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Constructing Archimedean copulas from diagonal sections. (English) Zbl 1242.62041
Summary: We introduce a family \(\mathcal F\) of functions called diagonal generators. These are convex functions with the properties of diagonal sections of archimedean copulas. We show that to each diagonal generator \(f\) there corresponds an Archimedean copula \(C^{f}\) with the asymptotic representation \[ C^{f}(u_{1},u_{2}) = \lim_{k\to \infty } f^{k} [f^{ - k}(u_{1})+f^{ - k}(u_{2}) - 1]. \] Moreover, the diagonal section of \(C^{f}\) equals \(f\).
We characterize archimedean copulas in terms of their asymptotic form. We construct a family \(\mathcal F_{F}\) of diagonal generators, induced by a regular distribution function \(F\). We study a differential equation (depending on a function parameter), whose solution is \(F\). We give four applications of diagonal generators: to concordance, quadrant dependence, measures of dependence and convergence of copulas.

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