Marshall–Olkin exponential distribution

Short description: Probability distribution in applied statistics

Marshall–Olkin exponentialSupport|  x∈[0,∞)b

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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] [4]

## Definition

Let {EB:∅≠B⊂{1,2,…,b}} be a set of independent, exponentially distributed random variables, where EB has mean 1/λB. Let

Tj=min⁡{EB:j∈B}, j=1,…,b.

The joint distribution of T=(T1,…,Tb) is called the Marshall–Olkin exponential distribution with parameters {λB,B⊂{1,2,…,b}}.

### Concrete example

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

E{1},E{2},E{3},E{1,2},E{1,3},E{2,3},E{1,2,3}

Then we have:

T1=min⁡{E{1},E{1,2},E{1,3},E{1,2,3}}T2=min⁡{E{2},E{1,2},E{2,3},E{1,2,3}}T3=min⁡{E{3},E{1,3},E{2,3},E{1,2,3}}

## 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. ↑ Barrera, J.; Cancela, H.; Moreno, E. (2015), "Topological optimization of reliable networks under dependent failures", Operations Research Letters 43 (2): 132-136, doi:10.1016/j.orl.2014.12.014
3. ↑ 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
4. ↑ 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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