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Asymptotic Conditional Distribution of Exceedance Counts

Part of:Distribution theoryStochastic processesNonparametric inference

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.
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Abstract

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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 processes62E20: Asymptotic distribution theory

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 1 , March 2012, pp. 270 - 291
Copyright
© Applied Probability Trust 

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