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Journal of Statistical Theory and Practice

, Volume 8, Issue 2, pp 343–366 | Cite as

The Marshall-Olkin Family of Distributions: Mathematical Properties and New Models

  • Gauss M. Cordeiro
  • Artur J. LemonteEmail author
  • Edwin M. M. Ortega
Article
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Abstract

Marshall and Olkin (1997) introduced a general method for obtaining more flexible distributions by adding a new parameter to an existing one, called the Marshall-Olkin family of distributions. In this article, we provide a comprehensive treatment of general mathematical and statistical properties of this family of distributions. Moreover, we propose three new distributions on the basis of the Marshall-Olkin scheme. We discuss maximum likelihood estimation with censored data and provide the observed information matrix. The usefulness of the new family is illustrated by means of an application to real data.

Keywords

Marshall-Olkin family Maximum likelihood estimation Mean deviation Moment Rényi entropy 
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References

  1. Bebbington, M., C. D. Lai, and R. Zitikis. 2007. A flexible Weibull extension. Reliability Eng. System Safety, 92, 719–726.CrossRefGoogle Scholar
  2. Bjerkedal, T. 1960. Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli. Am. J. Hyg., 72, 130–148.Google Scholar
  3. Caroni, C. 2010. Testing for the Marshall-Olkin extended form of the Weibull distribution. Stat. Papers, 51, 325–336.MathSciNetCrossRefGoogle Scholar
  4. Chen, Z. 2000. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Stat. Prob. Lett., 49, 155–162.MathSciNetCrossRefGoogle Scholar
  5. Chen, G., and N. Balakrishnan. 1995. A general purpose approximate goodness-of-fit test. J. Quality Technol., 27, 154–161.CrossRefGoogle Scholar
  6. Cordeiro, G. M., and A. J. Lemonte. 2013. On the Marshall-Olkin extended Weibull distribution. Stat. Papers, 54, 333–353.MathSciNetCrossRefGoogle Scholar
  7. Doornik, J. A. 2009. An object-oriented matrix language—Ox 6. London, UK: Timberlake Consultants Press.Google Scholar
  8. Economou, P., and C. Caroni. 2007. Parametric proportional odds frailty models. Commun. Stat. Simulation Comput., 36, 579–592.MathSciNetCrossRefGoogle Scholar
  9. Eugene, N., C. Lee, and F. Famoye. 2002. Beta-normal distribution and its applications. Commun. Stat. Theory Methods, 31, 497–512.MathSciNetCrossRefGoogle Scholar
  10. García, V. J., E. Gómez-Déniz, and F. J. Vázquez-Polo. 2010. A new skew generalization of the normal distribution: Properties and applications. Comput. Stat. Data Anal., 54, 2021–2034.MathSciNetCrossRefGoogle Scholar
  11. Ghitany, M. E. 2005. Marshall-Olkin extended Pareto distribution and its application. Int. J. Appl. Math., 18, 17–32.MathSciNetzbMATHGoogle Scholar
  12. Ghitany, M. E., F. A. Al-Awadhi, and L. A. Alkhalfan. 2007. Marshall-Olkin extended Lomax distribution and its application to censored data. Commun. Stat. Theory Methods, 36, 1855–1866.MathSciNetCrossRefGoogle Scholar
  13. Ghitany, M. E., E. K. Al-Hussaini, and R. A. AlJarallah. 2005. Marshall-Olkin extended Weibull distribution and its application to censored data. J. Appl. Stat., 32, 1025–1034.MathSciNetCrossRefGoogle Scholar
  14. Ghitany, M. E., and S. Kotz. 2007. Reliability properties of extended linear failure-rate distributions. Prob. Eng. Informational Sci., 21, 441–450.MathSciNetCrossRefGoogle Scholar
  15. Gómez-Déniz, E. 2010. Another generalization of the geometric distribution. Test, 19, 399–415.MathSciNetCrossRefGoogle Scholar
  16. Gupta, R. C., N. Kannan, and A. RayChoudhuri. 1997. Analysis of log-normal survival data. Math. Biosci., 139, 103–115.CrossRefGoogle Scholar
  17. Gupta, R. D., and D. Kundu. 1999. Generalized exponential distributions. Austr. N. Z. J. Stat., 41, 173–188.MathSciNetCrossRefGoogle Scholar
  18. Gupta, R. C., S. Lvin, and C. Peng. 2010. Estimating turning points of the failure rate of the extended Weibull distribution. Comput. Stat. Data Anal., 54, 924–934.MathSciNetCrossRefGoogle Scholar
