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Optimal rates of convergence in the Weibull model based on kernel-type estimators

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  • Mercadier, Cécile
  • Soulier, Philippe

Abstract

Let F be a distribution function in the maximal domain of attraction of the Gumbel distribution such that −log(1−F(x))=x1/θL(x) for a positive real number θ, called the Weibull tail index, and a slowly varying function L. It is well known that the estimators of θ have a very slow rate of convergence. We establish here a sharp optimality result in the minimax sense, that is when L is treated as an infinite dimensional nuisance parameter belonging to some functional class. We also establish the rate optimal asymptotic property of a data-driven choice of the sample fraction that is used for estimation.

Suggested Citation

  • Mercadier, Cécile & Soulier, Philippe, 2012. "Optimal rates of convergence in the Weibull model based on kernel-type estimators," Statistics & Probability Letters, Elsevier, vol. 82(3), pages 548-556.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:3:p:548-556
    DOI: 10.1016/j.spl.2011.11.022
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    1. Beirlant, J. & Bouquiaux, C. & Werker, B.J.M., 2006. "Semiparametric lower bounds for tail-index estimation," Other publications TiSEM 4f434455-72a7-4b68-b972-d, Tilburg University, School of Economics and Management.
    2. Mason, David M., 1983. "The asymptotic distribution of weighted empirical distribution functions," Stochastic Processes and their Applications, Elsevier, vol. 15(1), pages 99-109, June.
    3. Drees, Holger & Kaufmann, Edgar, 1998. "Selecting the optimal sample fraction in univariate extreme value estimation," Stochastic Processes and their Applications, Elsevier, vol. 75(2), pages 149-172, July.
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