Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T05:44:40.915Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Conditional Distribution of Exceedance Counts

Part of: Stochastic processes Nonparametric inference Distribution theory

Published online by Cambridge University Press:  04 January 2016

Michael Falk  and
Diana Tichy
Michael Falk*
Affiliation:
University of Würzburg
Diana Tichy*
Affiliation:
University of Würzburg
*
∗ Postal address: Institute of Mathematics, University of Würzburg, Emil-Fischer-Strasse 30, 97074 Würzburg, Germany.
∗ Postal address: Institute of Mathematics, University of Würzburg, Emil-Fischer-Strasse 30, 97074 Würzburg, Germany.
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.

We investigate the asymptotic distribution of the number of exceedances among d identically distributed but not necessarily independent random variables (RVs) above a sequence of increasing thresholds, conditional on the assumption that there is at least one exceedance. Our results enable the computation of the fragility index, which represents the expected number of exceedances, given that there is at least one exceedance. Computed from the first d RVs of a strictly stationary sequence, we show that, under appropriate conditions, the reciprocal of the fragility index converges to the extremal index corresponding to the stationary sequence as d increases.

Keywords

Exceedance over high thresholdfragility indexmultivariate extreme value theorypeaks-over-threshold approachcopulageneralized Pareto distribution (GPD)GPD copulaD-normextremal index

MSC classification

Primary: 62G32: Statistics of extreme values; tail inference
Secondary: 60G70: Extreme value theory; extremal processes 62E20: Asymptotic distribution theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 1 , March 2012 , pp. 270 - 291
Copyright
© Applied Probability Trust 

References

Aulbach, S., Bayer, V. and Falk, M. (2012). A multivariate piecing-together approach with an application to operational loss data. To appear in Bernoulli.CrossRefGoogle Scholar
Balkema, A. A. and de Haan, L. (1974). Residual life time at great age. Ann. Prob. 2, 792–804.CrossRefGoogle Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford University Press.CrossRefGoogle Scholar
Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes. John Wiley, Chichester.CrossRefGoogle Scholar
Buishand, T. A., de Haan, L. and Zhou, C. (2008). On spatial extremes: with application to a rainfall problem. Ann. Appl. Statist. 2, 624–642.CrossRefGoogle Scholar
De Haan, L. and Ferreira, A. (2006). Extreme Value Theory. Springer, New York.CrossRefGoogle Scholar
Deheuvels, P. (1978). Caractèrisation complète des lois extrème multivariées et de la convergene des types extrèmes. Publ. Inst. Statist. Univ. Paris 23, 1–36.Google Scholar
Deheuvels, P. (1984). Probabilistic aspects of multivariate extremes. In Statistical Extremes and Applications, ed. Tiago de Oliveira, J., Reidel, Dordrecht, pp. 117–130.CrossRefGoogle Scholar
Drees, H. and Huang, X. (1998). Best attainable rates of convergence for estimators of the stable tail dependence function. J. Multivariate Anal. 64, 25–47.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events (Appl. Math. 33). Springer, Berlin.CrossRefGoogle Scholar
Falk, M., Hüsler, J. and Reiss, R.-D. (2004). Laws of Small Numbers: Extremes and Rare Events, 2nd edn. Birkhäuser, Basel.CrossRefGoogle Scholar
Geluk, J. L., de Haan, L. and De Vries, C. G. (2007). Weak and strong financial fragility. Discussion Paper TI 2007–023/2, Tinbergen Institute.Google Scholar
Hofmann, D. (2009). Characterization of the D-norm corresponding to a multivariate extreme value distribution. , University of Würzburg.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
Huang, X. (1992). Statistics of bivariate extreme values. , Tinbergen Institute Research Series.Google Scholar
Nelsen, R. B. (2006). An Introduction to Copulas, 2nd edn. Springer, New York.Google Scholar
Pickands, J. III (1975). Statistical inference using extreme order statistics. Ann. Statist. 3, 119–131.Google Scholar
Reiss, R.-D. and Thomas, M. (2007). Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields, 3rd edn. Birkhäuser, Basel.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.CrossRefGoogle Scholar
Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.Google Scholar
Segers, J. (2003). Approximate distributions of clusters of extremes. Discussion Paper 2003–91, Tilburg University, The Netherlands.CrossRefGoogle Scholar
Smith, R. L. (1990). Max-stable processes and spatial extremes. Preprint, University of North Carolina.Google Scholar
Takahashi, R. (1988). Characterizations of a multivariate extreme value distribution. Adv. Appl. Prob. 20, 235–236.CrossRefGoogle Scholar
Yun, S. (2000). The distributions of cluster functionals of extreme events in a dth-order Markov chain. J. Appl. Prob. 37, 29–44.CrossRefGoogle Scholar