  19. Gupta, R. D., and C. Peng. 2009. Estimating reliability in proportional odds ratio models. Comput. Stat. Data Anal., 53, 1495–1510.MathSciNetCrossRefGoogle Scholar
  20. Hansen, B. E. 1994. Autoregressive conditional density estimation. Int. Econ. Rev., 35, 705–730.CrossRefGoogle Scholar
  21. Kenney, J. F., and E. S. Keeping. 1962. Mathematics of statistics, Part 1, 3rd ed., 101–102. Princeton, NJ: Princeton University.Google Scholar
  22. 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
  23. Lam, K. F., and T. L. Leung. 2001. Marginal likelihood estimation for proportional odds models with right censored data. Lifetime Data Anal., 7, 39–54.MathSciNetCrossRefGoogle Scholar
  24. Lemonte, A. J. 2013a. A new extension of the Birnbaum-Saunders distribution. Braz. J. Prob. Stat., 27, 133–149.MathSciNetCrossRefGoogle Scholar
  25. Lemonte, A. J. 2013b. A new exponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtub-shaped failure rate function. Comput. Stat. Data Anal., 62, 149–170.MathSciNetCrossRefGoogle Scholar
  26. Lemonte, A. J., W. Barreto-Souza, and G. M. Cordeiro. 2013. The exponentiated Kumaraswamy distribution and its log-transform. Braz. J. Prob. Stat., 27, 31–53.MathSciNetCrossRefGoogle Scholar
  27. Lemonte, A. J., and G. M. Cordeiro. 2011. The exponentiated generalized inverse Gaussian distribution. Stat. Prob. Lett., 81, 506–517.MathSciNetCrossRefGoogle Scholar
  28. Maiti, S. S., and M. Dey. 2012. Tilted normal distribution and its survival properties. J. Data Sci., 10, 225–240.MathSciNetGoogle Scholar
  29. Marshall, A. W., and I. Olkin. 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.MathSciNetCrossRefGoogle Scholar
  30. Marshall, A. W., and I. Olkin. 2007. Life distributions. Structure of nonparametric, semiparametric and parametric families. New York, NY: Springer.zbMATHGoogle Scholar
  31. Moors, J. J. A. 1998. A quantile alternative for kurtosis. J. R. Stat. Soc. D, 37, 25–32.Google Scholar
  32. Mudholkar, G. S., D. K. Srivastava, and M. Freimer. 1995. The exponentiated Weibull family. Technometrics, 37, 436–445.CrossRefGoogle Scholar
  33. Nadarajah, S., and S. Kotz. 2006. The exponentiated type distributions. Acta Appl. Math., 92, 97–111.MathSciNetCrossRefGoogle Scholar
  34. Nanda, A. K., and S. Das. 2012. Stochastic orders of the Marshall-Olkin extended distribution. Stat. Prob. Lett., 82, 295–302.MathSciNetCrossRefGoogle Scholar
  35. Prudnikov, A. P., Y. A. Brychkov, and O. I. Marichev. 1986. Integrals and series, vol. 1–3. Amsterdam, The Netherlands: Gordon and Breach.zbMATHGoogle Scholar
  36. R Development Core Team. 2012. R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing.Google Scholar
  37. Rao, G. S., M. E. Ghitany, and R. R. L. Kantam. 2011. An economic reliability test plan for Marshall-Olkin extended exponential distribution. Appl. Math. Sci., 5, 103–112.MathSciNetzbMATHGoogle Scholar
  38. Ristić, M. M., K. K Jose, and J. Ancy. 2007. A Marshall-Olkin gamma distribution and minification process. STARS, 11, 107–117.Google Scholar
  39. Rubio, F. J., and M. F. J. Steel. 2012. On the Marshall-Olkin transformation as a skewing mechanism. Comput. Stat. Data Anal., 56, 2251–2257.MathSciNetCrossRefGoogle Scholar
  40. Song, K. S. 2001. Rényi information, loglikelihood and an intrinsic distribution measure. J. Stat. Plan. Inference, 93, 51–69.CrossRefGoogle Scholar
  41. Xie, M., Y. Tang, and T. N. Goh. 2002. A modified Weibull extension with bathtub-shaped failure rate function. Reliability Eng. System Safety, 76, 279–285.CrossRefGoogle Scholar
  42. Zhang, T., and M. Xie. 2007. Failure data analysis with extended Weibull distribution. Commun. Stat. Simulation Comput., 36, 579–592.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2014

Authors and Affiliations

  • Gauss M. Cordeiro
    • 1
  • Artur J. Lemonte
    • 1
    • 3
    Email author
  • Edwin M. M. Ortega
    • 2
  1. 1.Departamento de EstaísticaUniversidade Federal de PernambucoRecifeBrazil
  2. 2.Departamento de Ciências Exatas, ESALQUniversidade de São PauloPiracicabaBrazil
  3. 3.Department of StatisticsUniversity of São PauloSão PauloBrazil

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