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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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