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Extended Marshall–Olkin Model and Its Dual Version

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Marshall ̶ Olkin Distributions - Advances in Theory and Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 141))

Abstract

We propose an extension of the generalized bivariate Marshall–Olkin model assuming dependence between the random variables involved. Probabilistic, aging properties, and survival copula representation of the extended model are obtained and illustrated by examples. Bayesian analysis is performed and possible applications are discussed. A dual version of extended Marshall–Olkin model is introduced and related stochastic order comparisons are presented.

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References

  1. Baglioni, A., Cherubini, U.: Within and between systemic country risk. Theory and evidence from the sovereign crisis in Europe. J. Econ. Dyn. Control 37, 1581–1597 (2013)

    Article  MathSciNet  Google Scholar 

  2. Balakrishnan, N., Lai, C.-D.: Continuous Bivariate Distributions, 2nd edn. Springer, New York (2009)

    MATH  Google Scholar 

  3. Barlow, R., Proschan, F.: Statistical Theory of Reliability and Life Testing. Silver Spring, Maryland (1981)

    Google Scholar 

  4. Barlow, R., Marshall, A., Proschan, F.: Properties of probability distributions with monotone hazard rate. Ann. Math. Stat. 34, 375–389 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bassan, B., Spizzichino, F.: Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. J. Multivar. Anal. 93, 313–339 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Block, H., Basu, A.: A continuous bivariate exponential extension. J. Am. Stat. Assoc. 69, 1031–1037 (1974)

    MATH  MathSciNet  Google Scholar 

  7. Dabrowska, D.: Kaplan-Meier estimate on the plane. Ann. Stat. 16, 1475–1489 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dimitrov, B., Kolev, N.: The BALM copula. Int. J. Stoch. Anal. 2013, 1–6 (2013)

    Article  MathSciNet  Google Scholar 

  9. Fang, R., Li, X.: A note on bivariate dual generalized Marshall-Olkin distributions with applications. Probab. Eng. Inf. Sci. 27, 367–374 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Freund, J.: A bivariate extension of the exponential distribution. J. Am. Stat. Assoc. 56, 971–977 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gumbel, E.: Bivariate exponential distributions. J. Am. Stat. Assoc. 55, 698–707 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gupta, R.C., Gupta, R.D., Gupta, P.L.: Modeling failure time data by Lehman alternatives. Commun. Stat. Theory Methods 27, 887–904 (1988)

    Article  Google Scholar 

  13. Jaschke, S., Siburg, K., Stoimenov, P.: Modelling dependence of extreme events in energy markets using tail copulas. J. Energy Mark. 5(4), 63–80 (2013)

    Google Scholar 

  14. Karlis, D.: ML estimation for multivariate shock models via an EM algorithm. Ann. Inst. Stat. Math. 55, 817–830 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Keilson, J., Sumita, U.: Uniform stochastic ordering and related inequalities. Can. J. Stat. 10, 181–198 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kundu, D., Dey, A.: Estimating the parameters of the Marshall-Olkin bivariate Weibull distribution by EM algorithm. Comput. Stat. Data Anal. 53, 956–965 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kundu, D., Gupta, R.: Modified Sarhan-Balakrishnan singular bivariate distribution. J. Stat. Plan. Inference 140, 525–538 (2010)

    MathSciNet  Google Scholar 

  18. Kundu, D., Gupta, A.: Bayes estimation for the Marshall-Olkin bivariate Weibull distribution. Comput. Stat. Data Anal. 57, 271–281 (2013)

    Article  MathSciNet  Google Scholar 

  19. Lehmann, E.: Some concepts of dependence. Ann. Math. Stat. 37, 1137–1153 (1966)

    Article  MATH  Google Scholar 

  20. Li, X.: Duality of the multivariate distributions of Marshall-Olkin type and tail dependence. Commun. Stat. Theory Methods 37, 1721–1733 (2008)

    Article  MATH  Google Scholar 

  21. Li, X., Pellerey, F.: Generalized Marshall-Olkin distributions and related bivariate aging properties. J. Multivar. Anal. 102, 1399–1409 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lunn, D., Spiegelhalter, D., Thomas, A., Best, N.: The BUGS project: evolution, critique and future directions (with discussion). Stat. Med. 28, 3049–3082 (2009)

    Article  MathSciNet  Google Scholar 

  23. Mai, J.-F., Scherer, M.: Simulating Copulas. Imperial College Press, London (2012)

    Book  MATH  Google Scholar 

  24. Marshall, A., Olkin, I.: A multivariate exponential distribution. J. Am. Stat. Assoc. 62, 30–41 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  25. McNeil, A., Frey, L., Embrechts, P.: Quantitative Risk Management. Princeton University Press, Princeton (2005)

