Abstract
In 1959, in response to a query of M. Fréchet, A. Sklar introduced copulas. These are functions that link multivariate distributions to their one-dimensional margins. Thus, if H is an n-dimensional cumulative distribution function with one-dimensional margins F1,…,Fn, then there exists an n-dimensional copula C (which is unique when F1,…,Fn are continuous) such that H(x1,…,xn) = C (F1(x1),…,Fn (xn)). During the years 1959 — 1974, most results concerning copulas were obtained in the course of the development of the theory of probabilistic metric spaces, principally in connection with the study of families of binary operations on the space of probability distribution functions. Then it was discovered that two-dimensional copulas could be used to define nonparametric measures of dependence for pairs of random variables. In the ensuing years the copula concept was rediscovered on several occasions and these functions began to play an ever-more-important role in mathematical statistics, particularly in matters involving questions of dependence, fixed marginals and functions of random variables that are invariant under monotone transformations. Today, in view of the fact that they are the higher dimensional analogues of uniform distributions on the unit interval, and as the result of the efforts of a diverse group of scholars, the significance, ubiquity and utility of copulas is being recognized. This paper is devoted to an historical overview and rather personal account of these developments and to a description of some recent results.
Research supported by ONR Contract N — 00014 — 90 — 1008.
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References
Aczel, J. (1949) Sur les opérations définies pour nombres réels, Bull. Soc. Math. France 76, 59–64.
Aczel, J. (1966) Lectures on Functional Equations and their Applications, Academic Press, New York.
Alsina, C. and Schweizer, B. (1988) Mixtures are not derivable, Found. of Physics Letters 1, 171–174.
Dall’Aglio, G., (1959) Sulla compatibilità delle fuzione di ripartizione doppia, Rend. Mat. 18, 385–413.
Dall’Aglio, G. (1960) Les fonctions extrêmes de la classe de Fréchet à 3 dimensions, Publ. Inst. Statist. Univ. Paris 9, 175–188.
Dall’Aglio, G. (1961) Osservazioni sulla convergenza in distribuzione e in probabilità, Giorn. Ist. Ital. Attuari 24, 94–108.
Dall’Aglio, G. (1972) Fréchet classes and compatibility of distribution functions, Symposia Math. 9, 131–150.
Darsow, W. F., Nguyen, Bao and Olsen, E. T., Copulas and Markov processes, to appear.
Deheuvels, P. (1978) Caractérisation complète des lois extrêmes multivariées et de la convergence des types extrêmes, Publ. Inst. Statist. Univ. Paris 23, 1–36.
Deheuvels, P. (1980) The decomposition of infinite order and extreme multivariate distributions, in Asymptotic Theory of Statistical Tests and Estimation (Proc. Adv. Internat. Sympos., Univ. North Carolina, Chapel Hill, N.C., 1979), pp. 259–286, Academic Press, New York.
Deheuvels, P. (1979) La fonction de dépendance empirique et ses propriétés. Un test non paramétrique d’indépendance, Acad. Roy. Belg. Bull. C1 Sci. (5) 65, 274–292.
Deheuvels, P. (1981) Multivariate tests of independence, in Analytical Methods in Probability Theory (Oberwolfach, 1980), pp. 42–50, Lecture Notes in Math., 861, Springer-Verlag, Berlin.
Deheuvels, P. (1983) Indépendance multivariée partielle et inégalités de Fréchet, in Studies in Probability and Related Topics, Papers in Honour of Octav Onicescu on his 90th Birthday, ed. by M. C. Demetrescu and M. Iosifescu, Editrice Nagard, Rome, pp. 145–155.
Féron, R. (1956) Sur les tableaux de corrélation dont les marges sont données, cas de l’espace à trois dimensions, Publ. Inst. Statist. Univ. Paris 5, 3–12.
Frank, M. J. (1975) Associativity in a class of operations on a space of distribution functions, Aequationes Math. 12, 121–144.
Frank, M. J. (1979) On the simultaneous associativity of F(x,y) and x + y — F(x,y), Aequationes Math. 19, 194–226.
Frank, M. J. and Schweizer, B. (1979) On the duality of generalized infimal and supremal convolutions, Rend. Mat. 12, 1–23.
Frank, M. J., Nelsen, R. B. and Schweizer, B. (1987) Best-possible bounds for the distribution of a sum — a problem of Kolmogorov, Probab. Th. Rel. Fields 74, 199–211.
