Estimating extreme bivariate quantile regions

J.H.J. Einmahl, L.F.M. de Haan, A. Krajina

Research output: Contribution to journalArticleScientificpeer-review

Abstract

When simultaneously monitoring two possibly dependent, positive risks one is often interested in quantile regions with very small probability p. These extreme quantile regions contain hardly any or no data and therefore statistical inference is difficult. In particular when we want to protect ourselves against a calamity that has not yet occurred, we need to deal with probabilities p < 1/n, with n the sample size. We consider quantile regions of the form {(x, y) ∈ (0, ∞ )2: f(x, y) ≤ β}, where f, the joint density, is decreasing in both coordinates. Such a region has the property that it consists of the less likely points and hence that its complement is as small as possible. Using extreme value theory, we construct a natural, semiparametric estimator of such a quantile region and prove a refined form of consistency. A detailed simulation study shows the very good statistical performance of the estimated quantile regions. We also apply the method to find extreme risk regions for bivariate insurance claims.
Original languageEnglish
Pages (from-to)121-145
JournalExtremes
Volume16
Issue number2
DOIs
Publication statusPublished - 2013

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Quantile
Extremes
Insurance
Monitoring
Extreme Quantiles
Extreme Value Theory
Statistical Inference
Sample Size
Complement
Likely
Simulation Study
Estimator
Dependent

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Einmahl, J.H.J. ; de Haan, L.F.M. ; Krajina, A. / Estimating extreme bivariate quantile regions. In: Extremes. 2013 ; Vol. 16, No. 2. pp. 121-145.
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Estimating extreme bivariate quantile regions. / Einmahl, J.H.J.; de Haan, L.F.M.; Krajina, A.

In: Extremes, Vol. 16, No. 2, 2013, p. 121-145.

Research output: Contribution to journalArticleScientificpeer-review

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