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On the impossibility of estimating densities in the extreme tail

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  • Beirlant, Jan
  • Devroye, Luc

Abstract

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.

Suggested Citation

  • Beirlant, Jan & Devroye, Luc, 1999. "On the impossibility of estimating densities in the extreme tail," Statistics & Probability Letters, Elsevier, vol. 43(1), pages 57-64, May.
    • Handle: RePEc:eee:stapro:v:43:y:1999:i:1:p:57-64
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    References listed on IDEAS

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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. Schmid, Friedrich & Schmidt, Axel, 2006. "Nonparametric estimation of the coefficient of overlapping--theory and empirical application," Computational Statistics & Data Analysis, Elsevier, vol. 50(6), pages 1583-1596, March.
    2. Jean‐Pierre Gauchi & Jean‐Charles Leblanc, 2002. "Quantitative Assessment of Exposure to the Mycotoxin Ochratoxin A in Food," Risk Analysis, John Wiley & Sons, vol. 22(2), pages 219-234, April.
    3. 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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