Abstract
We survey multivariate extreme value distributions. These are limiting distributions of maxima and/or minima, componentwise, after suitable normalization. A distribution is extreme value stable if and only if its margins are stable and its dependence function is stable. Thus it is possible, without loss of generality, to choose any stable marginal distribution deemed convenient. We use the negative exponential one. The class of multivariate stable negative exponential distributions is characterized by the fact that weighted minima of components have negative exponential distributions. We examine several representations and the relationships among them and we consider, in terms of them, joint densities and scalar measures of dependence. We also consider estimation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Emoto S. E. and Matthews P. C. (1987) A Weibull model for dependent censoring. Research Report 87–14. Dept. of Math. Univ. Maryland, Catonsville Md.
Esary J. D. and Marshall A. W. (1974) Multivariate distributions with exponential minimums. Ann. Statist. 2, 84–93.
Galambos J. (1987) The asymptotic theory of extreme order statistics. 2nd ed. Robert E. Krieger Publishing Co. Malabar Fla.
Gumbel E. J. (1958) Statistics of Extremes. Columbia Univ. Press, New York.
de Haan I. (1978) A characterization of multivariate extreme-value distributions. Sankhya Series A, 40, 85–88.
de Haan L. (1984) A spectral representation for max-stable processes. Ann. Probab. 12, 1194–1204.
de Haan L. (1985) Extremes in higher dimensions: the model and some statistics. 45th Session ISI, Amsterdam.
de Haan L. and Pickands J. (1986) Stationary Min-Stable Stochastic Processes. Probab. Th. Rel. Fields 72, 477–492.
de Haan L. and Resnick S. I. (1977) Limit theory for multivariate sample extremes. Z. Wahrsch. 40, 317–337.
Leadbetter M. R., Lindgren G. and Rootzen H. (1983) Extremes and related properties of random sequences and processes. Springer, New York.
Marshall A. W. and Olkin I. (1967) A generalized bivariate exponential distribution. J. Appl. Probab. 4, 291–302.
Marshall A. W. and Olkin I. (1983) Domains of attraction of multivariate extreme value distributions. Ann. Probab. 11, 168–177.
Pickands J. III (1981) Multivariate extreme value distributions. 43rd Session, ISI, Buenos Aires, 859–878.
Smith R. L. (1985) Statistics of extreme values. 45th Session, ISI, Amsterdam.
Tiago de Oliveira J. (1962/63) Structure theory of bivariate extremes: extensions. Estudos de Math. Estat. Econom. 7, 165–195.
Tiago de Oliveira J. (1975) Bivariate extremes. Extensions. 40th Session, ISI, Warsaw.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1989 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pickands, J. (1989). Multivariate Negative Exponential and Extreme Value Distributions. In: Hüsler, J., Reiss, RD. (eds) Extreme Value Theory. Lecture Notes in Statistics, vol 51. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3634-4_22
Download citation
DOI: https://doi.org/10.1007/978-1-4612-3634-4_22
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-96954-1
Online ISBN: 978-1-4612-3634-4
eBook Packages: Springer Book Archive