Copula Theory: an Introduction
Copula Theory: an Introduction
Fabrizio Durante and Carlo Sempi
Abstract In this survey we review the most important properties of copulas, several families of copulas that have appeared in the literature, and which have been applied in various fields, and several methods of constructing multivariate copulas.
This version: September 14, 2009
1 Historical introduction
The history of copulas may be said to begin with Fr´ chet [69]. He studied the fole lowing problem, which is stated here in dimension 2: given the distribution functions F1 and F2 of two random variables X1 and X2 defined on the same probability space Ω ,F,P , what can be said about the set Γ F1 ,F2 of the bivariate d.f.’s whose marginals are F1 and F2 ? It is immediate to note that the set Γ F1 ,F2 , now called the Fr´ chet class of F1 and F2 , is not empty since, if X1 and X2 are independent, e then the distribution function x1 ,x2 F x1 ,x2 = F1 x1 F2 x2 always belongs to Γ F1 ,F2 . But, it was not clear which the other elements of Γ F1 ,F2 were. Preliminary studies about this problem were conducted in [64, 70, 88] (see also [30, 181] for a historical overview). But, in 1959, Sklar obtained the deepest result in this respect, by introducing the notion, and the name, of a copula, and proving the theorem that now bears his name [191]. In his own words [193]:
Fabrizio Durante Department of Knowledge-Based Mathematical Systems, Johannes Kepler University Linz, Austria e-mail: fabrizio.durante@jku.at Carlo Sempi Dipartimento di Matematica “Ennio De Giorgi” Universit` del Salento, Lecce, Italy e-mail: carlo.sempi@unisalento.it a
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Fabrizio Durante and Carlo Sempi [...] In the meantime, Bert (Schweizer) and I had been making progress in our work on statistical metric spaces, to the extent that Menger suggested it would be worthwhile for us to communicate our results to Fr´ chet. We did: Fr´ chet was interested, and asked us to e e write an announcement for the Comptes Rendus [183].
Fabrizio Durante and Carlo Sempi
Abstract In this survey we review the most important properties of copulas, several families of copulas that have appeared in the literature, and which have been applied in various fields, and several methods of constructing multivariate copulas.
This version: September 14, 2009
1 Historical introduction
The history of copulas may be said to begin with Fr´ chet [69]. He studied the fole lowing problem, which is stated here in dimension 2: given the distribution functions F1 and F2 of two random variables X1 and X2 defined on the same probability space Ω ,F,P , what can be said about the set Γ F1 ,F2 of the bivariate d.f.’s whose marginals are F1 and F2 ? It is immediate to note that the set Γ F1 ,F2 , now called the Fr´ chet class of F1 and F2 , is not empty since, if X1 and X2 are independent, e then the distribution function x1 ,x2 F x1 ,x2 = F1 x1 F2 x2 always belongs to Γ F1 ,F2 . But, it was not clear which the other elements of Γ F1 ,F2 were. Preliminary studies about this problem were conducted in [64, 70, 88] (see also [30, 181] for a historical overview). But, in 1959, Sklar obtained the deepest result in this respect, by introducing the notion, and the name, of a copula, and proving the theorem that now bears his name [191]. In his own words [193]:
Fabrizio Durante Department of Knowledge-Based Mathematical Systems, Johannes Kepler University Linz, Austria e-mail: fabrizio.durante@jku.at Carlo Sempi Dipartimento di Matematica “Ennio De Giorgi” Universit` del Salento, Lecce, Italy e-mail: carlo.sempi@unisalento.it a
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2
Fabrizio Durante and Carlo Sempi [...] In the meantime, Bert (Schweizer) and I had been making progress in our work on statistical metric spaces, to the extent that Menger suggested it would be worthwhile for us to communicate our results to Fr´ chet. We did: Fr´ chet was interested, and asked us to e e write an announcement for the Comptes Rendus [183].