Marshall–Olkin exponential distribution

Short description: Probability distribution in applied statistics

Marshall–Olkin exponentialSupport|  [math]\displaystyle{ x\in [0, \infty)^b }[/math]

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In applied statistics, the Marshall–Olkin exponential distribution is any member of a certain family of continuous multivariate probability distributions with positive-valued components. It was introduced by Albert W. Marshall and Ingram Olkin.[1] One of its main uses is in reliability theory, where the Marshall–Olkin copula models the dependence between random variables subjected to external shocks. [2] [3]

## Definition

Let [math]\displaystyle{ \\{E_B : \varnothing \ne B\subset \\{1,2,\ldots,b\\}\\} }[/math] be a set of independent, exponentially distributed random variables, where [math]\displaystyle{ E_B }[/math] has mean [math]\displaystyle{ 1/\lambda_B }[/math]. Let

[math]\displaystyle{ T_j=\min\\{E_B:j\in B\\},\ \ j=1,\ldots,b. }[/math]

The joint distribution of [math]\displaystyle{ T=(T_1,\ldots,T_b) }[/math] is called the Marshall–Olkin exponential distribution with parameters [math]\displaystyle{ \\{\lambda _B,B\subset \\{1,2,\ldots,b\\}\\}. }[/math]

### Concrete example

Suppose b = 3\. Then there are seven nonempty subsets of { 1, ..., b } = { 1, 2, 3 }; hence seven different exponential random variables:

[math]\displaystyle{ E_{\\{1\\}}, E_{\\{2\\}}, E_{\\{3\\}}, E_{\\{1,2\\}}, E_{\\{1,3\\}}, E_{\\{2,3\\}}, E_{\\{1,2,3\\}} }[/math]

Then we have:

[math]\displaystyle{ \begin{align} T_1 & = \min\\{ E_{\\{1\\}}, E_{\\{1,2\\}}, E_{\\{1,3\\}}, E_{\\{1,2,3\\}} \\} \\\ T_2 & = \min\\{ E_{\\{2\\}}, E_{\\{1,2\\}}, E_{\\{2,3\\}}, E_{\\{1,2,3\\}} \\} \\\ T_3 & = \min\\{ E_{\\{3\\}}, E_{\\{1,3\\}}, E_{\\{2,3\\}}, E_{\\{1,2,3\\}} \\} \\\ \end{align} }[/math]

## References

1. ↑ Marshall, Albert W.; Olkin, Ingram (1967), "A multivariate exponential distribution", Journal of the American Statistical Association 62 (317): 30–49, doi:10.2307/2282907
2. ↑ Botev, Z.; L'Ecuyer, P.; Simard, R.; Tuffin, B. (2016), "Static network reliability estimation under the Marshall-Olkin copula", ACM Transactions on Modeling and Computer Simulation 26 (2): No.14, doi:10.1145/2775106
3. ↑ Durante, F.; Girard, S.; Mazo, G. (2016), "Marshall--Olkin type copulas generated by a global shock", Journal of Computational and Applied Mathematics 296: 638–648, doi:10.1016/j.cam.2015.10.022

* Xu M, Xu S. "An Extended Stochastic Model for Quantitative Security Analysis of Networked Systems". Internet Mathematics, 2012, 8(3): 288–320.

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