    MATH  Google Scholar 

  26. Meintanis, S.: Test of fit for Marshall-Olkin distributions with applications. J. Stat. Plan. Inference 137, 3954–3963 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. Mirhosseini, S., Amini, M., Kundu, D., Dolati, A.: On a new absolutely continuous bivariate generalized exponential distribution. To appear in Statistical Methods and Applications (2015)

    Google Scholar 

  28. Mohsin, M., Kazianka, H., Pilz, J., Gebhardt, A.: A new bivariate exponential distribution for modeling moderately negative dependence. Stat. Methods Appl. 23, 123–148 (2014)

    Article  MathSciNet  Google Scholar 

  29. Mulero, J., Pellerey, F.: Bivariate aging properties under Archimedean dependence structures. Commun. Stat. Theory Methods 39, 3108–3121 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. Muliere, P., Scarsini, M.: Generalization of Marshall-Olkin type of distributions. Ann. Inst. Stat. Math. 39, 429–441 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  31. Nelsen, R.: An Introduction to Copulas, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  32. Pinto, J.: Deepening the notions of dependence and aging in bivariate probability distributions. Ph.D. thesis, Sao Paulo University (2014)

    Google Scholar 

  33. Pinto, J., Kolev, N.: Sibuya-type bivariate lack of memory property. J. Multivar. Anal. 134, 119–128 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  34. Pinto, J., Kolev, N.: Copula representations for invariant dependence functions. In: Glau, K., Scherer, M., Zagst, R. (eds.) Innovations in Quantitative Risk Management. Springer Series in Probability & Statistics, vol. 99, pp. 411–421. Springer, New York (2015)

    Chapter  Google Scholar 

  35. Sankaran, P.G., Gleeja, V.L.: On bivariate reversed hazard rates. J. Jpn. Stat. Soc. 36, 213–224 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  36. Shaked, M., Shanthikumar, J.: Stochastic Orders. Springer, New York (2007)

    Book  MATH  Google Scholar 

  37. Shoaee, S., Khorram, E.: A new absolute continuous bivariate generalized exponential distribution. J. Stat. Plan. Inference 142, 2203–2220 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  38. Sibuya, M.: Bivariate extreme statistics I. Ann. Inst. Stat. Math. 11, 195–210 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  39. Singpurwalla, N.: Reliability Analysis: A Bayesian Perspective. Wiley, Chichester (2006)

    Google Scholar 

  40. Spiegelhalter, D., Best, N., Carlin, B., van der Linde, A.: Bayesian measures of model complexity and fit (with discussion). J. R. Stat. Soc. Ser. A 64, 583–639 (2002)

    Article  MATH  Google Scholar 

  41. Spiegelhalter, D., Thomas, A., Best, N., Lunn, D.: WinBugs Version 1.4, User Manual. Institute of Public Health and Department of Epidemiology and Public Health. Imperial College School of Medicine, London (2003)

    Google Scholar 

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Acknowledgments

The authors are grateful for the precise referee suggestions which highly improved earlier versions of the manuscript. The first author acknowledges the sponsorship of Central Bank of Brazil under the Graduate Program PPG. The second author is partially supported by FAPESP (2013/07375-0 and 2011/51305-0) and CNPq grants. We are thankful to Leandro Ferreira for the help with OpenBugs software.

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Authors and Affiliations

  1. Department of Statistics, University of São Paulo, Sao Paulo, Brazil

    Jayme Pinto & Nikolai Kolev

Authors
  1. Jayme Pinto
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  2. Nikolai Kolev
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Corresponding author

Correspondence to Jayme Pinto .

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Editors and Affiliations

  1. Department of Statistics, University of Bologna, Bologna, Italy

    Umberto Cherubini

  2. Free University of Bozen-Bolzano, Faculty of Economics and Management, Bolzano, Italy

    Fabrizio Durante

  3. University of Bologna, Department of Statistics, Bologna, Italy

    Sabrina Mulinacci

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Pinto, J., Kolev, N. (2015). Extended Marshall–Olkin Model and Its Dual Version. In: Cherubini, U., Durante, F., Mulinacci, S. (eds) Marshall ̶ Olkin Distributions - Advances in Theory and Applications. Springer Proceedings in Mathematics & Statistics, vol 141. Springer, Cham. https://doi.org/10.1007/978-3-319-19039-6_6

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