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Marshall–Olkin exponential distribution

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Probability distribution in applied statistics
Marshall–Olkin exponential
Support x ∈ [ 0 , ∞ ) b {\displaystyle x\in [0,\infty )^{b}}

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. One of its main uses is in reliability theory, where the Marshall–Olkin copula models the dependence between random variables subjected to external shocks.

Definition

Let { E B : ∅ ≠ B ⊂ { 1 , 2 , … , b } } {\displaystyle \{E_{B}:\varnothing \neq B\subset \{1,2,\ldots ,b\}\}} be a set of independent, exponentially distributed random variables, where E B {\displaystyle E_{B}} has mean 1 / λ B {\displaystyle 1/\lambda _{B}} . Let

T j = min { E B : j ∈ B } ,     j = 1 , … , b . {\displaystyle T_{j}=\min\{E_{B}:j\in B\},\ \ j=1,\ldots ,b.}

The joint distribution of T = ( T 1 , … , T b ) {\displaystyle T=(T_{1},\ldots ,T_{b})} is called the Marshall–Olkin exponential distribution with parameters { λ B , B ⊂ { 1 , 2 , … , b } } . {\displaystyle \{\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 } {\displaystyle E_{\{1\}},E_{\{2\}},E_{\{3\}},E_{\{1,2\}},E_{\{1,3\}},E_{\{2,3\}},E_{\{1,2,3\}}}

Then we have:

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 } } {\displaystyle {\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

  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, JSTOR 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, S2CID 16677453
  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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