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A family of bivariate exponential distributions and their copulas

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Abstract

In the present paper we derive a family of bivariate exponential distributions based on an extended lack of memory property of a class of univariate distributions. These bivariate exponential laws are time transformed exponential models possessing Archimedean copulas. The bivariate aging properties and various dependence relationships are characterized in terms of the univariate aging concepts of the baseline distributions from which the bivariate models are generated.

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References

  • Averous, J. and Dortet-Bernadit, J. (2005). Dependence of Archimedean copulas and ageing properties of their generators. Sankhya 66, 607–620.

    Google Scholar 

  • Bassan, B. and Spizzichino, F. (2005). Relations among univariate ageing, bivariate ageing and dependence for exchangeable lifetimes. J. Multivariate Anal. 93, 313–339.

    Article  MATH  MathSciNet  Google Scholar 

  • Bassan, B., Kochar, S.C. and Spizzichino, F. (2002). Some bivariate notions of IFR and DMRL and related properties. J. Appl. Probab. 39, 533–544.

    Article  MATH  MathSciNet  Google Scholar 

  • Block, H.W. and Basu, A.P. (1974). A continuous bivariate exponential extension. J. Amer. Statist. Assoc. 69, 1031–1037.

    MATH  MathSciNet  Google Scholar 

  • Cowan, R. (1987). A bivariate exponential distribution arising in random geometry. Ann. Inst. Statist. Math. 39, 103–111.

    Article  MATH  MathSciNet  Google Scholar 

  • Downton, F. (1970). Bivariate exponential distribution in reliability theory. J. R. Stat. Soc. 32, 408–417.

    MATH  MathSciNet  Google Scholar 

  • Durante, F., Foschi, R. and Spizzichino, F. (2010). Ageing functions and multivariate notions of NBU and IFR. Probab. Engrg. Inform. Sci. 24, 263–278.

    Article  MATH  MathSciNet  Google Scholar 

  • Freund, J. (1961). A bivariate extension of exponential distribution. J. Amer. Statist. Assoc. 56, 971–977.

    Article  MATH  MathSciNet  Google Scholar 

  • Friday, D.S. and Patil, G.P. (1977). A Bivariate Exponential Model with Applications to Reliability and Computer Generation of Random Variables. In Theory and Application of Reliability, (C.P. Tsokos and I.N. Shimi, eds.), vol. 1, pp. 527–549.

  • Gumbel, E.J. (1960). Bivariate exponential distributions. J. Amer. Statist. Assoc. 55, 698–707.

    Article  MATH  MathSciNet  Google Scholar 

  • Hawkes, A.G. (1972). A bivariate exponential distribution with applications to reliability. J. R. Stat. Soc. 34, 129–131.

    MATH  Google Scholar 

  • Hayakawa, Y. (1994). The construction of new bivariate exponential distributions from a Bayesian perspective. J. Amer. Statist. Assoc. 89, 1044–1049.

    Article  MATH  MathSciNet  Google Scholar 

  • Iyer, S.K., Manjunath, D. and Manivasakan, R. (2002). Bivariate exponential distribution using linear structures. Sankhya 64, 156–166.

    MATH  MathSciNet  Google Scholar 

  • Johnson, N.L. and Kotz, S. (1975). A vector valued multivariate hazard rate. J. Multivariate Anal. 5, 53–66.

    Article  MATH  MathSciNet  Google Scholar 

  • Kagan, A., Linnik, Y. and Rao, C.R. (1973). Characterization Problems in Mathematical Statistics. John Wiley, New York.

    MATH  Google Scholar 

  • Marshall, A.W. and Olkin, I. (1967). A multivariate exponential distribution. J. Amer. Statist. Assoc. 62, 30–44.

    Article  MATH  MathSciNet  Google Scholar 

  • Nagao, M. and Kadoya, M. (1971). Two-variate exponential distribution and its numerical table for engineering application. Disas. Prey. Res. Inst. 20, 183–215.

    Google Scholar 

  • Nelsen, R.B. (2006). An Introduction to Copulas, 2nd edn. Springer-Verlag.

  • Paulson, A.S. (1974). A characterization of the exponential distribution and a bivariate exponential distribution. Sankhya 35, 69–78.

    MathSciNet  Google Scholar 

  • Pellerey, F. (2008). On univariate an bivariate ageing for dependent lifetimes with Archimedean copulas. Kybernetrika 44, 795–806.

    MATH  MathSciNet  Google Scholar 

  • Raftery, A.E. (1985). Some properties of a new continuous bivariate exponential distribution. Statist. Decisions 2, 53–58. Supplement.

    MathSciNet  Google Scholar 

  • Regoli, G. (2009). A class of bivariate exponential distributions. J. Multivariate Anal. 100, 1261–1269.

    Article  MATH  MathSciNet  Google Scholar 

  • Ryu, K. (1993). An extension of the Marshall–Olkin’s bivariate exponential distribution. J. Amer. Statist. Assoc. 88, 1458–1465.

    Article  MATH  MathSciNet  Google Scholar 

  • Sarkar, S.K. (1987). A continuous bivariate exponential distribution. J. Amer. Statist. Assoc. 82, 667–675.

    Article  MATH  MathSciNet  Google Scholar 

  • Spizzichino, F. (2009). Concept of duality for multivariate survival models. Fuzzy Sets Syst. 160, 325–333.

    Article  MATH  MathSciNet  Google Scholar 

  • Spizzichino, F. (2010). Semi Copulas and Interpretation of Coincidence Between Stochastic Dependence and Ageing, Copula Theory and its Applications. In Lecture Notes in Statistics, (P. Jawarski, F. Durante, W. Hardle and T. Rychlik, eds.), Vol. 198, pp. 237–254. Springer-Verlag.

  • Tosch, T.J. and Holmes, P.T. (1980). A bivariate failure model. J. Amer. Statist. Assoc. 75, 415–417.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Statistics, Cochin University of Science and Technology, Cochin, 682 022, India

    N. Unnikrishnan Nair & P. G. Sankaran

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  1. N. Unnikrishnan Nair
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  2. P. G. Sankaran
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Correspondence to P. G. Sankaran.

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Unnikrishnan Nair, N., Sankaran, P.G. A family of bivariate exponential distributions and their copulas. Sankhya B 76, 1–18 (2014). https://doi.org/10.1007/s13571-013-0067-2

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  • DOI: https://doi.org/10.1007/s13571-013-0067-2

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