A scalar product for copulas
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Date
2007-10-25T11:55:28Z
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Abstract
We introduce a scalar product for n-dimensional copulas, based on the Sobolev
scalar product for W 1,2
-functions. The corresponding norm has quite remarkable
properties and provides a new geometric framework for copulas. We show that, in
the bivariate case, it measures invertibility properties with respect to the ∗-product
for copulas defined by Darsow et al. The unique copula of minimal norm is the null
element for the ∗-multiplication, whereas the copulas of maximal norm are precisely
the invertible elements.
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Keywords
Copula, Scalar product, Sobolev space