Fréchet, M. (1935) Généralisations du théorème des probabilités totales, Fund. Math. 25, 379–387.
Fréchet, M. (1951) Sur les tableaux de corrélation dont les marges sont données, Ann. Univ. Lyon 9, Sect. A, 53–77.
Fréchet, M. (1957) Les tableaux de corrélation et les programmes linéaires, Revue Inst. Int. Statist. 25, 23–40.
Galambos, J. (1978) The Asymptotic Theory of Extreme Order Statistics, John Wiley & Sons, New York.
Genest, C. (1987) Frank’s family of bivariate distributions, Biometrika 74, 549–555.
Genest, C. and MacKay, R. J. (1986) Copules archimédiennes et familles de lois bidimensionelles dont les marges sont données, Canadian J. Statist. 2, 145–159.
Genest, C. and MacKay, R. J. (1986) The joy of copulas: bivariate distributions with uniform marginals, Amer. Statist. 40, 280–283.
Genest, C. and Rivest, L. P. (1989) A characterization of Gumbel’s family of extreme value distributions, Statist. and Probab. Letters 8, 207–211.
Hoeffding, W. (1940) Masstabinvariante Korrelationstheorie, Schriften des Mathematischen Instituts und des Instituts fur Angewandte Mathematik der Universität Berlin 5, 179–233.
Kamiński, A., Mikusiński, P., Sherwood, H. and Taylor, M. D. (1988) Properties of a special class of doubly stochastic measures, Aequationes Math. 36, 212–229.
Kellerer, H. G. (1964) Masstheoretische Marginalprobleme, Math. Ann. 153, 168–198.
Kellerer, H. G.(1964) Verteilungsfunktionen mit gegebenen Marginalverteilungen, Z. Warsch. Verw. Geb. 3, 247–270.
Kimberling, C. H. (1973) Exchangeable events and completely monotonic sequences, Rocky Mountain J. Math. 3, 565–574.
Kimberling, C. H. (1974) A probabilistic interpretation of complete monotonicity, Aequationes Math. 10, 152–164.
Kimeldorf, G. and Sampson, A. R. (1975) One-parameter families of bivariate distributions with fixed marginals, Commun. Statist. 4, 293–301.
Kimeldorf, G. and Sampson, A. R. (1975) Uniform representations of bivariate distributions, Commun. Statist. 4, 617–627.
Kimeldorf, G. and Sampson, A. R. (1978) Monotone dependence, Ann. Statist. 6, 895–903.
Kimeldorf, G. and Sampson, A. R. (1987) Positive dependence orderings, Ann. Inst. Statist. Math. 39, 113–128.
Kimeldorf, G. and Sampson, A. R. (1989) A framework for positive dependence, Ann. Inst. Statist. Math. 41, 31–45.
Kotz, S. and Johnson, N. L. (1977) Propriétés de dépendance des distributions intérées généralisées à deux variables Farlie-Gumbel-Morgenstern, C. R. Acad. Sci. Paris 285A, 277–280.
Kruskal, W. H. (1958) Ordinal measures of association, J. Amer. Statist. Assoc. 53, 814–861.
Lehmann, E. L. (1966) Some concepts of dependence, Ann. Math. Statist. 37, 1137–1153.
Ling, C. H. (1965) Representation of associative functions, Publ. Math. Debrecen 12, 189–212.
Makarov, G. D. (1981) Estimates for the distribution function of a sum of two random variables when the marginal distributions are fixed, Theor. Probab. Appl. 26, 803–806.
Marshall, A. W. and Olkin, I. (1988) Families of multivariate distributions, J. Amer. Statist. Assoc. 83, 834–841.
Menger, K. (1942) Statistical metrics, Proc. Nat. Acad. Sci. U.S.A. 28, 535–537.
Moore, D. S. and Spruill, M. C. (1975) Unified large-sample theory of general chi-squared statistics for tests of fit, Ann. of Math. 65, 117–143.
Mostert, P. S. and Shields, A. L. (1957) On the structure of semigroups on a compact manifold with boundary, Ann. Statist. 3, 599–616.
Moynihan, R. (1978) On TT-semigroups of probability distribution functions, II, Aequationes Math. 17, 19–40.
Moynihan, R. and Schweizer, B. (1979) Betweenness relations in probabilistic metric spaces, Pacific J. Math. 81, 175–196.
Moynihan, R., Schweizer, B. and Sklar, A. (1978) Inequalities among binary operations on probability distribution functions, in General Inequalities 1, ed. by E. F. Beckenbach, Birkhäuser Verlag, Basel, pp. 133–149.
