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Kernel estimators of extreme level curves

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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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Authors and Affiliations

  1. Toulouse School of Economics (GREMAQ), University of Toulouse, Aile J.J. Laffont, 21 allée de Brienne, 31000, Toulouse, France

    Abdelaati Daouia

  2. INRIA Rhône-Alpes, Team Mistis, Inovallée, 655, av. de l’Europe, Montbonnot, 38334, Saint-Ismier cedex, France

    Laurent Gardes, Stéphane Girard & Alexandre Lekina

Authors
  1. Abdelaati Daouia
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  2. Laurent Gardes
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  3. Stéphane Girard
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  4. Alexandre Lekina
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Correspondence to Stéphane Girard.

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Daouia, A., Gardes, L., Girard, S. et al. Kernel estimators of extreme level curves. TEST 20, 311–333 (2011). https://doi.org/10.1007/s11749-010-0196-0

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