Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-25T14:20:25.552Z Has data issue: false hasContentIssue false
  • English
  • Français

Rare events, temporal dependence, and the extremal index

Part of: Nonparametric inference Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Johan Segers
Johan Segers*
Affiliation:
Tilburg University
*
∗Postal address: Department of Econometrics and Operations Research, Tilburg University, PO Box 90153, NL-5000 LE Tilburg, The Netherlands. Email address: jsegers@uvt.nl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Classical extreme value theory for stationary sequences of random variables can to a large extent be paraphrased as the study of exceedances over a high threshold. A special role within the description of the temporal dependence between such exceedances is played by the extremal index. Parts of this theory can be generalized not only to random variables on an arbitrary state space hitting certain failure sets, but even to a triangular array of rare events on an abstract probability space. In the case of M4 (maxima of multivariate moving maxima) processes, the arguments take a simple and direct form.

Keywords

Block maximumexceedanceextremal indexfailure setmixing conditionM4 processrare eventstationary sequence

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 62G32: Statistics of extreme values; tail inference
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 2 , June 2006 , pp. 463 - 485
Copyright
© Applied Probability Trust 2006 

References

Barbour, A. D., Novak, S. Y. and Xia, A. (2002). Compound Poisson approximation for the distribution of extremes. Adv. Appl. Prob. 24, 223–240.CrossRefGoogle Scholar
Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes: Theory and Applications. John Wiley, New York.CrossRefGoogle Scholar
De Haan, L. (1984). A spectral representation for max-stable processes. Ann. Prob. 12, 1194–1204.Google Scholar
Ferro, C. A. T. and Segers, J. (2003). Inference for clusters of extremes. J. R. Statist. Soc. B 65, 545–556.Google Scholar
Hsing, T. (1989). Extreme value theory for multivariate stationary sequences. J. Multivariate Anal. 29, 274–291.Google Scholar
Hsing, T., Hüsler, J. and Leadbetter, M. R. (1988). On the exceedance point process for a stationary sequence. Prob. Theory Relat. Fields 78, 97–112.CrossRefGoogle Scholar
Hüsler, J. (1990). Multivariate extreme values in stationary random sequences. Stoch. Process. Appl. 35, 99–108.Google Scholar
Hüsler, J. (1993). A note on exceedances and rare events of non-stationary sequences. J. Appl. Prob. 30, 877–888.CrossRefGoogle Scholar
Hüsler, J. and Schmidt, M. (1996). A note on the point processes of rare events. J. Appl. Prob. 33, 654–663.CrossRefGoogle Scholar
Leadbetter, M. R. (1974). On extreme values in stationary sequences. Z. Wahrscheinlichkeitsth. 28, 289–303.Google Scholar
Leadbetter, M. R. (1983). Extremes and local dependence of stationary sequences. Z. Wahrscheinlichkeitsth. 65, 291–306.Google Scholar
Loynes, R. M. (1965). Extreme values in uniformly mixing stationary stochastic processes. Ann. Math. Statist. 36, 993–999.CrossRefGoogle Scholar
Nandagopalan, S. (1994). On the multivariate extremal index. J. Res. Nat. Inst. Stand. Technol. 99, 543–550.Google Scholar
Novak, S. Y. (2002). Multilevel clustering of extremes. Stoch. Process. Appl. 97, 59–75.Google Scholar
O'Brien, G. L. (1987). Extreme values for stationary and Markov sequences. Ann. Prob. 15, 281–291.Google Scholar
Perfekt, R. (1997). Extreme value theory for a class of Markov chains with values in {R}d . Adv. Appl. Prob. 29, 138–164.Google Scholar
Smith, R. L. and Weissman, I. (1996). Characterization and estimation of the multivariate extremal index. Tech. Rep., University of North Carolina. Available at www.unc.edu/depts/statistics/postscript/rs/extremal.ps.Google Scholar
Zhang, Z. (2002). Multivariate extremes, max-stable process estimation and dynamic financial modeling. , University of North Carolina.Google Scholar
Zhang, Z. and Smith, R. L. (2004). The behavior of multivariate maxima of moving maxima processes. J. Appl. Prob. 41, 1113–1123.CrossRefGoogle Scholar