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Multivariate Generalized Marshall–Olkin Distributions and Copulas

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Abstract

The multivariate generalized Marshall–Olkin distributions, which include the multivariate Marshall–Olkin exponential distribution due to Marshall and Olkin (J Am Stat Assoc 62:30–41, 1967) and multivariate Marshall–Olkin type distribution due to Muliere and Scarsini (Ann Inst Stat Math 39:429–441, 1987) as special cases, are studied in this paper. We derive the survival copula and the upper/lower orthant dependence coefficient, build the order of these survival copulas, and investigate the evolution of dependence of the residual life with respect to age. The main conclusions developed here are both nice extensions of the main results in Li (Commun Stat Theory Methods 37:1721–1733, 2008a, Methodol Comput Appl Probab 10:39–54, 2008b) and high dimensional generalizations of some results on the bivariate generalized Marshall–Olkin distributions in Li and Pellerey (J Multivar Anal 102:1399–1409, 2011).

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References

  • Balakrishnan N, Lai CD (2009) Continuous bivariate distributions, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Barlow RE, Proschan F (1981) Statistical theory of reliability and life testing. To begin with, Silver Spring

  • Boyd SP, Vandenberghe L (2004) Convex optimization. Cambridge University Press, New York

    Book  MATH  Google Scholar 

  • Denuit M, Dhaene J, Goovaerts M, Kaas R (2005) Actuarial theory for dependent risks. Wiley, New York

    Book  Google Scholar 

  • Frees EW, Carriere J, Valdez E (1996) Annuity valuation with dependent mortality. J Risk Insur 63:229–261

    Article  Google Scholar 

  • Galambos J, Kotz S (1978) Characterizations of probability distributions. Springer, Berlin

    MATH  Google Scholar 

  • Jaworski P, Durante F, Härdle W, Rychlik T (2010) Copula theory and its applications. Springer, New York

    Book  MATH  Google Scholar 

  • Joe H (1993) Parametric families of multivariate distributions with given maginals. J Multivar Anal 46:262–282

    Article  MATH  MathSciNet  Google Scholar 

  • Joe H (1997) Multivariate models and dependence concepts. Chapman & Hall, London

    Book  MATH  Google Scholar 

  • Kotz S, Balakrishnan N, Johnson NL (2000) Continuous multivariate distributions. In: Models and applications, vol 1. Wiley, New York

    Google Scholar 

  • Lai CD, Xie M (2006) Stochastic ageing and dependence for reliability. Springer, New York

    MATH  Google Scholar 

  • Lawless JF (2003) Statistical models and methods for lifetime data, 2nd edn. Wiley, Hoboken

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Li H (2008b) Tail dependence comparison of survival Marshall–Olkin copulas. Methodol Comput Appl Probab 10:39–54

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Lu JC (1989) Weibull extension of the Freund and Marshall–Olkin bivariate exponential model. IEEE Trans Reliab 38:615–619

    Article  MATH  Google Scholar 

  • Mai J, Scherer M (2010) The Pickands representation of survival Marshall–Olkin copulas. Stat Probab Lett 80:357–360

    Article  MATH  MathSciNet  Google Scholar 

  • Mai J, Scherer M (2011) Reparameterizing Marshall–Olkin copulas with applications to sampling. J Stat Comput Simul 81:59–78

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Marshall AW, Olkin I (2007) Life distributions. Springer, New York

    MATH  Google Scholar 

  • McNeil AJ, Frey R, Embrechts P (2005) Quantitative risk management. Princeton University Press, Princeton, New York

    MATH  Google Scholar 

  • Muliere P, Scarsini M (1987) Characterization of a Marshall–Olkin type class of distributions. Ann Inst Stat Math 39:429–441

    Article  MATH  MathSciNet  Google Scholar 

  • Nelsen RB (2006) An introduction to copulas, 2nd edn, Springer, New York

    MATH  Google Scholar 

  • Sarhan AM, Hamilton DC, Smitha B, Kundub D (2010) The bivariate generalized linear failure rate distribution and its multivariate extension. Comput Stat Data Anal 55:644–654

    Article  Google Scholar 

  • Scarsini M (1984) One measure of concordance. Stochastica 8:201–218

    MATH  MathSciNet  Google Scholar 

  • Schmidt R (2002) Tail dependence for elliptically contoured distributions. Math Methods Oper Res 55:301–327

    Article  MATH  MathSciNet  Google Scholar 

  • Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New York

    Book  MATH  Google Scholar 

  • Wu C (1997) New characterization of Marshall–Olkin type distributions via bivariate random summation scheme. Stat Probab Lett 34:171–178

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    Jianhua Lin & Xiaohu Li

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  1. Jianhua Lin
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  2. Xiaohu Li
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Correspondence to Xiaohu Li.

Additional information

Supported by National Natural Science Foundation of China (11171278) and the Fundamental Research Funds for the Central Universities of China (2010121005).

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Lin, J., Li, X. Multivariate Generalized Marshall–Olkin Distributions and Copulas. Methodol Comput Appl Probab 16, 53–78 (2014). https://doi.org/10.1007/s11009-012-9297-4

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  • DOI: https://doi.org/10.1007/s11009-012-9297-4

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