Marshall–Olkin exponential distribution

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 $\{E_{B}:\varnothing \neq B\subset \{1,2,\ldots ,b\}\}$ be a set of independent, exponentially distributed random variables, where $E_{B}$ has mean $1/\lambda _{B}$ . Let

T_{j}=\min\{E_{B}:j\in B\},\ \ j=1,\ldots ,b.

The joint distribution of $T=(T_{1},\ldots ,T_{b})$ is called the Marshall–Olkin exponential distribution with parameters $\{\lambda _{B},B\subset \{1,2,\ldots ,b\}\}.$

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:

{\begin{aligned}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{aligned}}

References

Marshall, Albert W.; Olkin, Ingram (1967), "A multivariate exponential distribution", Journal of the American Statistical Association, 62 (317): 30–49, doi:10.2307/2282907, MR 0215400
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
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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[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, MR 0215400

[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