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Kernel estimators of extreme level curves

Abdelaati Daouia 1 Laurent Gardes 2 Stéphane Girard 2 Alexandre Lekina 2 
2 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology
Abstract : We address the estimation of extreme level curves of heavy-tailed distributions. This problem is equivalent to estimating quantiles when covariate information is available and when their order converges to one as the sample size increases. We show that, under some conditions, these so-called ''extreme conditional quantiles'' can still be estimated through a kernel estimator of the conditional survival function. Sufficient conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed estimators. Making use of this result, some kernel estimators of the conditional tail-index are introduced and a Weissman type estimator is derived, permitting to estimate extreme conditional quantiles of arbitrary large order. These results are illustrated through simulated and real datasets.
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Abdelaati Daouia, Laurent Gardes, Stéphane Girard, Alexandre Lekina. Kernel estimators of extreme level curves. Test, Spanish Society of Statistics and Operations Research/Springer, 2011, 20 (2), pp.311-333. ⟨10.1007/s11749-010-0196-0⟩. ⟨inria-00393588v4⟩

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