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Another proof of a slow convergence result of Birgé

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  • Devroye, Luc

Abstract

We give a short proof of the following result. Let fn be any density estimate based upon an i.i.d. sample drawn from a density f. For any monotone decreasing sequence {an} of positive numbers converging to zero with , a density f may be found such that for all n. This density may be picked from the class of densities on [0, 1] that are bounded by two. The proof of this fact simplifies an earlier proof by Birgé (1986) and extends a weaker lower bound by the author (1983).

Suggested Citation

  • Devroye, Luc, 1995. "Another proof of a slow convergence result of Birgé," Statistics & Probability Letters, Elsevier, vol. 23(1), pages 63-67, April.
    • Handle: RePEc:eee:stapro:v:23:y:1995:i:1:p:63-67
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    Cited by:

    1. Meister Alexander, 2008. "Uniform and individual convergence rates for convex density classes," Statistics & Risk Modeling, De Gruyter, vol. 26(1), pages 25-34, March.
    2. 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.

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