Journal
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
Volume 344, Issue 1, Pages 429-439Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jmaa.2008.02.045
Keywords
copula; scalar product; Sobolev space
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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 of copulas with respect to the *-operation introduced by Darsow et al. (1992). The unique copula of minimal norm is the null element for the *-operation, whereas the copulas of maximal norm are precisely the invertible elements. (c) 2008 Elsevier Inc. All rights reserved.
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