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Some integration-by-parts formulas involving 2-copulas. (English) Zbl 1322.60008
Cuadras, Carles M. (ed.) et al., Distributions with given marginals and statistical modelling. Papers presented at the meeting, Barcelona, Spain, July 17–20, 2000. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-0914-3/hbk). 153-159 (2002).
Summary: We note examples of probabilistic interpretations of integrals involving 2-copulas. We then use the theory of strong convergence of copulas to justify an integration-by-parts formula involving 2-copulas,
\[ \int_{I^2} f(A)\,dB=\int^1_0 f(t)\,dt-\int_{I^2}-f'(A)D_1AD_2B=\int^a_0 f(t)\,dt-\int_{I_2} f'(A)D_2AD_1B \] where \(A\) and \(B\) are arbitrary 2-copulas and \(f\) is continuously differentiable.
For the entire collection see [Zbl 1054.62002].

60E05 Probability distributions: general theory
62E10 Characterization and structure theory of statistical distributions
62H05 Characterization and structure theory for multivariate probability distributions; copulas
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