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Marshall–Olkin exponential distribution

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Marshall–Olkin exponential
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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

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Let be a set of independent, exponentially distributed random variables, where has mean . Let

The joint distribution of is called the Marshall–Olkin exponential distribution with parameters

Concrete example

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

Then we have:

References

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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, 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.