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We consider regularly varying random vectors. Our goal is to estimate in anon-parametric way some characteristics related to conditioning on an extremeevent, like the tail dependence coefficient. We introduce a quasi-spectraldecomposition that allow to improve efficiency of estimators. Asymptoticnormality of estimators is based on weak convergence of tail empiricalprocesses. Theoretical results are supported by simulation studies.

Multivariate Tail Estimation: Conditioning on an extreme event