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On the impossibility of estimating densities in the extreme tail

✍ Scribed by Jan Beirlant; Luc Devroye

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
100 KB
Volume
43
Category
Article
ISSN
0167-7152

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✦ Synopsis

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 inΓΏnitely many times di erentiable 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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In this paper we shall give an alternative derivation of the coe cient of tail dependence introduced by Ledford and Tawn [1996, Biometrika 83, 169 -187] and propose a consistent estimator, which is asymptotically normal.