Dependence of non-continuous random variables
Dependence of non-continuous random variables
Dependence of non-continuous random variables
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Fallacies <strong>Dependence</strong> Concordance Measures Problems & References AppendixSome Theory Behind: <strong>Dependence</strong> Structure in theNon-Continous CaseIn the <strong>continuous</strong> case: C is the cdf. <strong>of</strong> the transformed vector(F 1 (X),F 2 (Y )).Idea: In the <strong>non</strong>-<strong>continuous</strong> case, use a different transformation <strong>of</strong>the marginals:ψ(x,u) := P[X < x] + uP[X = x] = F(x−) + u∆F(x)Result: for a <strong>random</strong> vector (U,V) with uniform marginals whichis independent <strong>of</strong> (X,Y ), (ψ(X,U),ψ(Y ,V)) has uniformmarginals and the corresponding unique copula is a possible copula<strong>of</strong> (X,Y ).Moreover: different dependence structure <strong>of</strong> (U,V) leads todifferent possible copulas <strong>of</strong> (X,Y).J. Nešlehová ETH Zurich<strong>Dependence</strong> <strong>of</strong> <strong>non</strong>-<strong>continuous</strong> <strong>random</strong> <strong>variables</strong>
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