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Journal of Statistical Theory and Practice

, Volume 6, Issue 4, pp 745–759 | Cite as

The Kumaraswamy Birnbaum-Saunders Distribution

  • Helton Saulo
  • Jeremias Leão
  • Marcelo Bourguignon
Article

Abstract

Motivated by the recent work of Cordeiro and Castro (2011), we study the Kumaraswamy Birnbaum-Saunders (Kw-\({\mathcal B}{\mathcal S}\)) distribution. This distribution provides an enormous flexibility in modeling heavy-tailed and skewed data. We derive some mathematical properties of the Kw-\({\mathcal B}{\mathcal S}\) including moments, quantile function, average lifetime function, mean residual lifetime function, and order statistics. In addition, we discuss maximum likelihood estimation of the model parameters.

Keywords

Birnbaum-Saunders distribution Distribution theory Kumaraswamy Birnbaum-Saunders distribution 

AMS Subject Classification

33C90 62E99 

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References

  1. Amoroso, L. 1925. Ricerche intorno alla curva dei redditi. Ann. Math., 2, 123–159.MathSciNetzbMATHGoogle Scholar
  2. Birnbaum, Z. W., and S. C. Saunders. 1969. A new family of life distributions. J. Appl Probability, 6, 319–327.MathSciNetCrossRefGoogle Scholar
  3. Cordeiro, G. M., and M. Castro. 2011. A new family of generalized distributions. J. Stat. Comput. Simulation, 81(7), 883–898.MathSciNetCrossRefGoogle Scholar
  4. Cordeiro, G. M., A. J. Lemonte, and E. M. M. Ortega. 2011. An extended fatigue life distribution. Statistics. doi:10.1080/02331888.2011.617447CrossRefMathSciNetzbMATHGoogle Scholar
  5. Cordeiro, G. M., S. Nadarajah, and E. M. M. Ortega. 2012. The Kumaraswamy Gumbel distribution. Stat. Methods Appl., 21, 139–168.MathSciNetCrossRefGoogle Scholar
  6. Cordeiro, G. M., E. M. M. Ortega, and S. Nadarajah. 2010. The Kumaraswamy Weibull distribution with application to failure data. J. Franklin Inst., 347, 1399–1429.MathSciNetCrossRefGoogle Scholar
  7. Eaton, J. W., D. Bateman, and S. Hauberg. 2008. GNU Octave manual version 3. Bristol, Network Theory Limited.Google Scholar
  8. Eugene, N., C. Lee, and F. Famoye. 2002. Beta-normal distribution and its applications. Commun. Stat. Theory Methods 31, 497–512.MathSciNetCrossRefGoogle Scholar
  9. Good, I. J. 1953. The population frequencies of the species and the estimation of population parameters. Biometrika, 40, 237–260.MathSciNetCrossRefGoogle Scholar
  10. Greenwood, J. A., J. M. Landwehr, N. C. Matalas, and J. R. Wallis. 1979. Probability weighted moments: Definition and relation to parameters of several distributions expressable in inverse form. Water Resources Res., 15, 1049–1054.CrossRefGoogle Scholar
  11. Guess, F., and F. Proschan. 1988. Mean residual life: Theory and applications. In Handbook of statistics, reliability and qualiy control, ed. P. R. Krishnaiah and C. R. Rao, vol. 7, 215–224. Amsterdam, North-Holland.CrossRefGoogle Scholar
  12. Gupta, R. C., P. L. Gupta, and R. D. Gupta. 1998. Modeling failure time data by Lehman alternatives. Commun. Stat. Theory Methods, 27, 887–904.MathSciNetCrossRefGoogle Scholar
  13. Hoskings, J. R. M., and J. R. Wallis. 1987. Parameter and quantile estimation for the generalized Pareto distribution. Technometrics, 29, 339–349.MathSciNetCrossRefGoogle Scholar
  14. Jones, M. C., 2009. A beta-type distribution with some tractability advantages. Stat. Methodol., 6, 70–81.MathSciNetCrossRefGoogle Scholar
  15. Kenney, J. F., and E. S. Keeping. 1962. Mathematics of statistics, 3rd ed., vol. 1. Princeton, Van Nostrand.zbMATHGoogle Scholar
  16. Kumaraswamy, P. 1980. Generalized probability density-function for double-bounded random-processes. J. Hydrol., 462, 79–88.CrossRefGoogle Scholar
  17. Kundu, D., N. Kannan, and N. Balakrishnan. 2008. On the hazard function of Birnbaum-Saunders distribution and associated inference. Comput. Stat. Data Anal., 52, 2692–2702.MathSciNetCrossRefGoogle Scholar
  18. McDonald, J. B. 1984. Some generalized functions for the size distribution of income. Econometrica, 52, 647–663.CrossRefGoogle Scholar
  19. Moors, J. J. 1998. A quantile alternative for kurtosis. J. R. Stat. Soc. D, 37, 25–32.Google Scholar
  20. Nadarajah, S., G. M. Cordeiro, and E. M. M. Ortega. 2011. General results for the Kumaraswamy-G distribution. J. Stat. Comput. Simulation, doi:10.1080/00949655.2011.562504CrossRefMathSciNetzbMATHGoogle Scholar
  21. Pascoa, A. R. M., E. M. M. Ortega, and G. M. Cordeiro. 2011. The Kumaraswamy generalized gamma distribution with application in survival analysis. Stat. Methodol., 8, 411–433.MathSciNetCrossRefGoogle Scholar
  22. Rieck, J. R. 1999. A moment-generating function with application to the Birnbaum-Saunders distribtuion. Commun. Stat. Theory Methods, 28, 2213–2222.CrossRefGoogle Scholar
  23. Saunders, S. C. 1974. A family of random variables closed under reciprocation. J. Am. Stat. Assoc., 69, 533–539.CrossRefGoogle Scholar
  24. Team, R. D. C. 2011. R: A language and environment for statistical computing. Vienna, Austria, R Foundation for Statistical Computing.Google Scholar
  25. Watson, G. N. 1995. A treatise on the theory of Bessel function, 2nd ed. Cambridge, UK, Cambridge University Press.zbMATHGoogle Scholar

Copyright information

© Grace Scientific Publishing 2012

Authors and Affiliations

  • Helton Saulo
    • 1
  • Jeremias Leão
    • 2
  • Marcelo Bourguignon
    • 3
  1. 1.Departamento de EconomiaUniversidade Federal do Rio Grande do SulPorto AlegreBrazil
  2. 2.Departamento de EstatísticaUniversidade Federal do PiauíTeresinaBrazil
  3. 3.Departamento de EstatísticaUniversidade Federal de Pernambuco, Cidade UniversitáriaRecifeBrazil

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