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Ghoudi, Kilani; Khoudraji, Abdelhaq; Rivest, Louis-Paul

Statistical characteristics of copulas of bidimensional extreme value distributions. (Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles.) (French) Zbl 0899.62071

Can. J. Stat. 26, No. 1, 187-197 (1998).
Summary: Let \((X,Y)\) be a bivariate random vector whose distribution function \(H(x,y)\) belongs to the class of bivariate extreme-value distributions. If \(F_1\) and \(F_2\) are the marginals of \(X\) and \(Y\), then \(H(x,y)= C\{F_1(x), F_2(y)\}\), where \(C\) is a bivariate extreme-value dependence function. This paper gives the joint distribution of the random variables \(Z= \{\log F_1(X)\}/ \{\log F_1(X) F_2(Y)\}\) and \(W= C\{F_1(X),F_2(Y)\}\). Using this distribution, an algorithm to generate random variables having bivariate extreme-value distributions is presented. Furthermore, it is shown that for any bivariate extreme-value dependence function \(C\), the distribution of the random variable \(W= C\{F_1(X), F_2(Y)\}\) belongs to a monoparametric family of distributions. This property is used to derive goodness-of-fit statistics to determine whether a copula belongs to an extreme-value family.

Cited in 2 Reviews
Cited in 51 Documents

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62E10 Characterization and structure theory of statistical distributions

Keywords:

Galambos’s distribution; goodness of fit; Gumbel’s dependence function; jackknife variance estimator; multivariate extreme-value distributions; simulation; U-statistic; goodness-of-fit statistics; copula; extreme-value family
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References:

[1] Anderson, Environmental factors affecting reservoir safety. Statistics for the Environment (publié sous la direction de V. Barnett et de F. Turkman) pp 163– (1993)
[2] Barbe, On Kendall’s process, J. Multivariate Anal. 58 pp 197– (1996)
[3] Capéra à, A nonparametric estimation procedure for bivariate extreme value copulas, Biometrika 84 pp 567– (1997)
[4] Clayton, A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence, Biometrika 65 pp 141– (1978) · Zbl 0394.92021
[5] Coronel-Brizio, Tests of fit for the generalized extreme value distribution. Rapport technique, Université Simon-Fraser (1996) · Zbl 0875.62177
[6] Deheuvels, On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions, Statist. Probab. Lett. 12 pp 429– (1991) · Zbl 0749.62033
[7] Galambos, Order statistics from sample from multivariate distributions, J. Amer. Statist. Assoc. 70 pp 674– (1975) · Zbl 0315.62022
[8] Galambos, The Asymptotic Theory of Extreme Order Statistics. (1987) · Zbl 0634.62044
[9] Genest, Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données, Rev. Canad. Statist. 14 pp 145– (1986)
[10] Genest, A characterization of Gumbel’s family of extreme value distributions, Statist. Probab. Lett. 8 pp 207– (1989) · Zbl 0701.62060
[11] Genest, Statistical inference procedures for bivariate Archimedean copulas, J. Amer. Statist. Assoc. 88 pp 1034– (1993) · Zbl 0785.62032
[12] Geoffroy, Contribution à la théorie des valeurs extrěmes, Publ. Inst. Statist. Univ. Paris 7 pp 37– (1958)
[13] Hutchinson, Continuous Bivariate Distributions, Emphasizing Applications. (1990)
[14] Joe, Familles of min-stable multivariate exponential and multivariate extreme value distributions, Statist. Probab. Lett. 9 pp 75– (1990) · Zbl 0686.62035
[15] Joe, Multivariate extreme-value distributions with applications to environmental data, Rev. Canad. Statist. 22 pp 47– (1994)
[16] Johnson, Distributions in Statistics: Continuous Multivariate Distributions. (1972) · Zbl 0248.62021
[17] Khoudraji, Contributions à l’étude des copules et à la modération des copules aux valeurs extrěmes (1995)
[18] Lee, U-Statistics. Theory and Practice. (1990)
[19] Marshall, A generalized bivariate exponential distribution, J. Appl. Probab. 4 pp 291– (1967) · Zbl 0155.24001
[20] Nelsen, Advances in Probability Distributions with Given Marginals pp 51– (1991) · doi:10.1007/978-94-011-3466-8_3
[21] Pickands, J. (1981). Multivariate extreme value distributions. Proc. 43rd Session Internal. Statist. Inst. (Buenos Aires), 859-878. · Zbl 0518.62045
[22] Ross, Introduction to Stochastic Processes. (1985)
[23] Sibuya, Bivariate extreme statistics, Ann. Inst. Statist. Math. 11 pp 195– (1960) · Zbl 0095.33703
[24] Sklar, Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris 8 pp 229– (1959)
[25] Tawn, Bivariate extreme value theory: Models and estimation, Biometrika 75 pp 397– (1988) · Zbl 0653.62045
[26] Tawn, Modelling multivariate extreme value distributions, Biometrika 77 pp 245– (1990) · Zbl 0716.62051
[27] Tiago de Olivera, Extremal distributions, Rev. Fac. Ciencas Univ. Lisboa A8 pp 299– (1958)
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