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A biconvex form for copulas. (English) Zbl 1349.62175
Summary: We study the integration of a copula with respect to the probability measure generated by another copula. To this end, we consider the map $$[\cdot, \cdot] : \mathcal{C}\times\mathcal{C}\to\mathbb{R}$$ given by $[C,D]:=\int_{[0,1]^d} C(\mathbf{u})dQ^D(\mathbf{u})$ where $$\mathcal{C}$$ denotes the collection of all $$d$$-dimensional copulas and $$Q^D$$ denotes the probability measures associated with the copula $$D$$. Specifically, this is of interest since several measures of concordance such as Kendall’s tau, Spearman’s rho and Gini’s gamma can be expressed in terms of the map $$[\cdot, \cdot]$$. Quite generally, the map $$[\cdot, \cdot]$$ can be applied to construct and investigate measures of concordance.

##### MSC:
 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62H20 Measures of association (correlation, canonical correlation, etc.)
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##### References:
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