Introduction to Copulas

Abstract

In 1959, in response to a question by M. Frechet, A. Sklar introduced the notion of a copula and proved the basic theorem that now bears his name. In the next 15 years, all results regarding copulas were obtained in connection with problems stemming from the theory of probabilistic metric spaces. Things changed in 1974, when, quite by accident, I reread a paper by A. Renyi entitled “On Measures of Dependence” and realized that I could easily construct such measures by using copulas. Subsequently, E. F. Wolff and I took up the study of these measures. This led to our 1976 note in the Comptes rendus de l’Academie des Sciences, Paris, followed in 1981 by our paper in the Annals of Statistics. After the publication of these papers and the appearance, in 1983, of the book Probabilistic Metric Spaces, written jointly with A. Sklar, the pace quickened as more of our students and colleagues became involved. Moreover, since interest in questions of statistical dependence was increasing, others came to the subject from different directions. In 1986, the enticingly entitled article “The Joy of Copulas,” by C. Genest and R. C. MacKay, attracted more attention. By the end of the 1980s, the volume of work on copulas and related matters had grown to such an extent that G. Dall’Aglio organized a symposium on distributions with given marginals, which was held in Rome in 1990. At that symposium, I presented a paper entitled “Thirty Years of Copulas” in which I was able to survey virtually the entire field. The conference and the published proceedings that followed G. Dall’Aglio et al. 1991 led to more interest in and involvement with the subject. There followed conferences, with published proceedings, in Seattle in 1993 Ruschendorf et al. 1996 , Prague in 1996 Benes and Stepan 1997 , and Barcelona in 2000 Cuadras et al. 2002 . The century was capped by the publication of An Introduction to Copulas, by R. B. Nelsen 1999 , which has become the standard overview of the basics of the subject. The first years of this new century have witnessed an explosion in copula-centered activity—not only among those who do statistics, but also among those who use statistics. The growth of this activity has been most pronounced in the areas of finance and