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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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