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 . 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 . In his own words :
Fabrizio Durante Department of Knowledge-Based Mathematical Systems, Johannes Kepler University Linz, Austria e-mail: email@example.com Carlo Sempi Dipartimento di Matematica “Ennio De Giorgi” Universit` del Salento, Lecce, Italy e-mail: firstname.lastname@example.org a
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 . This began an exchange of letters with Fr´ chet, in the course of which he sent me several packets of reprints, mainly dealing e with the work he and his colleagues were doing on distributions with given marginals. These reprints, among the later arrivals of which I particularly single out that of Dall’Aglio , were important for much of our subsequent work. At the time, though, the most signiﬁcant reprint for me was that of F´ ron . e F´ ron, in studying three-dimensional distributions had introduced auxiliary functions, e deﬁned on the unit cube, that connected such distributions with their one-dimensional margins. I saw that similar functions could be deﬁned on the unit n-cube for all n 2 and would similarly serve to link n-dimensional distributions to their one-dimensional margins. Having worked out the basic properties of these functions, I wrote about them to Fr´ chet, in English. e He asked me to write a note about them in French. While writing this, I decided I needed a name for these functions. Knowing the word “copula” as a grammatical term for a word or expression that links a subject and predicate, I felt that this would make an appropriate name for a function that links a multidimensional distribution to its one-dimensional margins, and used it as such. Fr´ chet received my note, corrected one mathematical statement, e made some minor corrections to my French, and had the note published by the Statistical Institute of the University of Paris as Sklar .
The proof of Sklar’s theorem was not given in , but a sketch of it was provided in  (see also ), so that for a few years practitioners in the ﬁeld had to reconstruct it relying on the hand–written notes by Sklar himself; this was the case, for instance, of the second author. It should be also mentioned that some “indirect” proofs of Sklar’s theorem (without mentioning copula) were later discovered by Moore and Spruill  and Deheuvels  For about 15 years, all the...