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]
^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
^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.
February 17, 2024
marshall, olkin, exponential, distribution, applied, statistics, member, certain, family, continuous, multivariate, probability, distributions, with, positive, valued, components, introduced, albert, marshall, ingram, olkin, main, uses, reliability, theory, wh. 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 Marshall Olkin exponentialSupportx 0 b displaystyle x in 0 infty b Definition editLet E B B 1 2 b displaystyle E B varnothing neq B subset 1 2 ldots b nbsp be a set of independent exponentially distributed random variables where E B displaystyle E B nbsp has mean 1 l B displaystyle 1 lambda B nbsp Let T j min E B j B j 1 b displaystyle T j min E B j in B j 1 ldots b nbsp The joint distribution of T T 1 T b displaystyle T T 1 ldots T b nbsp is called the Marshall Olkin exponential distribution with parameters l B B 1 2 b displaystyle lambda B B subset 1 2 ldots b nbsp Concrete example edit 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 nbsp 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 amp min E 1 E 1 2 E 1 3 E 1 2 3 T 2 amp min E 2 E 1 2 E 2 3 E 1 2 3 T 3 amp min E 3 E 1 3 E 2 3 E 1 2 3 end aligned nbsp References edit 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 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 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 Retrieved from https en wikipedia org w index php title Marshall Olkin exponential distribution amp oldid 1197518766, wikipedia, wiki, book, books, library,