On the impossibility of estimating densities in the extreme tail
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.
Volume (Year): 43 (1999)
Issue (Month): 1 (May)
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- 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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