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On the impossibility of estimating densities in the extreme tail


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  • Beirlant, Jan
  • Devroye, Luc
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    We give a short proof of the following result. Let X1,...,Xn be independent and identically distributed observations drawn from a density f on the real line. Let fn be any estimate of the density gn of max(X1,...,Xn). We show that there exists a unimodal infinitely many times differentiable density f such that Thus, in the total variation sense, universally consistent density estimates do not exist. A similar result is derived concerning the supremum norm.

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    Bibliographic Info

    Article provided by Elsevier in its journal Statistics & Probability Letters.

    Volume (Year): 43 (1999)
    Issue (Month): 1 (May)
    Pages: 57-64

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    Handle: RePEc:eee:stapro:v:43:y:1999:i:1:p:57-64

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    Keywords: Density estimation Nonparametric estimation Lower bounds Rate of convergence Maxima Extreme values;


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    1. Devroye, Luc, 1995. "Another proof of a slow convergence result of Birgé," Statistics & Probability Letters, Elsevier, vol. 23(1), pages 63-67, April.
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    Cited by:
    1. El-Aroui, Mhamed-Ali & Diebolt, Jean, 2002. "On the use of the peaks over thresholds method for estimating out-of-sample quantiles," Computational Statistics & Data Analysis, Elsevier, vol. 39(4), pages 453-475, June.


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