A characterization of Gumbel’s family of extreme value distributions. (English) Zbl 0701.62060
Summary: A family of multivariate extremal distributions proposed by E. J. Gumbel [Publ. Inst. Statist. Univ. Paris 9, 171-173 (1960; Zbl 0093.153)] is characterized among those whose dependence function is an Archimedean copula. The domains of attraction of Gumbel distributions are also determined within this class.
MSC:
62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
62E10 | Characterization and structure theory of statistical distributions |
Keywords:
extreme value distribution; Gumbel’s family; regular variation; multivariate extremal distributions; dependence function; Archimedean copula; domains of attraction of Gumbel distributionsCitations:
Zbl 0093.153References:
[1] | Ali, M. M.; Mikhail, N. N.; Haq, M. S., A class of bivariate distributions including the bivariate logistic, J. Multivariate Anal., 8, 405-412 (1978) · Zbl 0387.62019 |
[2] | Berman, S. M., Convergence to bivariate limiting extreme value distributions, Ann. Inst. Statist. Math., 13, 217-223 (1961) · Zbl 0119.15103 |
[3] | Cook, R. D.; Johnson, M. E., A family of distributions for modelling non-elliptically symmetric multivariate data, J. Roy. Statist. Soc. Ser. B, 43, 210-218 (1981) · Zbl 0471.62046 |
[4] | Frank, M. J., On the simultaneous associativity of \(F\)(x, y) and \(x + y − \(F\)(\textit{x, y}), Aequationes Math., 19, 194-226 (1979) · Zbl 0444.39003 |
[5] | Galambos, J., The Asymptotic Theory of Extreme Order Statistics (1978), Wiley: Wiley New York · Zbl 0381.62039 |
[6] | Genest, C., Frank’s family of bivariate distributions, Biometrika, 74, 549-555 (1987) · Zbl 0635.62038 |
[7] | Genest, C.; MacKay, R. J., Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données, Rev. Canad. Statist., 14, 145-159 (1986) · Zbl 0605.62049 |
[8] | Gumbel, E. J., Distributions des valeurs extrêmes en plusieurs dimensions, Publ. Inst. Statist. Univ. Paris, 9, 171-173 (1960) · Zbl 0093.15303 |
[9] | Leadbetter, M. R.; Rootzén, H., Extremal theory for stochastic processes, Ann. Probab., 16, 431-478 (1988) · Zbl 0648.60039 |
[10] | Marshall, A. W.; Olkin, I., Domains of attraction of multivariate extreme value distributions, Ann. Probab., 11, 168-177 (1983) · Zbl 0508.60022 |
[11] | Nelsen, R. B., Properties of a one-parameter family of bivariate distributions with specified marginals, Comm. Statist., 15, 3277-3285 (1986) · Zbl 0616.62067 |
[12] | Schweizer, B.; Sklar, A., Probabilistic Metric Spaces (1983), North-Holland: North-Holland Amsterdam · Zbl 0546.60010 |
[13] | Sibuya, M., Bivariate extreme statistics, Ann. Inst. Statist. Math., 11, 195-210 (1960) · Zbl 0095.33703 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.