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Distorted copulas: constructions and tail dependence. (English) Zbl 1194.62075

Summary: Given a copula \(C\), we examine under which conditions on an order isomorphism \(\psi \) of \([0, 1]\) the distortion \(C_{\psi }: [0, 1]^{2} \rightarrow [0, 1]\), \(C_{\psi }(x, y) = \psi \{C[\psi ^{-1}(x), \psi ^{-1}(y)]\}\), is again a copula. In particular, when the copula \(C\) is totally positive of order 2, we give a sufficient condition on \(\psi \) that ensures that any distortion of \(C\) by means of \(\psi \) is again a copula. The presented results allow us to introduce in a more flexible way families of copulas exhibiting different behavior in the tails.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H20 Measures of association (correlation, canonical correlation, etc.)
60E05 Probability distributions: general theory

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