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The behavior of multivariate maxima of moving maxima processes

Part of: Stochastic processes Multivariate analysis

Published online by Cambridge University Press:  14 July 2016

Zhengjun Zhang  and
Richard L. Smith
Zhengjun Zhang*
Affiliation:
Washington University
Richard L. Smith*
Affiliation:
University of North Carolina
*
∗ Postal address: Department of Mathematics, Washington University, Saint Louis, MO 63130-4899, USA. Email address: zjz@math.wustl.edu
∗∗ Postal address: Department of Statistics, University of North Carolina, Chapel Hill, NC 27599-3260, USA
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Abstract

In the characterization of multivariate extremal indices of multivariate stationary processes, multivariate maxima of moving maxima processes, or M4 processes for short, have been introduced by Smith and Weissman. Central to the introduction of M4 processes is that the extreme observations of multivariate stationary processes may be characterized in terms of a limiting max-stable process under quite general conditions, and that a max-stable process can be arbitrarily closely approximated by an M4 process. In this paper, we derive some additional basic probabilistic properties for a finite class of M4 processes, each of which contains finite-range clustered moving patterns, called signature patterns, when extreme events occur. We use these properties to construct statistical estimation schemes for model parameters.

Keywords

Multivariate extremeextreme value theorymax-stable processmultivariate extremal index

MSC classification

Primary: 60G70: Extreme value theory; extremal processes 62H05: Characterization and structure theory 60G10: Stationary processes 60G52: Stable processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1113 - 1123
Copyright
Copyright © Applied Probability Trust 2004 

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References

Davis, R. A., and Resnick, S. I. (1989). Basic properties and prediction of Max-ARMA processes. Adv. Appl. Prob. 21, 781–803.CrossRefGoogle Scholar
Davis, R. A., and Resnick, S. I. (1993). Prediction of stationary max-stable processes. Ann. Appl. Prob. 3, 497–525.CrossRefGoogle Scholar
De Haan, L. (1984). A spectral representation for max-stable processes. Ann. Prob. 12, 1194–1204.Google Scholar
Deheuvels, P. (1983). Point processes and multivariate extreme values. J. Multivariate Anal. 13, 257–272.CrossRefGoogle Scholar
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, Berlin.CrossRefGoogle Scholar
Nandagopalan, S. (1990). Multivariate extremes and the estimation of the extremal index. , University of North Carolina.Google Scholar
Pickands, J. III (1981). Multivariate extreme value distributions. Bull. Inst. Internat. Statist. 49, 859–878.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.CrossRefGoogle Scholar
Smith, R. L., and Weissman, I. (1996). Characterization and estimation of the multivariate extremal index. Tech. Rep., University of North Carolina.Google Scholar
Zhang, Z. (2002). Multivariate extremes, max-stable process estimation and dynamic financial modeling. , University of North Carolina.Google Scholar
Zhang, Z. (2004a). Some results on approximating max-stable processes. Tech. Rep., Department of Mathematics, Washington University, Saint Louis.Google Scholar
Zhang, Z. (2004b). Quotient correlation: a sample-based alternative to Pearson's correlation. Tech. Rep., Department of Mathematics, Washington University, Saint Louis.Google Scholar
Zhang, Z., and Smith, R. L. (2003). On the estimation and application of max-stable processes. Tech. Rep., Department of Mathematics, Washington University, Saint Louis.Google Scholar