A look at copulas in a curved mirror
We extend the seminal work of Roger Nelsen on symmetry-related properties and the degree of asymmetry of copulas, by reattributing the role the diagonal plays as axis of symmetry to a continuous strictly increasing curve in the unit square. First, we make explicit the geometrical notion of symmetry of a function on the unit square with respect to a curve. Next, we provide a measure for quantifying to what extent a quasi-copula or copula can be regarded asymmetric with respect to a given curve. Finally, we derive a lower and upper bound on the degree of asymmetry a quasi-copula can possess with respect to a given curve and show that each bound is sharp within the class of copulas.
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