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Kernel regression with Weibull-type tails

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Abstract

We consider the estimation of the tail coefficient of a Weibull-type distribution in the presence of random covariates. The approach followed is non-parametric and consists of locally weighted estimation in narrow neighbourhoods in the covariate space. We introduce two families of estimators and study their asymptotic behaviour under some conditions on the conditional response distribution, the kernel function, the density function of the independent variables, and for appropriately chosen bandwidth and threshold parameters. We also introduce a Weissman-type estimator for estimating upper extreme conditional quantiles. The finite sample behaviour of the proposed estimators is examined with a simulation experiment. The practical applicability of the methodology is illustrated on a dataset of sea storm measurements.

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Acknowledgments

The authors kindly acknowledge Laurens de Haan and Ana Ferreira for providing the sea level data. This work was supported by a research Grant (VKR023480) from VILLUM FONDEN and an international project for scientific cooperation (PICS-6416). The authors are very grateful to the editor and two anonymous referees for their helpful and constructive comments on the preliminary versions of the paper.

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Authors and Affiliations

  1. Department of Statistics and Actuarial Science, University of Stellenbosch, Private Bag X1, Matieland, Stellenbosch, 7602, South Africa

    Tertius de Wet

  2. Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark

    Yuri Goegebeur & Michael Osmann

  3. Institut Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67084, Strasbourg Cedex, France

    Armelle Guillou

Authors
  1. Tertius de Wet
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  2. Yuri Goegebeur
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  3. Armelle Guillou
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  4. Michael Osmann
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Corresponding author

Correspondence to Yuri Goegebeur.

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de Wet, T., Goegebeur, Y., Guillou, A. et al. Kernel regression with Weibull-type tails. Ann Inst Stat Math 68, 1135–1162 (2016). https://doi.org/10.1007/s10463-015-0531-z

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  • DOI: https://doi.org/10.1007/s10463-015-0531-z

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