Dependence of non-continuous random variables

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Fallacies Dependence Concordance Measures Problems & References AppendixPathological Example2/3q1/3• (p,q) ∈ [0,1/3] × [0,1/3]:perfect positive dependence• (p,q) = (1/ √ 3,1/ √ 3):independence• (p,q) ∈ [2/3,1] × [2/3,1]:perfect negative dependence1/3 p 2/3J. Nešlehová ETH ZurichDependence of non-continuous random variables
Fallacies Dependence Concordance Measures Problems & References AppendixFurther Pitfalls• The copula is not unique; i.e. several possible copulas exist• The dependence structures of F are generally not the same asthe dependence structures of the possible copulasF(x,y) ≤ F ∗ (x,y) ∀x,y ∈ R ⇏ C(u,v) ≤ C ∗ (u,v) ∀u,v ∈ [0,1]• Possible copulas remain invariant under strictly increasing andcontinuous transformations, but do not necessarily change inthe same way as the unique copula in the “continuous” case ifat least one of the transformations is decreasing• Weak convergence of F n does not imply the point-wiseconvergence of the corresponding possible copulas• Any measure of association which depends only on the copulaof F is a constantJ. Nešlehová ETH ZurichDependence of non-continuous random variables
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Dependence of non-continuous random variables

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