An extended Birnbaum–Saunders model and its application in the study of environmental quality in Santiago, Chile

  • Filidor Vilca
  • Antonio Sanhueza
  • Víctor Leiva
  • George Christakos
Original Paper


In this article, we introduce, characterize and apply an extended version of the Birnbaum–Saunders model based on the Mudolkar–Hutson skew distribution. This model is appropriated for describing phenomena involving accumulation of some type, as is the case of environmental contamination. Specifically, we find the density, distribution function, and moments of the new model. In addition, we derive several properties and transformations related to this distribution. Furthermore, we propose an estimation method for the parameters of the model. Moreover, we conduct a study of its hazard rate focuses in environmental analysis. A computational implementation in R language of the obtained results is discussed. Finally, we present two examples with real data from environmental quality in Chile that illustrate the proposed methodology.


Hazard analysis Likelihood methods R software Skew distributions 



The authors wish to thank the referees for their helpful comments that greatly improved this article. This study was partially supported by a FAPESP grant from Brazil and by DIPUV 29-2006, FONDECYT 1080326, FONDECYT 1090265 and DIUFRO 080061 grants from Chile.


  1. Aarset MV (1987) How to identify a bathtub shaped hazard rate? IEEE Trans Reliab 36:106–108CrossRefGoogle Scholar
  2. Balakrishnan N, Leiva V, Sanhueza A, Cabrera E (2009a) Mixture inverse Gaussian distribution and its transformations, moments and applications. Statistics 43:91–104CrossRefGoogle Scholar
  3. Balakrishnan N, Leiva V, Sanhueza A, Vilca F (2009b) Estimation in the Birnbaum–Saunders distribution based on scale-mixture of normals and the EM-algorithm. Stat Oper Res Trans 33:171–192Google Scholar
  4. Birnbaum ZW, Saunders SC (1968) A probabilistic interpretation of Miner’s rule. SIAM J Appl Math 16:637–652CrossRefGoogle Scholar
  5. Birnbaum ZW, Saunders SC (1969a) A new family of life distributions. J Appl Probab 6:319–327CrossRefGoogle Scholar
  6. Birnbaum ZW, Saunders SC (1969b) Estimation for a family of life distributions with applications to fatigue. J Appl Probab 6:328–347CrossRefGoogle Scholar
  7. Cox DR, Oakes D (1984) Analysis of survival data. Chapman and Hall, LondonGoogle Scholar
  8. Gokhale S, Khare M (2004) A review of deterministic, stochastic and hybrid vehicular exhaust emission models. J Transp Manag 2:59–74Google Scholar
  9. Gómez HW, Olivares-Pacheco JF, Bolfarine H (2009) An extension of the generalized Birnbaum–Saunders distribution. Stat Probab Lett 79:331–338CrossRefGoogle Scholar
  10. Huang S, Qu Y (2006) The loss in power when the test of differential expression is performed under a wrong scale. J Comput Biol 13:786–97CrossRefGoogle Scholar
  11. Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions—vol 2. Wiley, New YorkGoogle Scholar
  12. Kass R, Raftery A (1995) Bayes factors. J Am Stat Soc 90:773–795Google Scholar
  13. Lee L, Helsel D (2005) Statistical analysis of water-quality data containing multiple detection limits: S-language software for regression on order statistics. Comput Geosci 31:1241–1248CrossRefGoogle Scholar
  14. Leiva V, Hernández H, Riquelme M (2006) A new package for the Birnbaum–Saunders distribution. R Journal 6:35–40. Google Scholar
  15. Leiva V, Barros M, Paula GA, Sanhueza A (2008) Generalized Birnbaum–Saunders distributions applied to air pollutant concentration. Environmetrics 19:235–249CrossRefGoogle Scholar
  16. Leiva V, Sanhueza A, Angulo JM (2009a) A length-biased version of the Birnbaum–Saunders distribution with application in water quality. Stoch Environ Res Risk Assess 23:299–307CrossRefGoogle Scholar
  17. Leiva V, Barros M, Paula GA (2009b) Generalized Birnbaum–Saunders models using R. Brazilian Statistical Association, São Paulo, Brazil (in English)Google Scholar
  18. Leiva V, Sanhueza A, Saunders SC (2009c) New developments and applications on life distributions under cumulative damage. CIMFAV Technical Report No. 2009.01.
  19. Leiva V, Vilca F, Balakrishnan N, Sanhueza A (2010) A skewed sinh-normal distribution and its properties and application to air pollution. Comm Stat Theor Methods 39:426–443Google Scholar
  20. Marshall AW, Olkin I (2007) Life distributions. Springer, New YorkGoogle Scholar
  21. Miner MA (1945) Cumulative damage in fatigue. J Appl Mech 12:159–164Google Scholar
  22. Mudholkar GS, Hutson AS (2000) The epsilon-skew-normal distribution for analyzing near-normal data. J Stat Plan Inference 83:291–309CrossRefGoogle Scholar
  23. Podlaski R (2008) Characterization of diameter distribution data in near-natural forests using the Birnbaum–Saunders distribution. Can J For 18:518–527Google Scholar
  24. Owen WJ (2006) A new three-parameter extension to the Birnbaum–Saunders distribution. IEEE Trans Reliab 55:475–479CrossRefGoogle Scholar
  25. R Development Core Team (2008) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, AustriaGoogle Scholar
  26. Rieck JR, Nedelman JR (1991) A log-linear model for the Birnbaum–Saunders distribution. Technometrics 33:51–60CrossRefGoogle Scholar
  27. Saunders SC (1974) A family of random variables closed under reciprocation. J Am Stat Soc 69:533–539Google Scholar
  28. Saunders SC (2007) Reliability, life testing and prediction of services lives. Springer, New YorkGoogle Scholar
  29. Spiegelhalter DJ, Best NG, Carlin BP, van der Linde A (2002) Bayesian measures of model complexity and fit. J R Stat Soc B 64:1–34CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Filidor Vilca
    • 1
  • Antonio Sanhueza
    • 2
  • Víctor Leiva
    • 3
  • George Christakos
    • 4
  1. 1.Departamento de EstatísticaUniversidade Estadual de CampinasSão PauloBrazil
  2. 2.Departamento de Matemática y EstadísticaUniversidad de La FronteraTemucoChile
  3. 3.Departamento de Estadística, CIMFAVUniversidad de ValparaísoValparaísoChile
  4. 4.Department of GeographySan Diego State UniversitySan DiegoUSA

Personalised recommendations