Marshall-Olkin Bivariate Distribution.


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The Marshall–Olkin bivariate distribution, introduced by Albert W. Marshall and Ingram Olkin [1], is a family of continuous multivariate probability distributions that is an extension of the bivariate family of distributions with an extra shape parameter α.

It is defined by the proper survival function

Marshall-Olkin

Where (X, Y) is a random vector with joint survival function F(x, y).

Uses of the Marshall–Olkin bivariate distribution

The Marshall–Olkin bivariate distribution gives a wider range of behavior than the bivariate family of distributions. Specifically, the extra parameter α can model real-life situations better than the basic model [2]. In the classical Marshall-Olkin model, two components are subjected to random shocks from three different sources. Many extensions of the basic distribution have been proposed, including:

  • Ryu [3], who extended the basic model to a bivariate absolutely continuous distribution, which does not have the Marshall-Olkin lack of memory property.
  • Aly & Abuelamayem [4], who developed the Multivariate Inverted Kumaraswamy Distribution: Derivation and Estimation as a new Marshall–Olkin bivariate distribution for efficient application in several fields.  Parameters are found with both maximum likelihood and Bayesian approaches, which the authors state could be applied to all Marshall–Olkin multivariate distributions.

Thus, the Marshall–Olkin bivariate distribution is one member of a wider family of generalized Marshall–Olkin distributions. It is the most popular of all the bivariate lifetime distributions [4].

References

[1] Marshall, A. N., Olkin, I., (1997) A new method for adding a parameter to a family of distributions with applications to the exponential and Weibull families, Biometrica 84, 641-652.

[2] Jose, K.K. (2011). Marshall-Olkin Family of Distributions and their applications in reliability theory, time series modeling and stress-strength analysis. Int. Statistical Inst.: Proc. 58th World Statistical Congress, Dublin (Session CPS005) p.3918

[3] Ryu, K. An extension of Marshall and Olkin bivariate exponential distribution. J. Amer. Statist. Assoc., 88 (1993), pp. 1458-1465

[4] Aly, H. & Abuelamayem, O. (2020). Multivariate Inverted Kumaraswamy Distribution: Derivation and Estimation. Mathematical Problems in Engineering.