Abstract
LetH be the limiting distribution of a vector of maxima for ad-dimensional stationary sequnce with multivariate extremal index. We giv necessary and sufficient conditions forH to have independent or totally dependent margins by using relations between the multivariate extremal index and the univariate extremal indexes.
A new functional family of multivariates extreme value distributions, containingH, is introduced. We apply the results to characterize the asymptotic independence of the maximum and the minimum and compute the multivariate extremal index of the Multivariate Maxima of Moving Maxima process.
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References
Castillo, E. (1988).Extreme Value Theory in Engineering. Academic Press, Inc., New York.
Chernick, M. R., Hsing, T., andMcCormick, W. P. (1991). Calculating the extremal index for a class of stationary sequences.Advances in Applied Probability, 23:835–850.
Davis, R. (1979). Maxima and minima of stationary sequences.Annals of Probability, 7:453–460.
Davis, R (1982). Lmit laws for the maximum and minimum of stationary sequences.Zetschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 61:31–42.
Deheuvels, P. R. L. (1978). Caractérisation complète des lois extrèmes multivariées et de la convergence aux types extrèmes.Publications de l'Institute Statistique de la Université de Paris, 23:1–36.
Galambos, J. (1978).The Asymptotic Theory of Extreme Order Statistics. John Wiley & Sons, New York.
Hsing, T. (1989). Extreme value theory for multivariate stationary sequences.Journal of Multivariate Analysis, 29:274–291.
Joe, H. (1987).Multivariate Models and DependenceConcepts. Chapman & Hall, London.
Leadbetter, M. R. (1983). Extremes and local dependence in stationary sequences.Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 65:291–306.
Leadbetter, M. R., andNanadagopalan, L. (1989). On exceedance point processes for stationary sequences under mild oscillation restrictions.Lecture Notes in Statistics, 51:69–80.
Marshall, A. W. andOlkin, I. (1983). Domains of attraction of multivariate extreme value distributions.Annals of Probability, 11:168–177.
Martins, A. P. andFerreira, H. (2003). Measuring the extremal dependence. Technical report, CEAUL, Lisbon.
Nandagopalan, S. (1990).Multivariate extremes and estimation of the extremal inder. PhD Thesis, Department of Statistics, University of North Carolina at Chapel Hill, NC, USA.
O'Brien, G. (1987). Extreme values for stationary and markov sequences.Annals of Probability, 15:281–291.
Pereira, L. andFerreira, H. (2001). The asymptotic locations of the maximum and minimum of stationary sequences.Journal of Statistical Planning and Inference, 104:287–295.
Resnick, S. I. (1987).Extreme Values, Regular Variation and Point Processes. Springer-Verlag. New York.
Smith, R. L. (1990). Max-stable processes and spatial extremes. Technical report, University of North Carolina at Chapel Hill, NC, USA. In
Smith, R. L. andWeissman, I. (1996). Characterization and estimation of the multivariate extremal index. Technical report, University of North Carolina at Chapel Hill, NC, USA.
Takahashi, R. (1988). Characterization of a multivariate extreme value distribution.Advances in Applied Probability, 20:235–236.
Weissman, I. andNovak, S. Y. (1998). On blocks and runs estimators of the extremal index.Journal of Statistical Planning and Inference, 66:281–288.
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Martins, A.P., Ferreira, H. The multivariate extremal index and the dependence structure of a multivariate extreme value distribution. TEST 14, 433–448 (2005). https://doi.org/10.1007/BF02595412
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DOI: https://doi.org/10.1007/BF02595412