Nelsen, R. B., (1986) Properties of a one-parameter family of bivariate distributions with specified marginals, Commun. Statist. — Theory Meth. 15, 3277–3285.
Nelsen, R. B. and Schweizer, B., Bounds on distribution functions for sums of squares and radial errors, to appear.
Renyi, A. (1959) On measures of dependence, Acta Math. Acad. Sci. Eungav. 10, 441–451.
Rüschendorf, L. (1982) Random variables with maximum sums, Adv. Appl. Probab. 14, 623–632.
Rüschendorf, L. (1985) Construction of multivariate distributions with given marginals, Ann Inst. Statist. Math. 37, 225–233.
Scarsini, M. (1984) On measures of concordance, Stochastica 8, 201–218.
Scarsini, M. (1984) Strong measures of concordance and convergence in probability, Riv. Mat. Sci. Econom. Soc. 7, 39–44.
Scarsini, M. (1988) Multivariate stochastic dominance with fixed dependence structure, Operations Res. Letters 7, 237–240.
Scarsini, M., Copulae of probability measures on product spaces, to appear.
Schweizer, B. and Sklar, A. (1958) Espaces métriques aléatoires, C. R. Acad. Sci. Paris 247, 2092–2094.
Schweizer, B. and Sklar, A. (1961) Associative functions and statistical triangle inequalities, Publ. Math. Debrecen 8, 169–186.
Schweizer, B. and Sklar, A. (1974) Operations on distribution functions not derivable from operations on random variables, Studia Math. 52, 43–52.
Schweizer, B. and Sklar, A. (1983) Probabilistic Metric Spaces, Elsevier North-Holland, New York.
Schweizer, B. and Wolff, E. F. (1976) Sur une mesure de dépendance pour les variables aléatoires, C. R. Acad. Sci. Paris 283A, 659–661.
Schweizer, B. and Wolff, E. F. (1981) On nonparametric measures of dependence for random variables, Ann Statist. 9, 879–885.
Šerstnev, A. N. (1963) On the notion of a random normed space, Dokl. Akad. Nauk SSSR 149, 280–283.
Šerstnev, A. N. (1964) On a probabilistic generalization of metric spaces, Kazan Goz. Univ. Učen. Zap. 124, 3–11.
Sherwood, H. and Taylor, M. D. (1988) Doubly stochastic measures with hairpin support, Probab. Th. Rel. Fields 78, 617–626.
Sklar, A. (1959) Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris 8, 229–231.
Sklar, A. (1973) Random variables, joint distribution functions and copulas, Kybernetika 9, 449–460.
Strassen, V. (1965) The existence of probability measures with given marginals, Ann. Math. Statist. 36, 423–439.
Stute, W. (1986) Conditional empirical processes, Ann. Statist. 14, 638–647.
Stute, W. (1986) On almost sure convergence of conditional empirical distribution functions, Ann. Statist. 14, 891–901.
Urbanik, K. (1964, 1973, 1984, 1986) Generalized convolutions I, II, III and IV, Stadia Math. 23, 217–245; 45, 57–70; 80, 167–189; and 83, 57–95.
Vitale, R., Stochastic dependence and a class of degenerate distributions, in Topics in Statistical Dependence, ed. by H. Block, A. R. Sampson and T. Savits, IMS Lecture Notes and Monograph Series, to appear.
Wald, A. (1943) On a statistical generalization of metric spaces, Proc. Nat. Acad. Sci. U.S.A. 29, 196–197.
Whitt, W. (1976) Bivariate distributions with given marginals, Ann. Statist. 4, 1280–1289.
Williamson, R. C., An extreme limit theorem for dependency bounds of normalized sums of random variables, Information Sciences, to appear.
Williamson, R. C. and Downs, T. (1990) Probabilistic arithmetic: numerical methods for calculating convolutions and dependency bounds, Int. J. Approximate Reasoning 4, 89–158.
Wolff, E. F. (1977) Measures of dependence derived from copulas, Ph.D. Thesis, Univ. Massachusetts, Amherst.
Wolff, E. F. (1981) N-dimensional measures of dependence, Stochastica 4, 175–188.
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Schweizer, B. (1991). Thirty Years of Copulas. In: Dall’Aglio, G., Kotz, S., Salinetti, G. (eds) Advances in Probability Distributions with Given Marginals. Mathematics and Its Applications, vol 67. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3466-8_2
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