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Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données. (Archimedean copulas and families of bivariate laws with given marginals). (French) Zbl 0605.62049
A distribution function H on [0,1]\(\times [0,1]\) with uniform marginals is called ”Archimedean” if it is of the form \(H(x,y)=\phi^{-1}(\phi (x)+\phi (y))\) for some continuous, strictly decreasing and convex function \(\phi\) : [0,1]\(\to [0,\infty]\). The authors discuss some examples for parametric families of Archimedean distribution functions, where the parameter is related to the dependence properties of the distributions. They, furthermore, characterize the property that a limit H of a sequence of Archimedean df’s again is Archimedean.
Reviewer: L.Rüschendorf

MSC:
62H05 Characterization and structure theory for multivariate probability distributions; copulas
26A48 Monotonic functions, generalizations
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[1] Abel, Recherche des fonctions de deux quantites variables indépendantes x et y, telles que f(x, y), qui ont la propriété que f (z, f(x, y)) est une fonction symétrique de z, x et y., J. Reine Angew. Math. 1 pp 11– (1826) · ERAM 001.0003cj
[2] Oeuvres complétes de N. H. Abel. Volume 1. Christiania, 1881, 61-65.
[3] Ali, A class of bivariate distributions including the bivariate logistic., J. Multivariate Anal. 8 pp 405– (1978) · Zbl 0387.62019
[4] Bourbaki, Eléments de mathématique. Fascicule IX, Livre IV: Fonctions d’une variable réelle (1949)
[5] Conway, D. A. (1979). Multivariate distributions with specified marginals. Rapport de recherches no 145, Dép. de statistique, Université Stanford.
[6] Cook, A family of distributions for modelling nonelliptically symmetric multivariate data., J. Roy. Statist. Soc. Ser. B 43 pp 210– (1981)
[7] Cooper, Notes on certain inequalities (II)., J. London Math. Soc. 2 pp 159– (1927) · JFM 53.0218.01
[8] Farlie, The performance of some correlation coefficients for a general bivariate distribution., Biometrika 47 pp 307– (1960) · Zbl 0102.14903
[9] Frank, On the simultaneous associativity of F(x, y) and x + y -F(x, y), Aequationes Math. 19 pp 194– (1979) · Zbl 0444.39003
[10] Fréchet, Sur les tableaux de correlation dont les marges sont données., Ann. Univ. Lyon Sér. 3 14 pp 53– (1951)
[11] Gumbel, Distributions à plusieurs variables dont les marges sont données., C. R. Acad. Sci. Paris 246 pp 2717– (1958) · Zbl 0084.35803
[12] Gumbel, Distributions des valeurs extrěmes en plusieurs dimensions., Publ. Inst. Statist. Univ. Paris 9 pp 171– (1960)
[13] Gumbel, Bivariate exponential distributions., J. Amer. Statist. Assoc. 55 pp 698– (1960) · Zbl 0099.14501
[14] Gumbel, J. Amer. Statist. Assoc. 56 pp 335– (1961)
[15] Kimberling, A probabilistic interpretation of complete monotonicity., Aequationes Math. 10 pp 152– (1974) · Zbl 0309.60012
[16] Kimeldorf, One-parameter families of bivariate distributions with fixed marginals., Comm. Statist. 4 pp 293– (1975) · Zbl 0296.62012
[17] Kimeldorf, Uniform representations of bivariate distributions., Comm. Statist. 4 pp 617– (1975) · Zbl 0312.62008
[18] Kruskal, Ordinal measures of association., J. Amer. Statist. Assoc. 53 pp 814– (1958) · Zbl 0087.15403
[19] Lehmann, Some concepts of dependence., Ann. Math. Statist. 37 pp 1137– (1966) · Zbl 0146.40601
[20] Ling, Representation of associative functions., Publ. Math. Debrecen 12 pp 189– (1965)
[21] Mardia, Multivariate Pareto distributions., Ann. Math. Statist. 33 pp 1008– (1962) · Zbl 0109.13303
[22] Mardia, A translation family of bivariate distributions and Frecheťs bounds., Sankhya Ser. A 32 pp 119– (1970) · Zbl 0205.46602
[23] Mardia, Families of Bivariate Distributions (1970)
[24] Morgenstern, Einfache Beispiele zweidimensionaler Verteilungen., Mitt. Math. Statist. 8 pp 234– (1956)
[25] Oakes, A model for association in bivariate survival data., J. Roy. Statist. Soc. Ser. B 44 pp 414– (1982) · Zbl 0503.62035
[26] Plackett, A class of bivariate distributions., J. Amer. Statist. Assoc. 60 pp 516– (1965)
[27] Roberts, Convex Functions (1973)
[28] Satterthwaite, A generalisation of Gumbeľs bivariate logistic distribution., Metrika 25 pp 163– (1978) · Zbl 0386.62037
[29] Schucany, Correlation structure in Farlie-Gumbel-Morgenstern distributions., Biometrika 65 pp 650– (1978) · Zbl 0397.62033
[30] Schweizer, Probabilistic Metric Spaces (1983) · Zbl 0546.60010
[31] Schweizer, On nonparametric measures of dependence for random variables., Ann. Statist. 9 pp 879– (1981) · Zbl 0468.62012
[32] Sklar, Fonctions de répartition à n dimensions et leurs marges., Publ. Inst. Statist. Univ. Paris 8 pp 229– (1959)
[33] Takahasi, Note on the multivariate Burr’s distribution., Ann. Inst. Statist. Math. 17 pp 257– (1965) · Zbl 0134.36703
[34] Yanagimoto, Partial orderings of permutations and monotonicity of a rank correlation statistic., Ann. Inst. Statist. Math. 21 pp 489– (1969) · Zbl 0208.44704
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