plans from truncated life tests from generalized Birnbaum–Saunders distribution. Communications in Statistics—Simulation and Computation 36, 643–656.
  • Bhatti, C. R. (2010). The Birnbaum–Saunders autoregressive conditional duration model. Mathematics and Computers in Simulation 80, 2062–2078.
  • Birnbaum, Z. W. and Saunders, S. C. (1969a). A new family of life distributions. Journal of Applied Probability 6, 319–327.
  • Birnbaum, Z. W. and Saunders, S. C. (1969b). Estimation for a family of life distributions with applications to fatigue. Journal of Applied Probability 6, 328–347.
  • Cook, R. D. (1986). Assessment of local influence (with discussion). Journal of the Royal Statistical Society, Ser. B 48, 133–169.
  • Cook, R. D. and Weisberg, S. (1982). Residuals and Influence in Regression. London: Chapman & Hall.
  • Cordeiro, G. M. and Lemonte, A. J. (2011). The $\beta$-Birnbaum–Saunders distribution: An improved distribution for fatigue life modeling. Computational Statistics and Data Analysis 55, 1445–1461.
  • Cox, D. R. and Hinkley, D. V. (1974). Theoretical Statistics. London: Chapman & Hall.
  • Cox, D. R. and Lewis, P. A. W. (1966). The Statistical Analysis of Series of Events. London: Methuem.
  • Desmond, A. F. (1985). Stochastic models of failure in random environments. Canadian Journal of Statistics 13, 171–183.
  • Desmond, A. F. (1986). On the relationship between two fatigue-life models. IEEE Transactions on Reliability 35, 167–169.
  • Díaz-García, J. A. and Leiva, V. (2005). A new family of life distributions based on the elliptically contoured distributions. Journal of Statistical Planning and Inference 128, 445–457.
  • Doornik, J. A. (2006). An Object-Oriented Matrix Language—Ox 4, 5th ed. London: Timberlake Consultants Press.
  • García, V. J., Gómez-Déniz, E. and Vázquez-Polo, F. J. (2010). A new skew generalization of the normal distribution: Properties and applications. Computational Statistics and Data Analysis 54, 2021–2034.
  • Garvan, F. (2002). The Maple Book. London: Chapman & Hall/CRC.
  • Gómes, H. W., Olivares-Pacheco, J. F. and Bolfarine, H. (2009). An extension of the generalized Birnbaum–Saunders distribution. Statistics and Probability Letters 79, 331–338.
  • Gómez-Déniz, E. (2010). Another generalization of the geometric distribution. Test 19, 399–415.
  • Guiraud, P., Leiva, V. and Fierro, R. (2009). A non-central version of the Birnbaum–Saunders distribution for reliability analysis. IEEE Transactions on Reliability 58, 152–160.
  • Jørgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. New York: Springer-Verlag.
  • Kundu, D., Kannan, N. and Balakrishnan, N. (2008). On the hazard function of Birnbaum–Saunders distribution and associated inference. Computational Statistics and Data Analysis 52, 2692–2702.
  • Leiva, V., Barros, M., Paula, G. A. and Sanhueza, A. (2008). Generalized Birnbaum–Saunders distributions applied to air pollutant concentration. Environmetrics 19, 235–249.
  • Leiva, V., Sanhueza, A. and Angulo, J. M. (2009). A length-biased version of the Birnbaum–Saunders distribution with application in water quality. Stochastic Environmental Research and Risk Assessment 23, 299–307.
  • Leiva, V., Vilca, F., Balakrishnan, N. and Sanhueza, A. (2010). A skewed sinh-normal distribution and its properties and application to air pollution. Communications in Statistics—Theory and Methods 39, 426–443.
  • Lemonte, A. J., Cribari-Neto, F. and Vasconcellos, K. L. P. (2007). Improved statistical inference for the two-parameter Birnbaum–Saunders distribution. Computational Statistics and Data Analysis 51, 4656–4681.
  • Lemonte, A. J., Simas, A. B. and Cribari-Neto, F. (2008). Bootstrap-based improved estimators for the two-parameter Birnbaum–Saunders distribution. Journal of Statistical Computation and Simulation 78, 37–49.
  • Marshall, A. W. and Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84, 641–652.
  • Nadarajah, S. (2008). A truncated inverted beta distribution with application to air pollution data. Stochastic Environmental Research and Risk Assessment 22, 285–289.
  • Nichols, M. D. and Padgett, W. J. (2006). A bootstrap control chart for Weibull percentiles. Quality and Reliability Engineering International 22, 141–151.
  • Nocedal, J. and Wright, S. J. (1999). Numerical Optimization. New York: Springer.
  • Owen, W. J. (2006). A new three-parameter extension to the Birnbaum–Saunders distribution. IEEE Transactions on Reliability 55, 475–479.
  • Press, W. H., Teulosky, S. A., Vetterling, W. T. and Flannery, B. P. (2007). Numerical Recipes in C: The Art of Scientific Computing, 3rd ed. Cambridge: Cambridge Univ. Press.
  • R Development Core Team (2010). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing.
  • Sarhan, A. M. and Balakrishnan, N. (2007). A new class of bivariate distributions and its mixture. Journal of Multivariate Analysis 98, 1508–1527.
  • Sigmon, K. and Davis, T. A. (2002). MATLAB Primer, 6th ed. London: Chapman & Hall/CRC.
  • Vilca, F. and Leiva, V. (2006). A new fatigue life model based on the family of skew-elliptical distributions. Communications in Statistics—Theory and Methods 35, 229–244.
  • Wolfram, S. (2003). The Mathematica Book, 5th ed. New York: Cambridge Univ. Press.
  • Wu, J. and Wong, A. C. M. (2004). Improved interval estimation for the two-parameter Birnbaum–Saunders distribution. Computational Statistics and Data Analysis 47, 809–821.
  • Xu, A. and Tang, Y. (2010). Reference analysis for Birnbaum–Saunders distribution. Computational Statistics and Data Analysis 54, 185–192.